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Intertemporally consistent shadow prices in an open economy
Journal
of Public
Economics
32 (1987) 263-285.
North-Holland
INTERTEMPORALLY CONSISTENT SHADOW IN AN OPEN ECONOMY
PRICES
Estimates for Cyprus Clive BELL Vanderbilt University, Nashville,
_ Shantayanan Harvard Received
DEVARAJAN*
University, Cambridge,
February
TN 37235, USA
1985, revised version
MA 02138, USA received October
1986
In the practical estimation of shadow prices, the assumption that the economy is on a steadystate growth path is often invoked, usually implicitly. Most developing countries are far from this trajectory. We derive shadow prices where the dynamic evolution of the economy is endogenously determined, important influences being initial conditions and the constraints on foreign borrowing. Hence, the resulting (dated) shadow prices of tradeables, nontradeables, labor and the accounting rate of interest (ARI) are mutually consistent. Estimates for the Republic of Cyprus show that derivations from steady-state growth can be significant, leading to large fluctuations in the AR1 and thereby making the timing of investment projects crucial.
1. Introduction When assessing whether a project is socially profitable, it is necessary to use prices that reflect social scarcities, and for various well-known reasons, these may be quite different from market prices. One way of carrying out such an assessment is to formulate a model of the whole economy, with *We thank participants at seminars at the University of Pennsylvania, the World Bank, HMvard University and New York University and, especially, two anonymous referees for helpful comments on an earlier draft. Sethu Palaniappan and Arjun Jannah provided invaluable assistance in programming and running the model. An earlier version of this paper was presented at the Annual Meeting of the Southern Economic Association, Washington, D.C. November 1983. Errors and opinions are, of course, solely our responsibility. The World Bank does not accept responsibility for the views expressed herein which are those of the author(s) and should not be attributed to the World Bank or its affiliated organizations. The findings, interpretations, and conclusions are the results of research supported by the Bank; they do not necessarily represent official policy of the Bank. The designations employed, the presentation of material, and any maps used in this document are solely for the convenience of the reader and do not imply the expression of any opinion whatsoever on the part of the World Bank or its affiliates concerning the legal status of any country, territory, city, area, or of its authorities, or concerning the delimitation of its boundaries, or national affiliation. 0047-2727/87/$3.50
0
1987, Elsevier Science Publishers
B.V. (North-Holland)
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social welfare as the maximand, and solve for the optimal allocation of resources. If this is a ‘first-best’ optimum (i.e., there are no constraints on the government’s ability to make lump sum transfers), then the Lagrange multipliers corresponding to the scarcity constraints in the model can be used to value the project’s inputs and outputs [see, for example, Bruno (1975)]. Yet in this case, there is no point in evaluating the social profitability of projects, for the Lagrange multipliers equal market prices. If, as is usual, the government is unable to achieve a ‘first-best’ optimum, a ‘second-best’ problem must be solved, with the constraints preventing achievement of the first-best optimum included. In this case, however, the associated Lagrange multipliers often have to be reexpressed to derive the appropriate ‘shadow prices’ for project evaluation [Dreze and Stern (1985)]. This second-best approach is the one taken in the celebrated manuals of Little and Mirrlees (1974) and UNIDO (1972). Drawing on the theoretical work of Diamond and Mirrlees (1971) they establish rules for deriving shadow prices under particular assumptions about the government’s opportunity set. The best known rule, which is robust to a wide range of assumptions about government behavior, is to set the shadow price of a tradeable good at its border price. Next, the shadow price of a nontradeable is set equal to its marginal shadow cost of production. With tradeable inputs valued at their border prices and labor at the shadow wage rate, the shadow prices of nontradeables can be calculated once the shadow wage rate is known. The shadow wage rate, in turn, is the value (at shadow prices) of the worker’s marginal product in alternative employment, plus the shadow cost of any increase in his wages (for example, if he were previously unemployed). At this point, intertemporal considerations intrude: to the extent that paying the worker a higher wage involves a shift in the uses of income from savings to consumption, it becomes necessary to estimate the premium on savings in the economy. This premium is, of course, related to the accounting rate of interest (ARI), the rate at which the project’s net benefits (estimated at shadow prices) are to be discounted over time. In principle, the savings premium, the AR1 and the entire set of (dated) shadow prices of nontradeables and labor are mutually and simultaneously determined. In practice, however, the savings premium and the AR1 are almost invariably derived first, often by rather rough and ready methods. The estimate of the savings premium is then fed into the calculation of the remaining shadow prices, essentially within a static framework. When a model is used to derive the AR1 and the savings premium, it typically has only one sector and one factor, so that the results are applicable to a manygood economy only when all sectors are growing at the same rate, that is, on a ‘balanced growth’ path. Clearly, this one-way recursive procedure for estimating shadow prices, which is almost universally adopted in practice, rests on rather strong assumptions.
C. Bell and S. Devarajan, Shadow prices
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In this paper, we estimate all shadow prices, the AR1 and the savings premium from a many-good, intertemporal optimising model, so that all these parameters are mutually and simultaneously determined. Nor is it assumed that there is balanced growth. We specify the constraints on the government’s instruments that prevent the economy from achieving a first-best optimum, and show how shadow prices can be derived from the Lagrange multipliers. There remains, however, an apparent difficulty with this procedure. Once the primal solution to the optimisation problem is known, no additional information is yielded by the dual because any known project that improves welfare will be included in the optimal solution to the primal. Hence, there are no projects left to be evaluated [Srinivasan (1983)]. Now, the use of shadow prices might not be redundant if there existed projects that were not contained in the set of feasible activities from which the optimal allocation was chosen. The existence of such projects stems from the government’s limited knowledge of future transformation possibilities, and proposals for the projects may appear unexpectedly at any time throughout the planning horizon. What is needed, therefore, is a set of shadow prices that will detect whether such projects will improve social welfare, given the optimal allocation of resources based on the knowledge available at the start of the plan. The shadow prices in the manuals and those in this paper possess this property. As Dreze (1982) shows more generally, the gradient of the welfare function at the optimum can serve this purpose. We estimate shadow prices for the Republic of Cyprus, a small, open economy with a diversified production structure and a literate population. Cyprus has a record of rapid growth since its independence, despite the events of 1974. As it entered its first five-year plan in 1982, Cyprus was facing various problems of misallocation, including some stemming from the structure of taxation and others from a strong appetite for foreign borrowing. Hence, the emphasis placed here on public finance and intertemporal borrowing possibilities is by no means accidental. The model is presented in section 2. In section 3, the dual solution to the model is used to derive shadow prices. We then show how, with certain simplifying assumptions, these shadow prices specialize to those derived from the manuals in practical work. A discussion of the results follows in section 4, and section 5 presents some concluding remarks. 2. The model The model is an extension of that used by Bell and Devarajan (1982, 1983a) to obtain a consistent and feasible Five Year Plan for Cyprus. What follows is a version simplified for the purposes at hand; those who hanker
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C. Bell and S. Devarajan, Shadow prices
after the more intricate details are referred to those papers. All variables are dated, but the time index (r) is specified only when the context demands it. We begin with the familiar material balance equations: X+M=AX+C+J+E+G,
(1)
where X is the (n x 1) vector of domestically produced commodities; M is the vector of imports; A is the matrix of (fixed) interindustry coefficients; C is the vector of private consumption; J is the vector of investment deliveries by sector of origin; E is the vector of exports; and G is the vector of government current consumption. Factor incomes are given by W=VX+6,
(2)
where V is the (4 x n) matrix IIUfjll>vfj being the payment made to the Sth factor by the jth activity when it is operating at unit intensity; and 6 is the (9 x 1) vector of the dummy variables 6,. These dummy variables are set initially to zero; but since we are interested in what happens when the government transfers one unit of income to any factor by employing it, each of the ds’s can also take the value unity. The incomes of domestic institutions (including public sector firms) from their ownership of factors are given by: Y=QW
(3)
where R is the (h x q) matrix which maps factor payments into institutions’ incomes from factors. Private consumption is given by a linear expenditure system: C=B(Z-?)(I-f)Y,
(4)
where B is the (n x h) matrix of expenditure coefficients;’ s*is the diagonal matrix of savings rates out of post-tax income; and Z is the diagonal matrix of tax rates on institutions’ incomes. Investment demand is determined by an accelerator relationship, J(z) = H[K(r + 1) -K(z)],
(5)
where H is the (n x n) matrix of capital coefficients and K(r) is the vector of full capacity output levels in period r.2 ‘In the full model, the Engel lines are not constrained to go through the origin. Note also that the expenditure coefficients for institutions other than households are equal to zero. 21n the full model, the expansion of capacity may require investments in as many as four years preceding the date on which the additional capacity becomes available, depending on the sector in question.
C. Bell and S. Devarajan,
Shadow prices
261
Eqs. (l)-(5) represent the basic behavioral and technical assumptions of the optimising model. In addition, we impose a set of constraints to reflect the resources available to the economy and the extent to which they can be shifted from one use to another. First, the demand for labor cannot exceed its supply in each skill category:3 L(z) X(z) 5 L(r),
where L(r) is the (1x n) matrix of labor coefficients, lrj(r); and L(r) is the exogenously given vector of labor supplies. The labor coefficients change over time according to a set of exogenously given productivity growth rates. Similarly, the domestic production of any commodity cannot exceed installed capacity:
Also, it is reasonable to suppose that, once installed, capital is not shiftable. With no depreciation, this implies K(z) 5 K(z + 1).
Note that in the first period, all installed capacities are given, whereas in all other years, capacity is endogenous. Hence, with the exception of the first year, the sole primary factors of production are the various categories of labor. There remains the matter of how any balance of payments deficit is to be financed, for borrowing in the international capital market is an opportunity open to Cyprus. In any year, the rate of interest on what is borrowed is assumed to depend only on the amount borrowed in that year4 and to keep computation manageable, this relation is linearised. Starting at some low positive rate where the current account just balances, the rate of interest is assumed to rise in steps with the amount borrowed until it reaches an upper limit. The model is solved with two variations. In the first, it is assumed that there is a perfectly elastic supply of foreign funds at the upper limit for the rate of interest. In the second, the possibility of credit rationing at a parametric interest rate, and/or a reluctance on the part of the government 31n the full model, we allow workers with higher skills to do lower skilled jobs. For example, professionals can do ordinary white collar jobs, etc. 4Clearly, other variables (like the total indebtedness) affect the interest rate on current borrowing. Our purpose here, though, is to introduce a rising marginal cost of foreign borrowing. In view of the difftculties of estimating the parameters of this schedule, the formulation adopted here has the merit of simplicity. Tourinho (1985) adopts a similar specification.
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Shadow
prices
to permit very heavy foreign indebtedness, is introduced. This is expressed by the constraint that the debt-service ratio (DSR) in any year be no greater than a parametrically given value, so that the amount that can be borrowed in any year is bounded from above, albeit endogenously. To complete the picture, if there should be a surplus after paying off principal and interest on past borrowings, that surplus is lent abroad at a parametrically given rate. Note that with an upward sloping supply schedule of foreign funds, Cyprus can exercise a (modest) amount of monopsony power in the course of choosing an optimal programme. It is assumed that the terms of borrowing, other than the rate of interest, p(r), do not vary. In the full model, a grace period of three years is followed by seven years in which to repay, in equal installments. To introduce these terms would complicate the task of exposition here without adding anything in the way of substance, so the formulation adopted below specifies that interest and principal be repayable in full one year after the loan is taken up. With this simplification, the oustanding foreign debt, D(r), takes the form D(z)=[l+p(z-l)].D(r-l)+p’[M(z)-E(r)],
2=1,...,7:
where p’ is the (1 x n) vector of world prices and p(O) and D(0) are given parametrically, Under these terms, the amount borrowed in any year is simply the oustanding debt. Hence,
P(4 = PC~(Lu, where p[.] is linearised exceed cr, the following
(10)
as described above. If, in addition, constraint must be added:
the DSR must not
(11) Objective
function
and terminal
conditions
We ignore considerations of equity among contemporaries, so that the level of instantaneous welfare depends only on aggregate consumption. Domestic savings are valuable only for the increase in future consumption they make possible. If domestic savings can be increased by other means only at very high cost in terms of consumption foregone, then the option of foreign borrowing may be highly attractive; but any indebtedness that remains at the end of the horizon entails a cut in consumption thereafter, and therefore entails a cost. Similarly, the capital stock bequeathed to the future at the end of the horizon is valuable only inasmuch as it permits sustainable consumption in the years that follow. Hence, the objective function consists of the discounted utility of consumption throughout the
C. Bell and S. Devarajan, Shadow prices
269
time horizon, together with a valuation of (i) the vector of capacities in the terminal year, K(T), and (ii) the level of foreign debt in the terminal year, D(T). The estimation of c1r and CY~, the values of the stocks K(T) and D(T), respectively, in terms of their corresponding flows of consumption, is given in the Appendix. While we can value K(T), there is nothing so far that ensures that there will be sufficient labor to operate all this plant and equipment. With a planning horizon of fifteen years, therefore, it seems best to impose the condition that the demand for labor generated by running all plants at full capacity in the terminal year should not exceed the labor supplies available at that time, namely,
L(T)K(T)<=L(T).
(12)
3. Shadow prices The optimisation problem described above can be summarised as follows: T U[u’C(z)]
c (1+0-l IX(r),E~r).K(r)) t=l
+
u*[Icr;lq T) + a,D( T)] (l+<)‘_’ ’
(13)
subject to (1)-(12), with G(z), Qr), K(l), D(O), p(O) parametrically given and 6(r) = 0. Note that all taxes are assumed to lie outside the planner’s sphere of control. We now consider the dual solution to this problem as a means to derive shadow prices. Let the dual variables associated with, respectively, the material balances, the consumption function, the labor constraints, the installed capacity constraints, the condition that capita1 be non-shiftable, the growth of indebtedness (9), the requirement that the dummy transfers from government to factors, 6(r), be zero, the debt-service constraint and the terminal year constraint (12) be denoted by rc, rr,, nl, rcK,nN, xD, 7c6,rc, and 1*. The dual cost conditions may be written down at once. In all periods, we have: n’(z). A +7$(T). Y v + 11,(T).L + 7&(z) 2 7c’(z),
(14)
744 - 4(4 2 P’(%
(15)
n’(z) + p’rco(z)- op’rcJr) >=0, - n’(f) - p’ . rcD(r)2 0,
for fully traded goods,
for fully traded goods,
(16a) (16b)
C. Bell and S. Devarajan, Shadow prices
210
n’(z-
l).H-r&(7-
+7&(7)~0,
l)-7c’(z).H-n;,(7) 7=2 )...) T-l,
(174
77,(7-1)+(1+&7,(7)-[1+~(7-1)]~71~(7)~0,
7=1,...,7;
(fga) (Igb)
where Y = B(I - 3)(I - 1)Q, ~(7) = U’(r)/( 1 + [)‘- ‘, and 2 is a diagonal matrix of post-terminal growth rates (see Appendix). From (16a) and (16b), it follows at once that for traded goods, if the DSR constraint is not binding (x,=0): 77’(z)= - p’ .7LD(T). If it does bind, and exports
are positive,
then from (16a):
72’(z)= -p’(77D(7)-cr77,(7-l)). Hence, the shadow prices of traded goods are in both cases proportional to their respective world prices - the celebrated ‘border price’ rule. Note that for a given good, (16a) and (16b) cannot both hold with equality, confirming the fact that it is never optimal to import and export the same good. Next, the fact that there is some installed capacity in all sectors in the first year, together with the non-shiftability condition and the absence of depreciation, implies that K(z)>0 in all years. Also, since it is plausible that at least one good will be produced in positive amounts in each year, then mild restrictions on the consumption parameters will ensure that all goods are consumed in strictly positive amounts in each year. Hence, the fundamental duality theorem of linear programming tells us that (15), (17a) and (17b) must hold as strict equalities.5 Similarly, for every good which is actually produced domestically, the relevant dual cost condition in (14) will hold as a strict equality. As (15) holds as an equality, we may substitute for 77:(z) in (14). Suppose there are m traded goods, so that in any year m of the n elements of n(7) are known up 5Although, by definition, there is neither domestic capacity nor the technology to produce non-competitive imports, so that (14) will be a strict inequality, an inspection of (17) reveals that the dual variables nK and xN for any good do not depend directly on the rest. The dual variable B for non-competitive imports is given by X(T)= --p” ndr).
C. Bell and S. Devarajan, Shadow prices
271
to a scalar multiple. As there are 1 categories of labor, rcl has I elements also. Now for any reasonable technology and consumers’ preferences, all nontraded goods must be produced in positive quantities. Hence, the (n-m) dual cost conditions for such goods must hold as strict equalities. It follows from (14) that, in any year, the number of fully traded goods produced at home will be I plus the number of strictly positive elements of n, not linearly dependent on rc. The intuition for this result is that domestic production of a traded good may be profitable at marginal cost if capacity is not fully utilized, while it remains unprofitable to expand capacity. Turning to rr,, when D(r) is unconstrained by (1 l), (18a) and (18b) must hold as strict equalities, so that: (19) that is, the penalty on a unit increase in the foreign debt in the terminal year discounted back to the first year. Moving backwards from the terminal year, (18a) reveals the intuitively obvious result that the absolute magnitude of rrD increases from year to year at the prevailing rate of interest on foreign borrowings. This leads naturally to a discussion of the numeraire and the AR1 which, by definition, is the rate at which the value of the numeraire is falling. Any good can be chosen as numeraire, so that, in principle, there are as many ARIs as there are goods [Dreze and Stern (1985)]. Common practice is to take uncommitted public income as numeraire. If the DSR constraint is not binding, this choice implies that the AR1 is the rate at which n, is falling, since in that case an increase in public income implies an equal decrease in foreign indebtedness. Moreover, the shadow prices of traded goods will also be falling at this rate, so that the AR1 is independent of whether public income, foreign indebtedness or a particular traded good is the numeraire. When the DSR constraint binds, this equivalence breaks down since 71, enters the picture and its value will change from period to period. As a choice is forced, we have selected a particular traded good (light manufactures) as numeraire, and our AR1 is defined accordingly. This should be borne in mind when examining the results presented in section 4. As rrN does not feature in the first and last periods, there are [3(T- l)n+ T(I+ 1) +I] dual variables all told (ignoring the T. n elements of {%(l), . . . , z,(T)}, which are given at once by (15)). Against this, there are at most [(2T - 1)n + T(m + 1) + I] equations. Hence, while restricting the number of fully traded goods to be at most equal to the number of labor types will ensure that all traded goods will be produced at home on some scale in all years, much greater disaggregation may be possible in practice. Are the dual variables derived above able to detect welfare-improving changes in the allocation of resources? As any vector of dated inputs and
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C. Bell and S. Devarajan, Shadow prices
outputs may be thought of as a project - it must be sufficiently small in scale _ which has its own particular investment requirements and operating technique, it certainly seems permissible to use the dual variables to evaluate the project’s social profitability, and, if profitable, to include it in the Plan characterised by the primal solution. For although the Plan is derived on the assumption that the technology available to the economy is described by the matrices {A, V,L(r)}, it may be augmented by a set of projects which use techniques other than those in {A, v,L(~)j, provided their collective measure is sufficiently small. Now, each unit of good i used by such a project in year r will impose a cost on the economy, as measured by the resulting fall in the value of the objective function, in the amount xi(z). Where labor and other incomereceiving factors6 are concerned, however, the dual variables need to be interpreted with some care. If, for example, the government were to employ one worker of type fat the going wage, w/, and all workers of this type were employed, there would be no change in private consumption. In this case, the value of output foregone (at shadow prices), which is measured by the dual variable associated with the constraint on the supply of the fth kind of labor, is the appropriate shadow wage, viz r~~,~. The difficulty arises if the worker in question is unemployed; for although 7~~ will then be zero, the act of employing him will result in a transfer of wf from the government to the worker, a transfer which will, in general, affect the value of the objective function. The source of the difficulty is that while the government cannot use lump sum taxes as policy instruments, it is able to make some transfers by employing workers who would otherwise be jobless by increasing the derived demand for labor from production activities. Hence, the appropriate shadow prices of factors must reflect the fact that additional private consumption is both inherently valuable to the economy inasmuch as it appears in the objective function, as well as potentially costly, as it entails potential absorption of resources, which is where the dual variables n,(s) enter the picture. To derive the shadow price of a change in income, therefore, suppose the government transfers a unit of income to factor J The dual cost condition associated with the constraint 6 = 0 is:
where Y,-= B(I--.?)(!Z)Q, is the fth column of Y, and nd,Jz) is the fth element of Q(T). As (15) holds as an equality, the shadow price of this
6Note that the latter are not primary in the sense of being non-produced factors of production. They receive value added from production because the institutions of the economy happen to be of a particular kind.
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transfer
is: _n,,f=n’(z)Yf-
U’/(l
+i)‘-‘.
(20)
Now, n’(r)YYs is the value of the resources absorbed by households’ consumption out of this unit transfer, more commonly known as the consumption conversion factor for factor income of type f, and U’/( 1 + [)‘- 1 is the social value of this additional consumption. Hence, a previously idle worker, who is now to be engaged at the going market wage wf, will have a shadow price equal to: [d(T)Yf
- U’/( 1 + [)I- ‘1 . Wf.’
This is, of course, an application of the well-known theorem that the gradient of the Lagrangian is equal to the gradient of the objective function at the optimum. It should be noted that this shadow price may be negative, which simply reflects the fact that transfers from the government to factors would improve welfare if they could be put into effect.8 For some purposes, it will be useful to normalise the shadow price of labor by the going wage. In the language of Little and Mirrlees (1974) this is the accounting ratio for an unemployed laborer who would otherwise receive no income, which is given by (20). To complete this section, we set out how conditions (14)419) are related to procedures commonly used in the estimation of shadow prices in practice. Suppose, first, that the accounting rate of interest in each year is known, either because the supply of foreign funds is perfectly elastic or because some independent piece of analysis has been done. Suppose, further, that there is steady state growth, so that nN=O (i.e., the non-shiftability constraints do not bind). As established above, (17a) holds with strict equality: [rC’(r-I)-n’(r)]H=rck(r), Furthermore,
in steady
state growth
r<7: x(r) will decrease
(21) at a rate equal
to the
‘If the worker was receiving unemployment benefits, they should be subtracted from the going wage. Moreover, the value of leisure is ignored here. “At first sight, it would appear that these shadow prices apply solely to projects in the public sector. Consider, therefore, a project which just breaks even at market prices, with the government purchasing all inputs (including factors) and selling the resulting output. The government will neither gain nor lose income by these transactions, though agents in the private sector will certainly claim the entire resulting value asIded. Hence, we have, in effect, a project in the private sector, the social profitability of which can be established by employing the shadow prices derived above. Furthermore, even in the case of strictly positive private profits, appropriate shadow prices are simply a linear combination of the above shadow prices and producer prices.
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C. Bell and S. Devarajan, Shadow prices
ARI, p(z). Substituting
into (21), we get
_
p(? - l)n’(z)H = Z;(T),
,r < T.
(22)
which expresses the appealing intuition that the shadow price of capacity in sector i should be the shadow rental on the set of capital goods needed to augment capacity in that sector by one unit. In steady state growth, capacity will be fully utilised, so nk>O. Substituting (15) and (22) into (14), we have (recalling that (15) holds with strict equality): n’(r)[A + Y V+p(?-
l)H] +rr;(z)Lzp’(T)[z-
YV].
(23)
As before, all non-tradeables will be produced in strictly positive amounts, so that the relevant dual cost conditions hold with strict equality. Note that the dual prices for traded goods are known from (18a), which holds as a strict equality, and (19). Then (23) can be solved for n,, and the particular goods that are produced at home are revealed at the same time. The latter number at most 1, for although rr,>O, we see at once from (22) that it is linearly dependent on rc. Thus, independent estimation of the AR1 and the assumption of steady state growth permit the entire set of dual prices to be solved for in a fairly straightforward way. In practical work, however, shadow prices are not typically derived by finding a nonnegative solution to a system of linear inequalities. Instead, the analyst has two alternatives. First, knowing the number of primary factors, he can specify the particular traded goods - exactly the same number as there are factors - which will be produced at home. The dual cost conditions in (23) for these goods will hold as strict equalities, so that rc and rrcIcan be derived by matrix inversion [Srinivasan (1983)]. Second, in defiance of neoclassical trade theory, the analyst makes no particular assumption about the number of traded good produced domestically. In this case, n, is estimated independently, as the set of marginal products (valued at shadow prices) of workers in alternative employment. That is, the analyst answers the question: where is the labor coming from? The set of shadow prices of nontradeables follows at once from the relevant subset in (23). Note that in neither of these cases can negative shadow prices be ruled out [Bhagwati et al. (1978); Bell and Devarajan (1983b)]. In any event, the AR1 cannot be satisfactorily estimated without invoking (14)+19). Also, if it is implausible to argue that the economy will enjoy steady-state growth in the future, then the chain of argument leading to (23) will break down. In that case, there is nothing for it but to solve for the entire set of dual variables directly. 4. Results We begin
by specifying
the form of the felicity function,
i.e., the instanta-
C. Bell and S. Devarajan, Shadow prices
275
neous value of the objective function. We assume an isoelastic utility function u(.) of per capita consumption, c, and a constant rate of pure time preference, r. The value of the elasticity of marginal utility of consumption, q, is somewhat controversial, although Stern (1977) provides a careful discussion of various methods of estimating it, which suggest that a value between - 1 and -2 is defensible in many circumstances. Accordingly, we use these extreme values for q which lead to felicity functions of the form:
(24)
If population is growing at the rate II, then,
U[u'C] = N,u(c) [( 1 + I+)/(1 + r)]‘, where N, is the population in the initial year. As I is projected to be growing at 1.2 percent per annum, [ = r -I is surely barely positive. In fact, we assume [=O.Ol. As it turned out, the results for the case q = 1 differed little from those for ~=2. For want of space, therefore, the latter are reported here, and even then rather sparingly.’ Recall from section 2 that two variants of the cost of foreign borrowing are used. Despite the rising cost of foreign borrowing over a certain range, the optimal solution in the absence of the DSR constraint is characterised by a healthy appetite for foreign capital. With the DSR constraint and o set at 15 percent, a level thought to be desirable during the preparatory work for the Five Year Plan, there was no feasible path. When 0 was relaxed to 25 percent, however, an optimal solution was obtained, with the DSR constraint binding from 1984 through 1987. With a three-year grace period on foreign loans, this implies a premium on foreign exchange in the years up to and including 1984. In addition, there is a premium on exports in the years 1984-1987. We begin with the accounting rate of interest (ARI). In the absence of the DSR constraint, it might be thought that the AR1 is the interest rate on the last dollar borrowed in the year in question. This is incorrect for two reasons. First, borrowing an additional dollar today will affect the cost of borrowing in the future if the marginal cost of borrowing is not constant, as in the present case. Second, there is a grace period of three years, so gains from. arbitrage are possible. Figure 1 demonstrates that such departures of % an earlier version of this paper, a linear objective function (q=O) was used. This gave rise to even more volatile shadow prices than the ones reported in this paper (see below). Nevertheless the qualitative behavior of the dual variables was unchanged.
276
C. Bell and S. Devarajan, Shadow prices
24
I
42-
0
I 79
81 0
83 without DSC Fig. 1. Accounting
85
87 + rate of interest
89
91
93
with DSC (in percent)
the AR1 from the marginal rate of interest can be significant. When the pressures exerted by the necessity to complete projects in the pipeline at the start of the planning period are significant, the AR1 exceeds the marginal cost of foreign borrowing. After falling back, the AR1 proceeds to rise again, reaching a peak of 12 percent in 1988. Only in 1992 does simple intuition hold, for with a grace period, borrowing at the ‘top’ rate (six percent) in 1992 has no future repercussions within the horizon. The imposition of the DSR constraint will, of course, produce still sharper movements of the ARI, for borrowing an extra dollar today may be very costly if the constraint should bind in the future. Thus, the AR1 begins to rise in 1983, because the DSR constraint binds from 1984 to 1987. It reaches a sharp peak in 1985, before declining once more to the ‘top’ rate in 1987. Moreover, the fact that the DSR constraint binds in the mid-1980s intensities the pressures from the ‘pipeline’ effect in earlier years, as can be seen from fig. 1. Turning to the associated shadow prices, we follow the standard literature on social cost-benefit analysis by presenting them in the form of accounting ratios (AR), that is, the ratio of the shadow to the market price of the good or factor in question. As can be seen from figs. 2 and 3, the ARs for labor are quite volatile. This volatility is not confined to the initial and closing years,
C. Bell and S. Devarajan,
211
Shadow prices
when the influence of the boundary conditions holds sway. It derives from the fact that when there is full employment, the AR is equal to the social value of the worker’s marginal product and when there is unemployment, to the net social cost of transferring to the worker his market wage out of the government’s coffers, a social cost which may be positive or negative, depending on the premium on government income. When the AR is negative, it would be socially profitable to make lump-sum transfers to these workers, if such transfers were possible. In fact, this arises only in 1978. We assume that skilled workers can do the jobs of their less-skilled couterparts, but not conversely. Thus unemployed professional workers can till vacancies for (clerical) workers if the latter are fully employed. To the extent that the two groups have different consumption and taxation patterns, their shadow wage rates will be identical only if, at the margin, professional workers are employed in a clerical occupation. As the differences in consumption and taxation patterns in this case are slight, the shadow wage rates of the two groups will differ little when there is unemployment in both categories taken together (see fig. 2). There is full employment of skilled and unskilled blue collar workers in the years 198CL86 and 1983-86, respectively. The paths of their ARs (fig. 3) reveal an apparently anomalous possibility in
0.9
A.
0.8 0.7 0.6
A
0.5 0.4
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.0 -0.7 -0.0 78 0
80
82
04
Profeasionmla
06 +
Fig. 2. Accounting
88
90
Other Whita Collar
ratios for labor.
92
C. Bell and S. Devarajan, Shadow prices
278 1.6
1.4 1.2 1 0.8 0.8 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 78 0
80
82
84
86 +
Skilled Blue Collar Fig. 3. Accounting
88
90
92
Unakill Blue Collar
ratios for labor.
that the shadow wage of unskilled workers exceeds that of their skilled counterparts in 1980, despite the fact that there is unemployment among the former but not the latter. The reason for this is that there is a high premium on government income in that year, so that the cost of transferring income to the unemployed is relatively high, and the constraint on skilled blue collar workers binds only slightly. Note that in 1983 through 1986, skilled workers are employed in unskilled jobs, so that their respective ARs are identical. Turning to goods, the path of the AR for a traded good can be inferred from that of the ARI, since a traded good has been chosen as numeraire (see section 3). The shadow prices of non-traded goods are the marginal shadow costs of their production. The latter cover traded and non-traded inputs and labor, all of whose shadow prices are mutually and simultaneously determined. When existing capacity is fully utilised, marginal cost also includes the cost of adding a unit to capacity in the year in question. To highlight the influence of binding capacity constraints, we present here the ARs for construction and transport and communication (see fig. 4). It turns out that capacity in the construction sector is binding only in the initial year. Moreover, it is the only non-tradeable sector experiencing full capacity
219
C. Bell and S. Devarajan, Shadow prices 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 70
80 0
82
Construction Fig. 4. Accounting
80
84 +
Trmsp
88
90
92
and Comm
ratios for nontradeables.
utilisation in the first year. As already pointed out, the shadow wage rates in that year are negative, so the negative shadow price for transport and communications, especially in light of its underutilised capacity, is not surprising. The shadow price of construction in 1978, however, is positive, reflecting the shadow rent on existing capacity in that sector. Thereafter, the AR for construction is below unity, except in 1980 when there is a high premium on public income and correspondingly high shadow wage rates. By contrast, in the transport and communication sector, there is excess capacity in the first seven years, and full utilisation afterwards. The import content of building new capacity in this sector is very high. Hence, when there is a substantial premium on public income, the shadow cost of creating a unit of additional capacity may be enough to raise the AR above unity, even when the shadow wage rates and ARs of some other non-tradeables are below unity. Moreover, as shown in the dual cost condition (23) the AR1 enters into the reckoning as the opportunity cost of committing the resources needed in earlier periods to create new capacity today. Thus, the current AR will be sensitive to the ARIs in the gestation years of new investment when existing capacity is fully utilised. In the case of transport and communications, these factors are sufficient to keep the AR of this sector above unity from 1986 to 1991.
280
C. Bell and S. Devarajan,
Shadow prices
The slight upturn in the ARs for labor and non-tradeables in 1988 is due to the terminal conditions, which specify the post-terminal growth rates by sector exogenously. As these growth rates were higher than those endogenously generated during the horizon, a slight premium on these sectors was generated in order to create the capacity to sustain the post-terminal growth rates. A sensitivity test where all post-terminal growth rates were scaled by one-half revealed a significant damping of this small ‘peak’ in 1988. Lastly, we compare the paths of various shadow prices in the presence and absence of a debt-service ratio (DSR) constraint (fig. 5 and 6). The most striking difference in those for the transport and communication sector is the sharp rise and fall between 1984 and 1987, when the DSR constraint binds. This is because there is a premium on exportables when that constraint binds and light manufacturing - an exportable - has been taken as numeraire for goods. In addition, transport and communication is an intensive user of noncompetitive imports, which by definition command no premium of this sort. Hence, the offsetting increase in its shadow cost of production is small. The introduction of a DSR constraint appears to smooth the trajectory of shadow wage rates, especially in the middle 1980s. As the DSR constraint binds in 1987, it is desirable not to undertake too much borrowing (and hence investment) three years earlier, in 1984. With less investment in 1984,
0
without
DSC
Fig. 5. Accounting
+
tith
DSC
ratios for labor (skilled blue collar).
C. Bell and S. Devarajan, Shadow prices
281
2.4 2.2 2 1.8 1.6
I.4 1.2 1 0.0 0.0 0.4 0.2 0 -0.2
f 70
,
I 80
a
I
, 82
without
Fig. 6. Accounting
I
I 84
DSC
ratios for nontradeables
I
I 88 + (transport
I 88
I
with
I
, 90
I 92
DSC
and communication).
demand for labor is lower in 1985, so that if labor is fully employed its shadow price will also be lower then. As the DSR constraint does not bind after 1987, investment can pick up in 1985, with the attendant effect of a stronger demand for labor in 1986. In the case of skilled blue collar workers, this is sufficient to cause the two paths to crossover in 1985-86.
5. Conclusions Practical work on the derivation of shadow prices proceeds on many simplifying assumptions. An important one, which is perhaps insufficiently emphasised, is that the economy is, and will remain, on a smooth path along which the intertemporal disturbances that feature so prominently in the above account are absent or negligible. While conceding that an optimising model with comparatively few goods, activities and non-produced factors will tend to exaggerate the magnitude of year-to-year movements, we contend that assuming a stable and narrow range for the key intertemporal variables may lead to serious errors. Indeed, the shadow prices derived here are, if anything, disconcertingly unstable. There are several causes of this instability. First, there are the boundary conditions, both in terms of the economy’s initial endowments and the
282
C. Bell and S. Devarajan, Shadow prices
valuation of terminal capital stock and foreign indebtedness. The Cypriot economy was certainly not close to a steady-state growth path in 1978. Moreover, with gestation lags of up to four years, it might be argued that a fifteen-year horizon was insufficient for the economy to attain such a path. We are not persuaded, however, that this is the whole story. As Dervis et al. (1984) show with a heavily aggregated, nonlinear model, even a 30-year horizon is not enough to generate a smooth pattern of shadow prices. Second, if a constraint on foreign borrowing is imposed and does bind, then the trajectories of the ARs and the AR1 may become very erratic indeed. In the present model, consumption is not easily controllable, so that the principal burden of year-to-year adjustment must fall on capital accumulation. In these circumstances, the AR1 will tend to be volatile and this volatility will spillover to the ARs of labor and non-traded goods, especially those that are capital intensive. Thus, discussions of what is an ‘acceptable’ level of the DSR are not to be taken lightly. For if a target level is set and successfully pursued, there may be profound differences in the resulting structure of shadow prices compared to that in which no such constraint is imposed. The profitability of a project may then depend critically on its timing. These two causes of instability in the paths of shadow prices illustrate a more general point. In practice, few - if any - countries enjoy steady-state growth and fewer still can get there by borrowing what they need at a parametrically given rate of interest. Therefore, the empirical estimation of shadow prices cannot be undertaken without reference to the macroeconomic setting in which the economy operates. In this paper, we have estimated shadow prices that reflect such macroeconomic considerations. While the intertemporal features of this setting led to a fluctuating pattern of shadow prices, we would argue that this is an inevitable consequence of a move towards more realism. Of course, to obtain these estimates, simplifying assumptions had to be made elsewhere. In addition, detailed project appraisal may require a finer disaggregation for non-traded goods than that employed here. In this spirit, our estimates of shadow prices should be seen not as a list to be pinned to the wall and consulted uncritically when evaluating projects, but rather as guides when discussing the implications of the general evolution of the economy for the choice of socially profitable projects.
Appendix: Valuation of terminal stocks Recall that in the objective function we specified the contribution of terminal capital K, and debt D, as U*[a,K,+cr,D,]/(l+r)T-‘. The parameters cr, and a2 represented the flows of sustainable consumption (our numeraire) generated by a unit of terminal capital stock and debt, respec-
C. Bell and S.
Devarajan,
Shadow
prices
283
tively, and U*( *) was the function that evaluated the utility of this flow. In this appendix, we show how al, a2 and U* were arrived at. A stock of capital in year 7; K,, will produce a flow of income UK T where u is the value added producible from a unit of K. A feasible post-terminal trajectory is one in which all of this is consumed every year forever (assuming no depreciation and no decline in the labor force). This would lead to a valuation of a1 =u. The form of Ii* would then be derived by (for the case where v]= - 1)
-“dr=N,~log!$
(Al) T
Note that we have assumed nc population growth. at a rate 2, the functional form of U* would be
With population
growing
On the one hand, this is a lower bound on the valuation of K, because it may be possible for the economy to save a fraction s of income and enjoy a stream of income that grows, rather than staying constant, at a rate g. In this case, U* would be given by (again with L=O),
U*(uK,)=N,T
uK,(l -s) N T
On the other hand, our original formulation (Al) is an overstatement of the benefits from K, because we assume the average productivity of capital, u, holds at the margin. Since marginal productivity is typically lower, we are overstating the value of K,. However, we could not use the marginal productivity (say au, where a is the share of capital in value added) because this would understate the level of consumption producible from K, in the terminal year. Taking these factors into account, we assume that the two effects offset each other and therefore choose the formulation (Al). As for az, the penalty on consumption of having a unit of terminal debt, we observe that the debt-service payment will be pT, the interest on that debt. Of course, this must be paid by the government which has limited tax instruments at its disposal. Assume the government levies an across-theboard income tax; the foreign exchange saved by the reduction in consumption is then used to make the debt-service payment. Manipulation with a
284
C. Bell and S. Devarajan,
simple, two-sector
Shadow prices
model yields the relationship: v,M+l
%=
-
(&+a,,)M+P
M=
1-P 1 -VN(l-P)-QN’
PT,
where
B=share of tradeables in consumption, vN= share of value added in nontradeables production, and aij = input-output coefficient where T= tradeables and N = non-tradeables. For Cyprus, the following estimates were used: jI=o.4,
vN= 0.6,
aNN = 0.14,
a,,=0.2.
These result in CQ= - 1.9pr. Given the uncertainty surrounding mates, we round off at t12= -2.0~~. Hence, our valuation of terminal stocks is given by
these
esti-
References Bell, C. and S. Devarajan, 1982, In search of a Five Year Plan for the Republic of Cyprus: Experiments with a semi-input-output model. World Bank, mimeo. Bell, C. and S. Devarajan, 1983a, Appraising development plans for a small, open economy, Proceedings of the Fourth IFAC/IFORS/IIASA Conference, Washington, D.C. Bell, C. and S. Devarajan, 1983b, Shadow prices for project evaluation under alternative macroeconomic specifications, Quarterly Journal of Economics 97, 457477. Bhagwati, J., T.N. Srinivasan and H. Wan, Jr., 1978, Value subtracted, negative shadow prices in project evaluation and immiserising growth: Three paradoxes in the presence of trade distortions, Economic Journal, March, 121L125. Bruno, M.. 1975, Planning models, shadow prices and project evaluation, in: C. Blitzer et al., eds., Economy-wide models and development planning (Oxford University Press, Oxford). Chenery, H. and A. MacEwan, 1966, Optimal patterns of growth and aid: The case of Pakistan, in: I. Adelman and E. Thorbecke, eds., The theory and design of economic development (Johns Hopkins University Press, Baltimore). Dervis, K., R. Martin and S. Van Wijnbergen, 1984, Policy analysis of shadow pricing, foreign borrowing and resource extraction in Egypt, World Bank Staff Working paper no. 622. Diamond, P. and J. Mirrlees, 1971, Optimal taxation and public production: I-production efficiency, American Economic Review 61, 8-27. Dreze, J.P., 1982, On the choice of shadow prices for project evaluation, Discussion paper no. 16, Development Economics Research Centre, University of Warwick, July. Dreze, J. and N. Stern, 1985, The theory of cost-benefit analysis, in: M. Feldstein and A. Auerbach, eds., Handbook of public economics (North-Holland, Amsterdam).
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Shadow prices
285
Little, I. and J. Mirrlees, 1974, Project appraisal and planning for developing countries (Heinemann Press, London). Srinivasan, T.N., 1983, General equilibrium theory, project evaluation and economic development, in: M. Gersovitz et al., Theory and experience of economic development: Essays in honour of Sir W. Arthur Lewis (Allen and Unwin Press, Herts). Stern, N.H., 1977, The marginal valuation of income, in: Artis and Nobay, eds. Studies in modern economic analysis (Blackwell, Oxford). Taylor, L., 1975, Theoretical foundations and technical implications, in: C. Blitzer et al., eds., Economy-wide models and development planning (Oxford University Press, London). Tourinho, 0. 1985, Optimal foreign borrowing in a multisector dynamic equilibrium model: A case study for Brazil, M.I.T. Energy Laboratories, Working paper 8S-009WP. UNIDO, 1972, Guidelines for project evaluation (United Nations, New York).
of Public
Economics
32 (1987) 263-285.
North-Holland
INTERTEMPORALLY CONSISTENT SHADOW IN AN OPEN ECONOMY
PRICES
Estimates for Cyprus Clive BELL Vanderbilt University, Nashville,
_ Shantayanan Harvard Received
DEVARAJAN*
University, Cambridge,
February
TN 37235, USA
1985, revised version
MA 02138, USA received October
1986
In the practical estimation of shadow prices, the assumption that the economy is on a steadystate growth path is often invoked, usually implicitly. Most developing countries are far from this trajectory. We derive shadow prices where the dynamic evolution of the economy is endogenously determined, important influences being initial conditions and the constraints on foreign borrowing. Hence, the resulting (dated) shadow prices of tradeables, nontradeables, labor and the accounting rate of interest (ARI) are mutually consistent. Estimates for the Republic of Cyprus show that derivations from steady-state growth can be significant, leading to large fluctuations in the AR1 and thereby making the timing of investment projects crucial.
1. Introduction When assessing whether a project is socially profitable, it is necessary to use prices that reflect social scarcities, and for various well-known reasons, these may be quite different from market prices. One way of carrying out such an assessment is to formulate a model of the whole economy, with *We thank participants at seminars at the University of Pennsylvania, the World Bank, HMvard University and New York University and, especially, two anonymous referees for helpful comments on an earlier draft. Sethu Palaniappan and Arjun Jannah provided invaluable assistance in programming and running the model. An earlier version of this paper was presented at the Annual Meeting of the Southern Economic Association, Washington, D.C. November 1983. Errors and opinions are, of course, solely our responsibility. The World Bank does not accept responsibility for the views expressed herein which are those of the author(s) and should not be attributed to the World Bank or its affiliated organizations. The findings, interpretations, and conclusions are the results of research supported by the Bank; they do not necessarily represent official policy of the Bank. The designations employed, the presentation of material, and any maps used in this document are solely for the convenience of the reader and do not imply the expression of any opinion whatsoever on the part of the World Bank or its affiliates concerning the legal status of any country, territory, city, area, or of its authorities, or concerning the delimitation of its boundaries, or national affiliation. 0047-2727/87/$3.50
0
1987, Elsevier Science Publishers
B.V. (North-Holland)
264
C. Bell and S. Devarajan, Shadow prices
social welfare as the maximand, and solve for the optimal allocation of resources. If this is a ‘first-best’ optimum (i.e., there are no constraints on the government’s ability to make lump sum transfers), then the Lagrange multipliers corresponding to the scarcity constraints in the model can be used to value the project’s inputs and outputs [see, for example, Bruno (1975)]. Yet in this case, there is no point in evaluating the social profitability of projects, for the Lagrange multipliers equal market prices. If, as is usual, the government is unable to achieve a ‘first-best’ optimum, a ‘second-best’ problem must be solved, with the constraints preventing achievement of the first-best optimum included. In this case, however, the associated Lagrange multipliers often have to be reexpressed to derive the appropriate ‘shadow prices’ for project evaluation [Dreze and Stern (1985)]. This second-best approach is the one taken in the celebrated manuals of Little and Mirrlees (1974) and UNIDO (1972). Drawing on the theoretical work of Diamond and Mirrlees (1971) they establish rules for deriving shadow prices under particular assumptions about the government’s opportunity set. The best known rule, which is robust to a wide range of assumptions about government behavior, is to set the shadow price of a tradeable good at its border price. Next, the shadow price of a nontradeable is set equal to its marginal shadow cost of production. With tradeable inputs valued at their border prices and labor at the shadow wage rate, the shadow prices of nontradeables can be calculated once the shadow wage rate is known. The shadow wage rate, in turn, is the value (at shadow prices) of the worker’s marginal product in alternative employment, plus the shadow cost of any increase in his wages (for example, if he were previously unemployed). At this point, intertemporal considerations intrude: to the extent that paying the worker a higher wage involves a shift in the uses of income from savings to consumption, it becomes necessary to estimate the premium on savings in the economy. This premium is, of course, related to the accounting rate of interest (ARI), the rate at which the project’s net benefits (estimated at shadow prices) are to be discounted over time. In principle, the savings premium, the AR1 and the entire set of (dated) shadow prices of nontradeables and labor are mutually and simultaneously determined. In practice, however, the savings premium and the AR1 are almost invariably derived first, often by rather rough and ready methods. The estimate of the savings premium is then fed into the calculation of the remaining shadow prices, essentially within a static framework. When a model is used to derive the AR1 and the savings premium, it typically has only one sector and one factor, so that the results are applicable to a manygood economy only when all sectors are growing at the same rate, that is, on a ‘balanced growth’ path. Clearly, this one-way recursive procedure for estimating shadow prices, which is almost universally adopted in practice, rests on rather strong assumptions.
C. Bell and S. Devarajan, Shadow prices
265
In this paper, we estimate all shadow prices, the AR1 and the savings premium from a many-good, intertemporal optimising model, so that all these parameters are mutually and simultaneously determined. Nor is it assumed that there is balanced growth. We specify the constraints on the government’s instruments that prevent the economy from achieving a first-best optimum, and show how shadow prices can be derived from the Lagrange multipliers. There remains, however, an apparent difficulty with this procedure. Once the primal solution to the optimisation problem is known, no additional information is yielded by the dual because any known project that improves welfare will be included in the optimal solution to the primal. Hence, there are no projects left to be evaluated [Srinivasan (1983)]. Now, the use of shadow prices might not be redundant if there existed projects that were not contained in the set of feasible activities from which the optimal allocation was chosen. The existence of such projects stems from the government’s limited knowledge of future transformation possibilities, and proposals for the projects may appear unexpectedly at any time throughout the planning horizon. What is needed, therefore, is a set of shadow prices that will detect whether such projects will improve social welfare, given the optimal allocation of resources based on the knowledge available at the start of the plan. The shadow prices in the manuals and those in this paper possess this property. As Dreze (1982) shows more generally, the gradient of the welfare function at the optimum can serve this purpose. We estimate shadow prices for the Republic of Cyprus, a small, open economy with a diversified production structure and a literate population. Cyprus has a record of rapid growth since its independence, despite the events of 1974. As it entered its first five-year plan in 1982, Cyprus was facing various problems of misallocation, including some stemming from the structure of taxation and others from a strong appetite for foreign borrowing. Hence, the emphasis placed here on public finance and intertemporal borrowing possibilities is by no means accidental. The model is presented in section 2. In section 3, the dual solution to the model is used to derive shadow prices. We then show how, with certain simplifying assumptions, these shadow prices specialize to those derived from the manuals in practical work. A discussion of the results follows in section 4, and section 5 presents some concluding remarks. 2. The model The model is an extension of that used by Bell and Devarajan (1982, 1983a) to obtain a consistent and feasible Five Year Plan for Cyprus. What follows is a version simplified for the purposes at hand; those who hanker
266
C. Bell and S. Devarajan, Shadow prices
after the more intricate details are referred to those papers. All variables are dated, but the time index (r) is specified only when the context demands it. We begin with the familiar material balance equations: X+M=AX+C+J+E+G,
(1)
where X is the (n x 1) vector of domestically produced commodities; M is the vector of imports; A is the matrix of (fixed) interindustry coefficients; C is the vector of private consumption; J is the vector of investment deliveries by sector of origin; E is the vector of exports; and G is the vector of government current consumption. Factor incomes are given by W=VX+6,
(2)
where V is the (4 x n) matrix IIUfjll>vfj being the payment made to the Sth factor by the jth activity when it is operating at unit intensity; and 6 is the (9 x 1) vector of the dummy variables 6,. These dummy variables are set initially to zero; but since we are interested in what happens when the government transfers one unit of income to any factor by employing it, each of the ds’s can also take the value unity. The incomes of domestic institutions (including public sector firms) from their ownership of factors are given by: Y=QW
(3)
where R is the (h x q) matrix which maps factor payments into institutions’ incomes from factors. Private consumption is given by a linear expenditure system: C=B(Z-?)(I-f)Y,
(4)
where B is the (n x h) matrix of expenditure coefficients;’ s*is the diagonal matrix of savings rates out of post-tax income; and Z is the diagonal matrix of tax rates on institutions’ incomes. Investment demand is determined by an accelerator relationship, J(z) = H[K(r + 1) -K(z)],
(5)
where H is the (n x n) matrix of capital coefficients and K(r) is the vector of full capacity output levels in period r.2 ‘In the full model, the Engel lines are not constrained to go through the origin. Note also that the expenditure coefficients for institutions other than households are equal to zero. 21n the full model, the expansion of capacity may require investments in as many as four years preceding the date on which the additional capacity becomes available, depending on the sector in question.
C. Bell and S. Devarajan,
Shadow prices
261
Eqs. (l)-(5) represent the basic behavioral and technical assumptions of the optimising model. In addition, we impose a set of constraints to reflect the resources available to the economy and the extent to which they can be shifted from one use to another. First, the demand for labor cannot exceed its supply in each skill category:3 L(z) X(z) 5 L(r),
where L(r) is the (1x n) matrix of labor coefficients, lrj(r); and L(r) is the exogenously given vector of labor supplies. The labor coefficients change over time according to a set of exogenously given productivity growth rates. Similarly, the domestic production of any commodity cannot exceed installed capacity:
Also, it is reasonable to suppose that, once installed, capital is not shiftable. With no depreciation, this implies K(z) 5 K(z + 1).
Note that in the first period, all installed capacities are given, whereas in all other years, capacity is endogenous. Hence, with the exception of the first year, the sole primary factors of production are the various categories of labor. There remains the matter of how any balance of payments deficit is to be financed, for borrowing in the international capital market is an opportunity open to Cyprus. In any year, the rate of interest on what is borrowed is assumed to depend only on the amount borrowed in that year4 and to keep computation manageable, this relation is linearised. Starting at some low positive rate where the current account just balances, the rate of interest is assumed to rise in steps with the amount borrowed until it reaches an upper limit. The model is solved with two variations. In the first, it is assumed that there is a perfectly elastic supply of foreign funds at the upper limit for the rate of interest. In the second, the possibility of credit rationing at a parametric interest rate, and/or a reluctance on the part of the government 31n the full model, we allow workers with higher skills to do lower skilled jobs. For example, professionals can do ordinary white collar jobs, etc. 4Clearly, other variables (like the total indebtedness) affect the interest rate on current borrowing. Our purpose here, though, is to introduce a rising marginal cost of foreign borrowing. In view of the difftculties of estimating the parameters of this schedule, the formulation adopted here has the merit of simplicity. Tourinho (1985) adopts a similar specification.
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C. Bell and S. Devarajan,
Shadow
prices
to permit very heavy foreign indebtedness, is introduced. This is expressed by the constraint that the debt-service ratio (DSR) in any year be no greater than a parametrically given value, so that the amount that can be borrowed in any year is bounded from above, albeit endogenously. To complete the picture, if there should be a surplus after paying off principal and interest on past borrowings, that surplus is lent abroad at a parametrically given rate. Note that with an upward sloping supply schedule of foreign funds, Cyprus can exercise a (modest) amount of monopsony power in the course of choosing an optimal programme. It is assumed that the terms of borrowing, other than the rate of interest, p(r), do not vary. In the full model, a grace period of three years is followed by seven years in which to repay, in equal installments. To introduce these terms would complicate the task of exposition here without adding anything in the way of substance, so the formulation adopted below specifies that interest and principal be repayable in full one year after the loan is taken up. With this simplification, the oustanding foreign debt, D(r), takes the form D(z)=[l+p(z-l)].D(r-l)+p’[M(z)-E(r)],
2=1,...,7:
where p’ is the (1 x n) vector of world prices and p(O) and D(0) are given parametrically, Under these terms, the amount borrowed in any year is simply the oustanding debt. Hence,
P(4 = PC~(Lu, where p[.] is linearised exceed cr, the following
(10)
as described above. If, in addition, constraint must be added:
the DSR must not
(11) Objective
function
and terminal
conditions
We ignore considerations of equity among contemporaries, so that the level of instantaneous welfare depends only on aggregate consumption. Domestic savings are valuable only for the increase in future consumption they make possible. If domestic savings can be increased by other means only at very high cost in terms of consumption foregone, then the option of foreign borrowing may be highly attractive; but any indebtedness that remains at the end of the horizon entails a cut in consumption thereafter, and therefore entails a cost. Similarly, the capital stock bequeathed to the future at the end of the horizon is valuable only inasmuch as it permits sustainable consumption in the years that follow. Hence, the objective function consists of the discounted utility of consumption throughout the
C. Bell and S. Devarajan, Shadow prices
269
time horizon, together with a valuation of (i) the vector of capacities in the terminal year, K(T), and (ii) the level of foreign debt in the terminal year, D(T). The estimation of c1r and CY~, the values of the stocks K(T) and D(T), respectively, in terms of their corresponding flows of consumption, is given in the Appendix. While we can value K(T), there is nothing so far that ensures that there will be sufficient labor to operate all this plant and equipment. With a planning horizon of fifteen years, therefore, it seems best to impose the condition that the demand for labor generated by running all plants at full capacity in the terminal year should not exceed the labor supplies available at that time, namely,
L(T)K(T)<=L(T).
(12)
3. Shadow prices The optimisation problem described above can be summarised as follows: T U[u’C(z)]
c (1+0-l IX(r),E~r).K(r)) t=l
+
u*[Icr;lq T) + a,D( T)] (l+<)‘_’ ’
(13)
subject to (1)-(12), with G(z), Qr), K(l), D(O), p(O) parametrically given and 6(r) = 0. Note that all taxes are assumed to lie outside the planner’s sphere of control. We now consider the dual solution to this problem as a means to derive shadow prices. Let the dual variables associated with, respectively, the material balances, the consumption function, the labor constraints, the installed capacity constraints, the condition that capita1 be non-shiftable, the growth of indebtedness (9), the requirement that the dummy transfers from government to factors, 6(r), be zero, the debt-service constraint and the terminal year constraint (12) be denoted by rc, rr,, nl, rcK,nN, xD, 7c6,rc, and 1*. The dual cost conditions may be written down at once. In all periods, we have: n’(z). A +7$(T). Y v + 11,(T).L + 7&(z) 2 7c’(z),
(14)
744 - 4(4 2 P’(%
(15)
n’(z) + p’rco(z)- op’rcJr) >=0, - n’(f) - p’ . rcD(r)2 0,
for fully traded goods,
for fully traded goods,
(16a) (16b)
C. Bell and S. Devarajan, Shadow prices
210
n’(z-
l).H-r&(7-
+7&(7)~0,
l)-7c’(z).H-n;,(7) 7=2 )...) T-l,
(174
77,(7-1)+(1+&7,(7)-[1+~(7-1)]~71~(7)~0,
7=1,...,7;
(fga) (Igb)
where Y = B(I - 3)(I - 1)Q, ~(7) = U’(r)/( 1 + [)‘- ‘, and 2 is a diagonal matrix of post-terminal growth rates (see Appendix). From (16a) and (16b), it follows at once that for traded goods, if the DSR constraint is not binding (x,=0): 77’(z)= - p’ .7LD(T). If it does bind, and exports
are positive,
then from (16a):
72’(z)= -p’(77D(7)-cr77,(7-l)). Hence, the shadow prices of traded goods are in both cases proportional to their respective world prices - the celebrated ‘border price’ rule. Note that for a given good, (16a) and (16b) cannot both hold with equality, confirming the fact that it is never optimal to import and export the same good. Next, the fact that there is some installed capacity in all sectors in the first year, together with the non-shiftability condition and the absence of depreciation, implies that K(z)>0 in all years. Also, since it is plausible that at least one good will be produced in positive amounts in each year, then mild restrictions on the consumption parameters will ensure that all goods are consumed in strictly positive amounts in each year. Hence, the fundamental duality theorem of linear programming tells us that (15), (17a) and (17b) must hold as strict equalities.5 Similarly, for every good which is actually produced domestically, the relevant dual cost condition in (14) will hold as a strict equality. As (15) holds as an equality, we may substitute for 77:(z) in (14). Suppose there are m traded goods, so that in any year m of the n elements of n(7) are known up 5Although, by definition, there is neither domestic capacity nor the technology to produce non-competitive imports, so that (14) will be a strict inequality, an inspection of (17) reveals that the dual variables nK and xN for any good do not depend directly on the rest. The dual variable B for non-competitive imports is given by X(T)= --p” ndr).
C. Bell and S. Devarajan, Shadow prices
271
to a scalar multiple. As there are 1 categories of labor, rcl has I elements also. Now for any reasonable technology and consumers’ preferences, all nontraded goods must be produced in positive quantities. Hence, the (n-m) dual cost conditions for such goods must hold as strict equalities. It follows from (14) that, in any year, the number of fully traded goods produced at home will be I plus the number of strictly positive elements of n, not linearly dependent on rc. The intuition for this result is that domestic production of a traded good may be profitable at marginal cost if capacity is not fully utilized, while it remains unprofitable to expand capacity. Turning to rr,, when D(r) is unconstrained by (1 l), (18a) and (18b) must hold as strict equalities, so that: (19) that is, the penalty on a unit increase in the foreign debt in the terminal year discounted back to the first year. Moving backwards from the terminal year, (18a) reveals the intuitively obvious result that the absolute magnitude of rrD increases from year to year at the prevailing rate of interest on foreign borrowings. This leads naturally to a discussion of the numeraire and the AR1 which, by definition, is the rate at which the value of the numeraire is falling. Any good can be chosen as numeraire, so that, in principle, there are as many ARIs as there are goods [Dreze and Stern (1985)]. Common practice is to take uncommitted public income as numeraire. If the DSR constraint is not binding, this choice implies that the AR1 is the rate at which n, is falling, since in that case an increase in public income implies an equal decrease in foreign indebtedness. Moreover, the shadow prices of traded goods will also be falling at this rate, so that the AR1 is independent of whether public income, foreign indebtedness or a particular traded good is the numeraire. When the DSR constraint binds, this equivalence breaks down since 71, enters the picture and its value will change from period to period. As a choice is forced, we have selected a particular traded good (light manufactures) as numeraire, and our AR1 is defined accordingly. This should be borne in mind when examining the results presented in section 4. As rrN does not feature in the first and last periods, there are [3(T- l)n+ T(I+ 1) +I] dual variables all told (ignoring the T. n elements of {%(l), . . . , z,(T)}, which are given at once by (15)). Against this, there are at most [(2T - 1)n + T(m + 1) + I] equations. Hence, while restricting the number of fully traded goods to be at most equal to the number of labor types will ensure that all traded goods will be produced at home on some scale in all years, much greater disaggregation may be possible in practice. Are the dual variables derived above able to detect welfare-improving changes in the allocation of resources? As any vector of dated inputs and
212
C. Bell and S. Devarajan, Shadow prices
outputs may be thought of as a project - it must be sufficiently small in scale _ which has its own particular investment requirements and operating technique, it certainly seems permissible to use the dual variables to evaluate the project’s social profitability, and, if profitable, to include it in the Plan characterised by the primal solution. For although the Plan is derived on the assumption that the technology available to the economy is described by the matrices {A, V,L(r)}, it may be augmented by a set of projects which use techniques other than those in {A, v,L(~)j, provided their collective measure is sufficiently small. Now, each unit of good i used by such a project in year r will impose a cost on the economy, as measured by the resulting fall in the value of the objective function, in the amount xi(z). Where labor and other incomereceiving factors6 are concerned, however, the dual variables need to be interpreted with some care. If, for example, the government were to employ one worker of type fat the going wage, w/, and all workers of this type were employed, there would be no change in private consumption. In this case, the value of output foregone (at shadow prices), which is measured by the dual variable associated with the constraint on the supply of the fth kind of labor, is the appropriate shadow wage, viz r~~,~. The difficulty arises if the worker in question is unemployed; for although 7~~ will then be zero, the act of employing him will result in a transfer of wf from the government to the worker, a transfer which will, in general, affect the value of the objective function. The source of the difficulty is that while the government cannot use lump sum taxes as policy instruments, it is able to make some transfers by employing workers who would otherwise be jobless by increasing the derived demand for labor from production activities. Hence, the appropriate shadow prices of factors must reflect the fact that additional private consumption is both inherently valuable to the economy inasmuch as it appears in the objective function, as well as potentially costly, as it entails potential absorption of resources, which is where the dual variables n,(s) enter the picture. To derive the shadow price of a change in income, therefore, suppose the government transfers a unit of income to factor J The dual cost condition associated with the constraint 6 = 0 is:
where Y,-= B(I--.?)(!Z)Q, is the fth column of Y, and nd,Jz) is the fth element of Q(T). As (15) holds as an equality, the shadow price of this
6Note that the latter are not primary in the sense of being non-produced factors of production. They receive value added from production because the institutions of the economy happen to be of a particular kind.
273
C. Bell and S. Devarajan, Shadow prices
transfer
is: _n,,f=n’(z)Yf-
U’/(l
+i)‘-‘.
(20)
Now, n’(r)YYs is the value of the resources absorbed by households’ consumption out of this unit transfer, more commonly known as the consumption conversion factor for factor income of type f, and U’/( 1 + [)‘- 1 is the social value of this additional consumption. Hence, a previously idle worker, who is now to be engaged at the going market wage wf, will have a shadow price equal to: [d(T)Yf
- U’/( 1 + [)I- ‘1 . Wf.’
This is, of course, an application of the well-known theorem that the gradient of the Lagrangian is equal to the gradient of the objective function at the optimum. It should be noted that this shadow price may be negative, which simply reflects the fact that transfers from the government to factors would improve welfare if they could be put into effect.8 For some purposes, it will be useful to normalise the shadow price of labor by the going wage. In the language of Little and Mirrlees (1974) this is the accounting ratio for an unemployed laborer who would otherwise receive no income, which is given by (20). To complete this section, we set out how conditions (14)419) are related to procedures commonly used in the estimation of shadow prices in practice. Suppose, first, that the accounting rate of interest in each year is known, either because the supply of foreign funds is perfectly elastic or because some independent piece of analysis has been done. Suppose, further, that there is steady state growth, so that nN=O (i.e., the non-shiftability constraints do not bind). As established above, (17a) holds with strict equality: [rC’(r-I)-n’(r)]H=rck(r), Furthermore,
in steady
state growth
r<7: x(r) will decrease
(21) at a rate equal
to the
‘If the worker was receiving unemployment benefits, they should be subtracted from the going wage. Moreover, the value of leisure is ignored here. “At first sight, it would appear that these shadow prices apply solely to projects in the public sector. Consider, therefore, a project which just breaks even at market prices, with the government purchasing all inputs (including factors) and selling the resulting output. The government will neither gain nor lose income by these transactions, though agents in the private sector will certainly claim the entire resulting value asIded. Hence, we have, in effect, a project in the private sector, the social profitability of which can be established by employing the shadow prices derived above. Furthermore, even in the case of strictly positive private profits, appropriate shadow prices are simply a linear combination of the above shadow prices and producer prices.
274
C. Bell and S. Devarajan, Shadow prices
ARI, p(z). Substituting
into (21), we get
_
p(? - l)n’(z)H = Z;(T),
,r < T.
(22)
which expresses the appealing intuition that the shadow price of capacity in sector i should be the shadow rental on the set of capital goods needed to augment capacity in that sector by one unit. In steady state growth, capacity will be fully utilised, so nk>O. Substituting (15) and (22) into (14), we have (recalling that (15) holds with strict equality): n’(r)[A + Y V+p(?-
l)H] +rr;(z)Lzp’(T)[z-
YV].
(23)
As before, all non-tradeables will be produced in strictly positive amounts, so that the relevant dual cost conditions hold with strict equality. Note that the dual prices for traded goods are known from (18a), which holds as a strict equality, and (19). Then (23) can be solved for n,, and the particular goods that are produced at home are revealed at the same time. The latter number at most 1, for although rr,>O, we see at once from (22) that it is linearly dependent on rc. Thus, independent estimation of the AR1 and the assumption of steady state growth permit the entire set of dual prices to be solved for in a fairly straightforward way. In practical work, however, shadow prices are not typically derived by finding a nonnegative solution to a system of linear inequalities. Instead, the analyst has two alternatives. First, knowing the number of primary factors, he can specify the particular traded goods - exactly the same number as there are factors - which will be produced at home. The dual cost conditions in (23) for these goods will hold as strict equalities, so that rc and rrcIcan be derived by matrix inversion [Srinivasan (1983)]. Second, in defiance of neoclassical trade theory, the analyst makes no particular assumption about the number of traded good produced domestically. In this case, n, is estimated independently, as the set of marginal products (valued at shadow prices) of workers in alternative employment. That is, the analyst answers the question: where is the labor coming from? The set of shadow prices of nontradeables follows at once from the relevant subset in (23). Note that in neither of these cases can negative shadow prices be ruled out [Bhagwati et al. (1978); Bell and Devarajan (1983b)]. In any event, the AR1 cannot be satisfactorily estimated without invoking (14)+19). Also, if it is implausible to argue that the economy will enjoy steady-state growth in the future, then the chain of argument leading to (23) will break down. In that case, there is nothing for it but to solve for the entire set of dual variables directly. 4. Results We begin
by specifying
the form of the felicity function,
i.e., the instanta-
C. Bell and S. Devarajan, Shadow prices
275
neous value of the objective function. We assume an isoelastic utility function u(.) of per capita consumption, c, and a constant rate of pure time preference, r. The value of the elasticity of marginal utility of consumption, q, is somewhat controversial, although Stern (1977) provides a careful discussion of various methods of estimating it, which suggest that a value between - 1 and -2 is defensible in many circumstances. Accordingly, we use these extreme values for q which lead to felicity functions of the form:
(24)
If population is growing at the rate II, then,
U[u'C] = N,u(c) [( 1 + I+)/(1 + r)]‘, where N, is the population in the initial year. As I is projected to be growing at 1.2 percent per annum, [ = r -I is surely barely positive. In fact, we assume [=O.Ol. As it turned out, the results for the case q = 1 differed little from those for ~=2. For want of space, therefore, the latter are reported here, and even then rather sparingly.’ Recall from section 2 that two variants of the cost of foreign borrowing are used. Despite the rising cost of foreign borrowing over a certain range, the optimal solution in the absence of the DSR constraint is characterised by a healthy appetite for foreign capital. With the DSR constraint and o set at 15 percent, a level thought to be desirable during the preparatory work for the Five Year Plan, there was no feasible path. When 0 was relaxed to 25 percent, however, an optimal solution was obtained, with the DSR constraint binding from 1984 through 1987. With a three-year grace period on foreign loans, this implies a premium on foreign exchange in the years up to and including 1984. In addition, there is a premium on exports in the years 1984-1987. We begin with the accounting rate of interest (ARI). In the absence of the DSR constraint, it might be thought that the AR1 is the interest rate on the last dollar borrowed in the year in question. This is incorrect for two reasons. First, borrowing an additional dollar today will affect the cost of borrowing in the future if the marginal cost of borrowing is not constant, as in the present case. Second, there is a grace period of three years, so gains from. arbitrage are possible. Figure 1 demonstrates that such departures of % an earlier version of this paper, a linear objective function (q=O) was used. This gave rise to even more volatile shadow prices than the ones reported in this paper (see below). Nevertheless the qualitative behavior of the dual variables was unchanged.
276
C. Bell and S. Devarajan, Shadow prices
24
I
42-
0
I 79
81 0
83 without DSC Fig. 1. Accounting
85
87 + rate of interest
89
91
93
with DSC (in percent)
the AR1 from the marginal rate of interest can be significant. When the pressures exerted by the necessity to complete projects in the pipeline at the start of the planning period are significant, the AR1 exceeds the marginal cost of foreign borrowing. After falling back, the AR1 proceeds to rise again, reaching a peak of 12 percent in 1988. Only in 1992 does simple intuition hold, for with a grace period, borrowing at the ‘top’ rate (six percent) in 1992 has no future repercussions within the horizon. The imposition of the DSR constraint will, of course, produce still sharper movements of the ARI, for borrowing an extra dollar today may be very costly if the constraint should bind in the future. Thus, the AR1 begins to rise in 1983, because the DSR constraint binds from 1984 to 1987. It reaches a sharp peak in 1985, before declining once more to the ‘top’ rate in 1987. Moreover, the fact that the DSR constraint binds in the mid-1980s intensities the pressures from the ‘pipeline’ effect in earlier years, as can be seen from fig. 1. Turning to the associated shadow prices, we follow the standard literature on social cost-benefit analysis by presenting them in the form of accounting ratios (AR), that is, the ratio of the shadow to the market price of the good or factor in question. As can be seen from figs. 2 and 3, the ARs for labor are quite volatile. This volatility is not confined to the initial and closing years,
C. Bell and S. Devarajan,
211
Shadow prices
when the influence of the boundary conditions holds sway. It derives from the fact that when there is full employment, the AR is equal to the social value of the worker’s marginal product and when there is unemployment, to the net social cost of transferring to the worker his market wage out of the government’s coffers, a social cost which may be positive or negative, depending on the premium on government income. When the AR is negative, it would be socially profitable to make lump-sum transfers to these workers, if such transfers were possible. In fact, this arises only in 1978. We assume that skilled workers can do the jobs of their less-skilled couterparts, but not conversely. Thus unemployed professional workers can till vacancies for (clerical) workers if the latter are fully employed. To the extent that the two groups have different consumption and taxation patterns, their shadow wage rates will be identical only if, at the margin, professional workers are employed in a clerical occupation. As the differences in consumption and taxation patterns in this case are slight, the shadow wage rates of the two groups will differ little when there is unemployment in both categories taken together (see fig. 2). There is full employment of skilled and unskilled blue collar workers in the years 198CL86 and 1983-86, respectively. The paths of their ARs (fig. 3) reveal an apparently anomalous possibility in
0.9
A.
0.8 0.7 0.6
A
0.5 0.4
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.0 -0.7 -0.0 78 0
80
82
04
Profeasionmla
06 +
Fig. 2. Accounting
88
90
Other Whita Collar
ratios for labor.
92
C. Bell and S. Devarajan, Shadow prices
278 1.6
1.4 1.2 1 0.8 0.8 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 78 0
80
82
84
86 +
Skilled Blue Collar Fig. 3. Accounting
88
90
92
Unakill Blue Collar
ratios for labor.
that the shadow wage of unskilled workers exceeds that of their skilled counterparts in 1980, despite the fact that there is unemployment among the former but not the latter. The reason for this is that there is a high premium on government income in that year, so that the cost of transferring income to the unemployed is relatively high, and the constraint on skilled blue collar workers binds only slightly. Note that in 1983 through 1986, skilled workers are employed in unskilled jobs, so that their respective ARs are identical. Turning to goods, the path of the AR for a traded good can be inferred from that of the ARI, since a traded good has been chosen as numeraire (see section 3). The shadow prices of non-traded goods are the marginal shadow costs of their production. The latter cover traded and non-traded inputs and labor, all of whose shadow prices are mutually and simultaneously determined. When existing capacity is fully utilised, marginal cost also includes the cost of adding a unit to capacity in the year in question. To highlight the influence of binding capacity constraints, we present here the ARs for construction and transport and communication (see fig. 4). It turns out that capacity in the construction sector is binding only in the initial year. Moreover, it is the only non-tradeable sector experiencing full capacity
219
C. Bell and S. Devarajan, Shadow prices 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 70
80 0
82
Construction Fig. 4. Accounting
80
84 +
Trmsp
88
90
92
and Comm
ratios for nontradeables.
utilisation in the first year. As already pointed out, the shadow wage rates in that year are negative, so the negative shadow price for transport and communications, especially in light of its underutilised capacity, is not surprising. The shadow price of construction in 1978, however, is positive, reflecting the shadow rent on existing capacity in that sector. Thereafter, the AR for construction is below unity, except in 1980 when there is a high premium on public income and correspondingly high shadow wage rates. By contrast, in the transport and communication sector, there is excess capacity in the first seven years, and full utilisation afterwards. The import content of building new capacity in this sector is very high. Hence, when there is a substantial premium on public income, the shadow cost of creating a unit of additional capacity may be enough to raise the AR above unity, even when the shadow wage rates and ARs of some other non-tradeables are below unity. Moreover, as shown in the dual cost condition (23) the AR1 enters into the reckoning as the opportunity cost of committing the resources needed in earlier periods to create new capacity today. Thus, the current AR will be sensitive to the ARIs in the gestation years of new investment when existing capacity is fully utilised. In the case of transport and communications, these factors are sufficient to keep the AR of this sector above unity from 1986 to 1991.
280
C. Bell and S. Devarajan,
Shadow prices
The slight upturn in the ARs for labor and non-tradeables in 1988 is due to the terminal conditions, which specify the post-terminal growth rates by sector exogenously. As these growth rates were higher than those endogenously generated during the horizon, a slight premium on these sectors was generated in order to create the capacity to sustain the post-terminal growth rates. A sensitivity test where all post-terminal growth rates were scaled by one-half revealed a significant damping of this small ‘peak’ in 1988. Lastly, we compare the paths of various shadow prices in the presence and absence of a debt-service ratio (DSR) constraint (fig. 5 and 6). The most striking difference in those for the transport and communication sector is the sharp rise and fall between 1984 and 1987, when the DSR constraint binds. This is because there is a premium on exportables when that constraint binds and light manufacturing - an exportable - has been taken as numeraire for goods. In addition, transport and communication is an intensive user of noncompetitive imports, which by definition command no premium of this sort. Hence, the offsetting increase in its shadow cost of production is small. The introduction of a DSR constraint appears to smooth the trajectory of shadow wage rates, especially in the middle 1980s. As the DSR constraint binds in 1987, it is desirable not to undertake too much borrowing (and hence investment) three years earlier, in 1984. With less investment in 1984,
0
without
DSC
Fig. 5. Accounting
+
tith
DSC
ratios for labor (skilled blue collar).
C. Bell and S. Devarajan, Shadow prices
281
2.4 2.2 2 1.8 1.6
I.4 1.2 1 0.0 0.0 0.4 0.2 0 -0.2
f 70
,
I 80
a
I
, 82
without
Fig. 6. Accounting
I
I 84
DSC
ratios for nontradeables
I
I 88 + (transport
I 88
I
with
I
, 90
I 92
DSC
and communication).
demand for labor is lower in 1985, so that if labor is fully employed its shadow price will also be lower then. As the DSR constraint does not bind after 1987, investment can pick up in 1985, with the attendant effect of a stronger demand for labor in 1986. In the case of skilled blue collar workers, this is sufficient to cause the two paths to crossover in 1985-86.
5. Conclusions Practical work on the derivation of shadow prices proceeds on many simplifying assumptions. An important one, which is perhaps insufficiently emphasised, is that the economy is, and will remain, on a smooth path along which the intertemporal disturbances that feature so prominently in the above account are absent or negligible. While conceding that an optimising model with comparatively few goods, activities and non-produced factors will tend to exaggerate the magnitude of year-to-year movements, we contend that assuming a stable and narrow range for the key intertemporal variables may lead to serious errors. Indeed, the shadow prices derived here are, if anything, disconcertingly unstable. There are several causes of this instability. First, there are the boundary conditions, both in terms of the economy’s initial endowments and the
282
C. Bell and S. Devarajan, Shadow prices
valuation of terminal capital stock and foreign indebtedness. The Cypriot economy was certainly not close to a steady-state growth path in 1978. Moreover, with gestation lags of up to four years, it might be argued that a fifteen-year horizon was insufficient for the economy to attain such a path. We are not persuaded, however, that this is the whole story. As Dervis et al. (1984) show with a heavily aggregated, nonlinear model, even a 30-year horizon is not enough to generate a smooth pattern of shadow prices. Second, if a constraint on foreign borrowing is imposed and does bind, then the trajectories of the ARs and the AR1 may become very erratic indeed. In the present model, consumption is not easily controllable, so that the principal burden of year-to-year adjustment must fall on capital accumulation. In these circumstances, the AR1 will tend to be volatile and this volatility will spillover to the ARs of labor and non-traded goods, especially those that are capital intensive. Thus, discussions of what is an ‘acceptable’ level of the DSR are not to be taken lightly. For if a target level is set and successfully pursued, there may be profound differences in the resulting structure of shadow prices compared to that in which no such constraint is imposed. The profitability of a project may then depend critically on its timing. These two causes of instability in the paths of shadow prices illustrate a more general point. In practice, few - if any - countries enjoy steady-state growth and fewer still can get there by borrowing what they need at a parametrically given rate of interest. Therefore, the empirical estimation of shadow prices cannot be undertaken without reference to the macroeconomic setting in which the economy operates. In this paper, we have estimated shadow prices that reflect such macroeconomic considerations. While the intertemporal features of this setting led to a fluctuating pattern of shadow prices, we would argue that this is an inevitable consequence of a move towards more realism. Of course, to obtain these estimates, simplifying assumptions had to be made elsewhere. In addition, detailed project appraisal may require a finer disaggregation for non-traded goods than that employed here. In this spirit, our estimates of shadow prices should be seen not as a list to be pinned to the wall and consulted uncritically when evaluating projects, but rather as guides when discussing the implications of the general evolution of the economy for the choice of socially profitable projects.
Appendix: Valuation of terminal stocks Recall that in the objective function we specified the contribution of terminal capital K, and debt D, as U*[a,K,+cr,D,]/(l+r)T-‘. The parameters cr, and a2 represented the flows of sustainable consumption (our numeraire) generated by a unit of terminal capital stock and debt, respec-
C. Bell and S.
Devarajan,
Shadow
prices
283
tively, and U*( *) was the function that evaluated the utility of this flow. In this appendix, we show how al, a2 and U* were arrived at. A stock of capital in year 7; K,, will produce a flow of income UK T where u is the value added producible from a unit of K. A feasible post-terminal trajectory is one in which all of this is consumed every year forever (assuming no depreciation and no decline in the labor force). This would lead to a valuation of a1 =u. The form of Ii* would then be derived by (for the case where v]= - 1)
-“dr=N,~log!$
(Al) T
Note that we have assumed nc population growth. at a rate 2, the functional form of U* would be
With population
growing
On the one hand, this is a lower bound on the valuation of K, because it may be possible for the economy to save a fraction s of income and enjoy a stream of income that grows, rather than staying constant, at a rate g. In this case, U* would be given by (again with L=O),
U*(uK,)=N,T
uK,(l -s) N T
On the other hand, our original formulation (Al) is an overstatement of the benefits from K, because we assume the average productivity of capital, u, holds at the margin. Since marginal productivity is typically lower, we are overstating the value of K,. However, we could not use the marginal productivity (say au, where a is the share of capital in value added) because this would understate the level of consumption producible from K, in the terminal year. Taking these factors into account, we assume that the two effects offset each other and therefore choose the formulation (Al). As for az, the penalty on consumption of having a unit of terminal debt, we observe that the debt-service payment will be pT, the interest on that debt. Of course, this must be paid by the government which has limited tax instruments at its disposal. Assume the government levies an across-theboard income tax; the foreign exchange saved by the reduction in consumption is then used to make the debt-service payment. Manipulation with a
284
C. Bell and S. Devarajan,
simple, two-sector
Shadow prices
model yields the relationship: v,M+l
%=
-
(&+a,,)M+P
M=
1-P 1 -VN(l-P)-QN’
PT,
where
B=share of tradeables in consumption, vN= share of value added in nontradeables production, and aij = input-output coefficient where T= tradeables and N = non-tradeables. For Cyprus, the following estimates were used: jI=o.4,
vN= 0.6,
aNN = 0.14,
a,,=0.2.
These result in CQ= - 1.9pr. Given the uncertainty surrounding mates, we round off at t12= -2.0~~. Hence, our valuation of terminal stocks is given by
these
esti-
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