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Cost-benefit rules in general disequilibrium
Journal
of Public
Economics
Publishing
COST-BENEFIT
RULES IN GENERAL
Per-Olov Uniuersity
Received
February
DISEQUILIBRIUM
JOHANSSON*
of Lime& S-901 87 Umeb, Sweden
1981, revised version
received June 1981
The consequences of imbalances in markets for goods and factors are the prime issues in many project evaluations. This is so in evaluations of measures like migration, manpower training programmes and plant locations. In the present paper a general disequilibrium model of a small open economy is used to generate evaluation rules to be used when markets do not clear through price adjustments.
1. Introduction The traditional way to generate the rules the government should use when evaluating the efficiency gains from public sector projects is by using a general equilibrium model of the economy.’ This approach is reasonable if one wants to derive rules to be used in a tax-distorted economy since the markets still can be assumed to clear via price adjustments. However, in many empirical evaluations the consequences of imbalances in markets for goods and factors are the prime issues. Apparently, such distortions cannot be handled within the framework of a general equilibrium model. The purpose of this paper is to use a general disequilibrium model to generate project evaluation rules to be used in the presence of market imbalances. Regardless of how one feels about disequilibrium models as sensible explanations of the real world, the fact is that a great deal of the literature on cost-benefit analysis applications implicitly presupposes some form of disequilibrium fixed-price behavior. This is particularly true of project evaluation in developing countries; but, it is more generally true of the manner in which unemployed labor gets treated in cost-benefit analysis. *This paper is a result of research begun in 1979 as a part of a study at the University of Stockholm addressed to the question of what rules the government should use when evaluating energy projects. The study is supported by the Namnden f6r energiproduktionsforskning. I am indebted to Associate Professor Roland Andersson and Professor Peter Bohm for stimulation and assistance given during the period that I was engaged in the study. I would also like to thank two referees for helpful comments. Financial support has been provided by the Swedish Council for Social Science Research. ‘Compare, for example, Boadway (1975, 1978) and Lesourne (1975).
0047-2727/82/000&0000/$02.75
0
1982 North-Holland
122
P-O. Johansson, Costpbenefit rules in general disequilibrium
However, the existing literature is basically partial equilibrium,2 while this paper uses a general (dis-)equilibrium approach which takes account of all of the interactions between markets as well as the functioning of the individual markets. This approach also makes it possible to combine imbalances in several markets - as the practitioner often has to do - and to treat them in a consistent way. Moreover, cost-benefit analysts have always had difficulty in dealing with macroeconomic issues. This difficulty arose because of the lack of a satisfactory link between Keynesian macroeconomics and Walrasian or, for more practical cost-benefit analysts, Marshallian ~ microeconomics. Models of the kind developed in this paper provide such a link and can be used to derive a project evaluation model with a Keynesian textbook multiplier that has an explicitly choice-theoretic basis. The basic model used considers the case of the small open economy which takes the world market prices as given. The model can be viewed as composed of live ‘commodities’: an export good, an import good, a nontraded home good, homogeneous labor, and money.3 There is one ‘representative’ household (to concentrate on efficiency issues), three privately-owned firms and one state-owned firm in the model. The household demands all three goods and money and supplies labor. The privately-owned (profit-maximizing) firms produce export goods, imports and home goods. respectively, with homogeneous labor as the only variable input. Since the purpose is to study the impacts of market imbalances, and these typically will appear in the home goods market - the country is a price-taker in markets for traded goods - and in the labor market, the state-owned firm is assumed to produce home goods with labor as the only variable input. The government budget constraint is assumed to be of the simplest form for the purposes of the study any profits (deficits) in the public sector are disposed of (financed) by lump-sum payments. Of course, it would be interesting to have distorting taxes and subsidies in the model, but in order to concentrate solely on the market imbalance issue this generalization is not included in this paper; moreover, it demands so much space that it must be the subject of a separate paper. In section 2 I present a general equilibrium version of the model and derive shadow pricing rules for a (small) public project when all relative prices are flexible so that markets clear via price adjustments. The relative prices will then be fixed one by one in the following sections. Since the model contains live ‘commodities’, it is however possible to ‘See any standard text book in cost-benefit analysis. Niklasson (1971, 1976) has of disequilibrium model to generate project evaluation rules, and his ‘manual’ is an important contribution (but unfortunately not translated into English). However, diffuse as regards the properties of the model. This also constitutes the background Compare also Lesourne (1975). ‘A similar model is used in Johansson and Lofgren (1980).
used a kind original and Niklasson is to my paper.
P-O. Johansson, Cost-benefit rules in general disequilibrium
123
construct a lot of disequilibrium variations (24), but to keep the problem tractable I will only consider the following cases. First, the wage rate is fixed and in section 3 this is assumed to result in effective excess supply of labor (unemployment) while all other markets still clear through price adjustments. This fundamental disequilibrium situation is called orthodox Keynesian unemployment. Then, in addition, the price of home goods is fixed, while the exchange still is flexible, and this is assumed to generate one or the other of the following three fundamental disequilibrium situations.. In section 4 the wage rate and the price of home goods are assumed to be fixed and such that both the labor market and the market for home goods are in effective excess supply (Keynesian unemployment). In section 5 these two prices are fixed in such a way that the home goods market is in effective excess demand while the labor market still is in effective excess supply (classical unemployment). Then, in section 6, the price of home goods and the wage rate are fixed so that there is effective excess demand in both these markets (repressed inflation). Finally, in section 7 even the exchange rate is fixed, and I have a complete fixed-price model. In section 8 it is briefly indicated what rules are generated if the stateowned firm instead produces exportables or importables, and section 9 contains some concluding remarks. 2. The general equilibrium model The behavior of the representative privately-owned firm under a free market, i.e. when the firm perceives no restriction on trade, is described by the demand and supply functions:
L;:=Ijf(p,w-l),
(+) 1”= lgpi ew ‘),
(+I
yq= y;(& ew - ‘), (+)
(1)
where P,, = the price of home goods, pi = the exogenously given world market price in foreign currency of export goods (i=x) and imports (i=m), e= the exchange rate, w = the wage rate, Id=demand for labor, y”=supply of goods, and the algebraic signs beneath the functions indicate the assumed signs of the partial derivatives.
P-O.Johansson, Cost-benefit rules in general disequilibrium
124
These equations are derived from the maximization of profit, rc,(i = h, x, m), subject to the strictly concave and twice continuously differentiable production function FA$). The demand and supply functions are both notional and effective in this context since it is assumed that the firms perceive no restrictions on trade.4 The state-owned firm produces home goods, GE, with labor as the only variable input. It is generally assumed that the state-owned firm perceives that it can purchase all labor which it needs and sell all the output which it supplies at the existing levels of prices. On the other hand, it is not necessarily true that the state-owned firm maximizes the profit, rcg, i.e. p,, Fb(li)$ w, where F,($) is the production function, since the level of production of the firm may be used as an economic political mean. The behavior of the household is derived from the maximization of the strictly quasi-concave and twice continuously differentiable utility function u = U(Y;t,Y5 AL Ad, 1”) subject
to the budget
(2)
constraint
(3) where yy = the demand for the jth good (i= h, x,m), Md= the stock demand for cash, K a price index, tid = Md rep I, iii,= the initial stock of cash, and 1” = the supply of labor. The resulting behavior functions can be written as:
Ad = Ad(Mo + 71, ph,
p,
e, pme, w, K):
where rc=rcn,+rrh+nn,+n,. The first argument of these demand and supply functions is an income term, while the remaining arguments contain both a substitution and an income effect. In section 3 I return to the properties of the behavior functions and make explicit assumptions concerning the arguments. Observe, however, that the real balance effect is assumed to be so small that it is not explicitly included hereafter. i.e. it is assumed that K= 1 in the initial situation and that 4For the concept Barro and Grossman
of notional (1971).
demand
compared
to effective
demand,
see Clower
(1965) or
P-O. Johansson, Cost-benefit rules in general disequilibrium
125
the small project which will be considered leaves the general price level approximately unchanged.5 Another possible interpretation is that the individual real balance effects ~ in the case of several individuals - sum to zero. If I substitute for the profit argument in the budget constraint of the household I obtain:
md+ e(p, yi - fi, yk + P, y: - Px y:) + Ph(Y;t- Yi - GtJ + w( l;t+ 1;+ If + 1; - I”) = 0,
(5)
excess where md= Md-MO, which states that the value of all (notional) Hence, if the foreign demands sum to zero identically in prices (Walras’ law). exchange market, the market for home goods and the labor market are in equilibrium, then the flow demand for cash or savings, md, will be zero. Thus, there are three independent equilibrium conditions and three flexible relative prices; ph, e and w while the price of money = 1. This completes the description of the model. Now, I will use the model to obtain a monetary measure, dW, of the welfare change from increased production of home goods by the state-owned firm. Therefore, I totally differentiate the utility function, divide through by the marginal utility of money, A, and substitute the first-order utility maximization conditions, to obtain: d W= dull. = C(uj/l.)dy;
+ (u,,,Ji)dMd + (u,/i)dl”
= ph dyi + pXedy: + p, edyi + dMd - wdl”.
(6)
Next, substitute for the profit terms in the differentiated budget constraint of the household (observe that, for example, ph dy:- wdlz =0 from the profitmaximization conditions, and that all markets are in equilibrium). Substitutions into (6) give d W= du/3. = ph dG; - w dl;.
(7)
Eq. (7) can be used as a monetary measure of the welfare change if the changes in each of the variables concerned are sufficiently small for the marginal utility of money to be treated as a constant throughout the ‘For an analysis of the effects on the homogeneity properties of the demand functions that follow from the lack of a real balance effect, compare Howitt and Patinkin (1980a, 1980b). A real balance effect, however small, preserves the lack of money illusion.
126
P-O. Johansson,
Cost-henejit
rules in general disequilibrium
movement.6 In this case an appropriate rule for project evaluation could be to evaluate all outputs and inputs at their domestic market prices. This monetary measure of the welfare change which represents the has been arrived at in the change in profits in the public sector ‘traditional’ way, and is ‘in spirit’ identical to those obtained by Boadway (1975, 1978) Lesourne (1975) and Niklasson (1971, 1976) just to mention a few.7
3. Orthodox Keynesian unemployment When there is orthodox Keynesian unemployment the price vector (pi,, e, W,&?,) is assumed to be such that all markets except the labor market clear in the Walrasian sense, i.e. through price adjustments, while there is effective excess supply of labor - observe that the relative price between labor and money is c0nstant.s Hence, the household perceives a binding constraint upon its sales of labor, I: (A bar above a price indicates that it is exogenously given, while a bar above a quantity (with the exception of M,) indicates a perceived restriction.) Now, substitute l”=lin the utility function and in the budget constraint of the household; compare eqs. (2) and (3) in section 2 above. Maximization of utility then results in the following effective demand functions:
(8) The visible differences between (4) and (8) are that (8) contains the wage income, i.e. both price and quantity, as a term in the income effect argument of all four equations, while the wage rate does not appear as a separate argument. This is so because both the wage rate and the perceived restriction in the labor market are now decision parameters for the household. Also, as a rule T should enter as a separate argument because a change in T normally affects the marginal utilities of goods and money. To simplify the exposition I have eliminated the separate influence of r by assuming that there exists a monotone transformation such that u = H’(y;f, y:, y;,
Md)+ P(1”).
(9)
61f the changes are discrete, A W is a line integral and the value of W is path-dependent, i.e. depends upon the particular path of integration. Compare, for example, Silberberg (1972). ‘Of course, the measures may differ in the sense that different authors include different terms, as they consider different problems, but this does not alter the basic similarities. sThis seems to be the disequilibrium situation considered by Niklasson (1976, pp. 72274) and by Lesourne (1975, pp. 3S36).
The behavior equations of the privately-owned firms are the same as under general equilibrium, i.e. notional and effective demand and supply functions coincide. Regarding the market equilibrium conditions under this regime, it should be observed that although there is unemployment (underemployment) due to the constant relative price between labor and money, it holds that T=r,+& +c+r,=$+ I:+ I$+$; the household does the best possible and sells the amount of labor that is demanded. Thus, equilibrium in the balance of trade and in the market for home goods implies that the flow demand for cash is zero; compare eq. (5) in section 2 above. The monetary measure of the welfare change following from a small increase in the level of production of the state-owned firm can now be derived in an analogous way to eq. (7) in section 2. After some calculations 1 obtain: d W= du/A =
ph
dG; - Wdc + (p/1) d;
(10)
where p/A = W+ u,/i. The only visible difference between this measure and the one obtained in the general equilibrium case is that a term containing the wage rate, W, and the marginal disutility of effort in terms of the numeraire, u,/l, has been added. This is so because the household now perceives a binding constraint upon its sales of labor.’ In itself (10) could be used as a basis for shadow pricing rules under orthodox Keynesian unemployment. That is: (a) evaluate all outputs and inputs at domestic market prices, and (b) add a term representing the net change in the level of employment (dT) evaluated by the difference between the marginal disutility of effort and the wage rate. Eq. (10) is actually a straightforward application of Harberger’s general formula for applied welfare economics [Harberger (1971)]. Consider the last term. W is the demand price for labor while u,/1 is the supply price. Therefore, ti+u,/l is the marginal distortion on the labor market. Owing to lack of data many Swedish cost-benefit practitioners like Dahlberg (1972) and Johansson (1978) assume that the supply price of otherwise unemployed is equal to zero. This means that the supply price is put equal to the real opportunity cost in the absence of non-market production. In this case eq. (10) reduces to eq. (3.19) in Lesourne’s (1975) famous book in cost-benefit analysis. Finally, let us compare the partial equilibrium rules with those that can be obtained from eq. (10). The partial equilibrium view is to treat all labor ‘The household maximizes utility constraint F=r Thus, p is the Lagrange eqs. (20) below.
subject to the budget constraint and the additional multiplier associated with the constraint (T-I’). Compare
128
P-O. Johansson,
Cost-benefit
rules in general disequilibrium
employed in a marginal project as coming from the unemployed, which means that 1: is put equal to l To check this rule I can use the two independent equilibrium conditions, e.g. the market for home goods and the flow demand for cash, to derive the signs of 6p J&T~ and Se/6Gi. It turns out that pi,, the price of home goods, will decrease, while the sign of the effect on e, the exchange rate, is ambiguous. lo This means that an increased production by the state-owned firm will displace the private supply of home goods, while the sign of the effect on production and employment in the export sector and in the import sector is ambiguous. Thus, the sign of the effect on total employment is ambiguous. This means that even if the ‘individual’ supply price of unemployed resources is assumed to be zero and equal to the ‘individual’ real opportunity cost (in the absence of non-market production), the aggregate real opportunity cost might very well exceed the market wage; this is the case if d/CO in eq. (10) above. This result is rather different from the partial equilibrium one. The same result is possible to obtain if the assumption of a representative household is abandoned. If there are several households in the model there will be a redistribution of employment and unemployment between different households. However, when evaluating the welfare effects of policy changes in which several persons are affected, things are considerably complicated. The disputes over this issue are well known in the literature, and I have nothing to add to it. The reader is referred to Arrow (19.51) Boadway (1974) and Smith and Stephen (1975).
4. Keynesian unemployment Keynesian unemployment is defined as a situation where the priceewage constellation is such that there is effective excess supply in both the market for home goods and in the labor market. The market for foreign exchange is assumed to clear in the Walrasian sense due to a flexible exchange rate, while both the price of home goods and the nominal wage rate are fixed. Consistent trading is created through quantity adjustments in the market for home goods and in the labor market, and through a flexible exchange rate. If the privately-owned firm in the home goods sector perceives that demand is deficient, it will maximize profit by solving the program
(11) subject
to
Fh(c3= Y, “‘To sign the determinant of the equation substitutes. Compare Johansson (1980).
(12) system
I have to assume
normal
goods,
and net
P-O. Johansson, Cost-benefit rules in general disequilibrium
and the resulting
effective demand
129
for labor is given by
1;= FL l(Y).
(13)
The behavior equations of the household and of the firm producing exportables and importables, respectively, are the same as under orthodox Keynesian unemployment. The monetary welfare measure of an increased production of home goods by the state-owned firm contains, in addition to the terms obtained under orthodox Keynesian unemployment, the net change in the profit of the privately-owned firm producing home goods since MR > MC: d W= du/A = Pi, dGs, - Wdl; + Pi, dys, - % dl; + (p/A) dl: Note that P, dyi- Wdl;f has a Harberger interpretation. P, dyi- Wdlz is the marginal benefit less the marginal cost of an additional unit of labor and so is like a marginal distortion. To further examine this expression I have to solve, for example, the following equation system (I have substituted for rr in the effective demand functions of the household):
-
F,(l;t)- Gs,= 0; md(M,+ j&G; + Ph F,(l;)
+ Cpi eyf(p, eti I), & j, e, p, e) = 0. Observe that the labor market is in ‘equilibrium’ in the sense discussed section 3 above. Thus, by virtue of Walras’ law it follows that the balance trade is in equilibrium if the two markets in (14) are in equilibrium. Simple calculations reveal that Sl$SG; = - 1/F6;
Fb=6F,(1~)/61~;6e/6Gs,=0,
(14) in of
(15)
so employment
in the privately-owned part of the home goods sector will decrease, while the exchange rate, and as a consequence also employment in firms producing traded goods, wiil be left unchanged. Thus, the impact on monetary welfare of a small change in CL is: d W= du/A = P,, dGs, - Wdl; + &, dy; - Wdl; +(/L/A) dT = &, dGs, - Wdl; + &,(6F~c%;)(d@dGs,)
dGs, - Wdl; + (p/A) dT
=j& dG; - Wdl; - & dGs, - Wdl; + (p/A) di = - w(dl; + dl:) +(/~/l/1) di= [ - W+ (p/i)] dT= @,/A) dl
(16)
130
P-O. Johunsson,
Cost-benefit
rules in general disequilibrium
That is to say, since there is effective excess supply of home goods there will only be a redistribution of the production of such goods between the privately-owned and the state-owned firm. This result is a counterpart to the famous balanced-budget theorem, i.e. there is a one-to-one relationship between the decrease in the level of production of the privately-owned firm and the increase in the level of production of the state-owned firm. This is so hccause the initial change is on the supply side while the problem is that demand is deficient. The conventional supplyydemand figure in which the price of the good is fixed above the market clearing level gives an idea of this, So, the change in W is due to the difference, if any, in the marginal productivities between the two firms: dlf = - l/Fh and dli = l/F;. The effect on welfare is in a sense similar to the one obtained under full employment. That is, if the supply price of labor is positive (-u,>O), then welfare will increase only if the state-owned firm produces at a point where the marginal productivity, FL, exceeds the marginal productivity, Fh, of the privatelyowned firm, so that total employment decreases. In the general equilibrium case welfare will increase if the state-owned firm produces at a point where the value of the marginal product exceeds the wage, i.e. where pi, Fi> w. But this implies that Fb>Fh since the privately-owned firms produce where p,,Fh = w. To sum up, an appropriate rule for project evaluation under Keynesian unemployment could be: use the marginal productivities to calculate how a redistribution of the production of home goods between the private and the public sector affects employment. Next, multiply by ihe negative of the supply price of unemployed labor. This simple calculation gives the net change in the welfare. 5. Classical unemployment Classical unemployment is present when there is effective excess demand in the market for home goods, and effective excess supply in the labor market. In a situation like this, the firms perceive no restrictions on trade, while the household perceives restrictions on trade in both the market for home goods and in the labor market: Y;t=Yh
and
l’=l
(17)
The labor market is in excess supply while the market for home goods is in excess demand. It can be shown that the resulting effective demand functions can be written (after substitutions for n): YP= yWo
+
Px4 + P,
eyfP, e,&A
Md = Md(A, + pxey",+ p, ey",, p, e, pme).
i=x,m,
(18)
P-O. Johansson,
CostFbenej?t
rules in yeneral
disequilibrium
131
Since the home goods market is in excess demand, the household buys all that is produced. This and ‘additive separability’ in the sense used in section 3 above explains the absence of (J&Y:+ &, Gi) in the income argument and the absence of j& as a separate argument. The behavior functions of the privately-owned firms are the same as in sections 2 and 3 since firms perceive that they can buy and sell ‘unlimited’ quantities at the ruling prices [yE(p, W- l), yf(pi eF ‘)I. Since I retain the assumption of a flexible exchange rate and fixed prices of both labor and home goods, short-run equilibrium is now determined from the balance of payments condition:
Pxey”,k e*- ‘I+ P, &Xi% ew
‘)
- Cpi eyf (it?, + Qi eyf, & e, p, e) = 0.
(19)
However, it is obvious that 6e/6Gi = 0, i.e. a change in the level of production of the state-owned firm leaves the exchange rate constant. To obtain a welfare measure of the change in GL I have to replace both the price of home goods and the wage rate by shadow prices, say (Pi, +a/A) and (W-p/A), respectively. Too see this let the household maximize the utility function (2) subject to the budget constraint (3) and the two additional binding constraints yt = y,, and 1”= 1 The relevant first-order conditions are
Ui =
Apie,
i=x,m,
(20)
where a,1 and ,u are Lagrange multipliers. Thus, a/A may be understood to be the difference between the willingness to pay for an additonal unit of home goods and the market price; compare the conventional supply-demand figure in which the price of the good is fixed below the market clearing level. Regarding the interpretation of the imbalance in the labor market, the reader is referred to section 3 above. Next, after some substitutions I obtain: d W= du/;l =&
+ U/A)(dG”, + dy”,) - w(dl; + did,)
and an appropriate rule for project evaluation under classical unemployment could be: (a) use the marginal willingness to pay to evaluate rationed goods,
132
P-O. Johansson,
Cost-bent@
rules in general disequilibrium
and (b) use the supply price [ - u,/jL = W--/A] to obtain the cost of employing an otherwise unemployed (underemployed). In terms of compensating variations, (a/A) and (p/A) may be understood to be lump sums that leave an individual on the initial level of satisfaction after a change - evaluated at market prices - in the consumption of rationed goods and a reduction of underemployment, respectively. The labor market appears in eq. (21) in a way which is consistent with the partial equilibrium view discussed in section 3 above. That is, all labor employed in the marginal project is treated as coming from the unemployed. To point out another similarity, assume that the short-run equilibrium on the home goods market is close to the unconstrained one so that x z 0. Then, eq. (21) generates a rule which coincides with the partial equilibrium rule that the production of a marginal project can be treated as a net addition to the quantity traded so that the prevailing price measures the benefit. In other words, if there is classical unemployment it is possible to use the general disequilibrium model to generate a cost-benefit rule which is consistent with the partial equilibrium view. Finally, observe that there will be no income-induced multiplier effects under classical unemployment. The reason for this is that the levels of production of privately-owned firms are governed by the relative prices, and not by effective demand.’ 1 Moreover, since the household is rationed in the home goods market, it will use the new income (drcg+Wd$) to buy all that is produced of the good by the state-owned firm, and this implies in fact that there will be no multiplier effects even if I add a goods market where there is effective excess supply (‘Keynesian unemployment’). In a technical sense this is due to the fact that GE and 1: do not appear in the income term in the effective demand functions - compare eqs. (18) -~ since (pi, J$, + Pi, GT,- p,, J+,) and (~ 1: - lt - 1:- L”,+ 1) always arc equal to zero.
6. Repressed inflation Repressed inflation is present when the price vector is such that there is effective excess demand in both the market for home goods and in the labor market. This means that firms are constrained in the labor market and the household is constrained in the market for home goods. Consistent trading is created through quantity adjustments in the market for home goods and in the labor market, and through a flexible exchange rate. There are, however, four firms and I have to specify how the insufficient supply of labor is allocated between the firms. It is easily understood that the properties of this regime may depend on the kind of rationing scheme that is “Also observe that all relative prices are left unchanged. The exchange rate may change if the assumption of a ‘sufficient degree of additive separability’ is relaxed.
however
P-O. Johansson, Cost-benefit rules in general disequilibrium
133
postulated, and whether agents have the possibility to manipulate the scheme by emitting signals that are inconsistent with their budget restrictions. The behavior of the household under this regime is determined from the decision problem: maximize the utility function (2) subject to the budget constraint (3) and the additional binding constraint in the home goods market (yi = j&J. The resulting effective behavior functions can be written:
Md=Md(MO+ 71--ijhjh,&e,p,e,KQ, 1”= I”(R, +
7c-
jh y,,p, e, p, e, W).
(24
To derive the behavior of the firms, I have to specify a rationing scheme. Under this regime the total effective demand for labor [Id= l~+Z&‘& W-l) + Ii@, eW ‘) + lk&, eti- ‘)I exceeds the total effective supply of labor and the firms perceive a constraint on their demands in the labor market. A simple way to aliocate the insufficient supply of labor is to assume that the stateowned firm, whose level of production is exogenously given, gets all the labor which it needs, while each privately-owned firm gets a given share, bj, of the remaining supply of labor. This simple scheme, which will be used to illustrate the shadow pricing rules, yields y; = y$) = F,Q = F,(b, 0,
j=h,x,m,
(23)
where J= the perceived restriction on the amount of labor available for production of the jth good, ctj=cbjT=T= I”- Ii, i.e. the private sector uses up all the labor that is available. The monetary measure of the welfare change is d W= du/A = (Pi, + CC/~)(dGs, + dy;) - ti(dZ; + d&) +
c(p, edy: - WdS;).
(24)
All privately-owned firms now appear in the formula. This is so because they are rationed in the labor market and hence produce where the value of the marginal product exceeds the wage rate. Thus, one possible formulation of the project evaluation rules could be: (a) evaluate the net change in the level of production of rationed home goods at the marginal willingness to pay; (b) evaluate the net change in labor used in this sector at the market wage; and (c) evaluate changes in other sectors with the difference between the value of the marginal product and the market wage.
134
P-O. Johansson,
Costpbeneft
rules in yeneral disequilibrium
Finally, given the rationing scheme used it can be shown that SqSGiO. (These results were obtained by solving the equation system r+ li- 1”=O, md = 0.) So, employment and production will decrease in all privately-owned firms. This also explains that the domestic currency will depreciate; the domestic production of traded goods is reduced since Sl
not be equilibrium
(25) Thus, there will be an accumulation or a decumulation of cash. The assumption of a fixed exchange rate adds, however, nothing to the shadow pricing results obtained in the previous sections. Of course, an accumulation or a decumulation of cash generates long-run effects, but these are not discussed in this paper. Note, however, that the rules which can be derived support Boadway’s (1978) view that the Little-Mirrlees (1968) and the Dasgupta-Sen-Marglin (1972) measures do not correspond to the net social gain of a project under a fixed exchange rate regime. The Little-Mirrlees approach takes as a measure of welfare change the net contribution of the project to foreign exchange earnings. It follows straightforwardly from eq. (27) in section 8 below that such a measure does not correspond to the change in welfare. Consider also eqs. (16) and (21) in sections 4 and 5, respectively. In these cases increased public production of home goods leaves the exchange rate unchanged even under flexible exchange rates. This means that the net contribution to foreign exchange earnings is zero if the exchange rate is fixed. Nevertheless, the project affects welfare. According to the Dasgupta-Sen-Marglin approach non-traded goods are evaluated at domestic willingness to pay while traded goods are evaluated by a shadow exchange rate. This shadow exchange rate takes account of direct project purchases of traded goods. Therefore, this approach is partial equilibrium and will, as a rule, not generate the same measures as a general (dis-) equilibrium approach. This is most easily seen by introducing a tariff on imports, but I will not tire the reader by going through the calculations; the reader is referred to Boadway (1978) for a discussion. 8. Government production of traded goods So far I have
assumed
that
the state-owned
firm
produces
non-traded
P-O. Johansson, Cost-benefit rules in general disequilibrium
135
home goods. It is of course also possible to use the model to examine the case when the firm in question supplies (demands) exportables and/or importables. However, in the present context, where interest is focussed on market imbalances, the only interesting case seems to appear when there is some kind of ‘distortion’ in the foreign market, i.e. a tariff, a quota or some other distortion which makes the domestic and world prices differ. But, as far as I can see, this case adds little, if anything, to the results obtained above. There is, however, an exceptional case. Under Keynesian unemployment and fixed exchange rates, increased government production of traded goods will generate real income-induced multiplier effects in the home goods sector. Formally, by differentiating the equilibrium condition for the home goods market in eqs. (14) in section 4, remembering that the exchange rate, e, now is fixed, one obtains: sl~sGl=piey;f,/(l--p,y~,)Fh,
i=x,m,
(26)
so that
where P,, y& is the marginal propensity to consume home goods Fh = 6ysJ@. The following monetary welfare change measure is obtained:
(2 0), and
dW=du/l=j&edG;-(w-p/L)dl$+ji,dys,-(w-p/A)dl;
= pi FdG;/( 1 - P,, y;,) + (uJn) df.
(27)
So, the domestic market value of the direct change in output has to be multiplied by the ‘textbook’ Keynesian multiplier. At the same time, there will appear an employment multiplier. Hence, to obtain the welfare cost, the direct and the income-induced changes in employment should be multiplied by the supply price of otherwise unemployed. had difficulty in dealing with Cost-benefit analysts have always macroecomic issues. This difficulty arose because of the lack of a satisfactory link between microeconomics and Keynesian macroeconomics. To illustrate, empirical studies often include real multiplier effects.12 But these studies are based on the traditional microeconimic model which does not generate real multiplier effects. Eq. (27) shows that models of the kind used here provide “See
Bohm (1972), Dahlberg
(1972) and Johansson
(1978)
136
P-O. Johansson, Cost-benefit rules in general disequilibrium
‘the’ missing link between microeconomics and Keynesian macroeconomics. Eq. (27), moreover, provides a rationale for the mentioned practice to include real multiplier effects. Observe, however, that if the exchange rate is flexible, then the sign of the effect on production and employment in the home goods sector will be ambiguous. This is so because the exchange rate appreciates when the stateowned firm increases its production of traded goods, and this displaces private domestic supply of such goods and lowers the price of traded goods relative to the fixed price of non-traded home goods. 9. Concluding remarks In order to concentrate solely on the market disequilibrium issue I have ignored several other problems raised in both manuals and elsewhere: the small open economy assumption, the qualitative distinction between traded and non-traded goods, imbalances or other distortions on the capital market, and the shadow price of labor in a dual economy. Another problem which has been left out is how the cost-benefit practitioner should decide which regime is being experienced; the paper clearly shows that the kind of regime prevailing is essential for the kind of benefits and costs to be included. I have no simple recommendations to make, but in many instances already an examination of the sectors directly affected by a project should give valuable information. For example, in countries like Great Britain and Sweden the textile industry, the steel industry and the ship yards seem to be working under conditions which reminds of Keynesian unemployment; there is an ‘excess production’ and a high level of unemployment in these sectors. However, regardless of these practical problems I believe that the cost-benefit analysts could arrive at theoretically more well-founded and also empirically more realistic studies by making use of disequilibrium models. References Arrow, Kenneth J., 1951, Social choice and individual values (John Wiley & Sons, New York). Barro. Robert J. and Herschel I. Grossman, 1971. A general disequilibrium model of income and employment, American Economic Review 61, 82-63. Boadway, Robin W., 1974, The welfare foundations of cost-benelit analysis, Economic Journal 35, 926939. Boadway, Robin W., 1974, The welfare foundations of cost-benefit analysis, Economic Journal Studies XLII, 361-373. Boadwav. Robin W., 1978, A note on the treatment of foreign exchange in project evaluation, Economica 45, 391-399. Bohm, Peter, 1972, A cost-benefit analysis of mining at Stekenjokk (SOU, Stockholm) [in Swedish]. Clower, Robert, 1965, The Keynesian counterrevolution: A theoretical approach, in: Frank H. Hahn and Frank P. R. Brechling, eds., The theory of interest rates (Macmillan, London) 103125.
P-O. Johansson,
Cost-benefit
rules in general disequilibrium
137
Dahlberg, Ake, 1972, Manpower training programmes ~ consequences for the individual and society, Umed Economic Studies 3 [in Swedish]. Dasgupta, Partha, Amartya Sen and Stephen A. Marglin, 1972, Guidelines for project evaluation (United Nations Industrial Organisation, Vienna). Harberger, Arnold C., 1971, Three basic postulates for applied welfare economics: An interpretive essay, Journal of Economic Literature 9, 785-797. Howitt, Peter W. and Don Patinkin, 1980a, Utility function transformations and money illusion: Comment, American Economic Review 70, 819-822. Howitt, Peter W. and Don Patinkin, 1980b, Utility function transformations and money illusion: A further comment, American Economic Review 70, 826828. Johansson, Per-Olov, 1978, Economics of employment a study of the effects of a plant location, Umea Economic Studies 53 [in Swedish]. Johansson, Per-Olov, 1980, On potential employment effects of firm oriented labor market measures in a small open economy, Umel Economic Studies 74. Johansson, Per-Olov and Karl-Gustaf Lofgren, 1980, The effects of tariffs and real wages on employment in a Barro-Grossman model of an open economy, Scandinavian Journal of Economics 82, 1677183. Lesourne, Jacques, 1975, Cost-benefit analysis and economic theory (North-Holland, Amsterdam). Little, Ian M.D. and James A. Mirrlees, 1968, Manual of industrial project analysis in developing countries, vol 1 (Organisation for Economic Cooperation and Development, Paris). Niklasson, Harald, 1971,‘An examination of the theoretical foundations for the use of costbenefit analysis in evaluations of e.g. labor market policy measures; Lund,’ mimeo [[in Swedish].\ Niklasson, Harald, 1976, Costtbenefit analysis in theory and practice,l Samhallsvetenskapliga Studier fran Vaxjo,iSerie B, no. 1, Vaxjij [in Swedish], Silberberg, Eugene, 1972, Duality and the many consumer’s surpluses, American Economic Review 62, 942-952. Smith, Ben and Frank H. Stephen, 1975, Cost-benefit analysis and compensation criteria: A note, Economic Journal 85,902-905.
of Public
Economics
Publishing
COST-BENEFIT
RULES IN GENERAL
Per-Olov Uniuersity
Received
February
DISEQUILIBRIUM
JOHANSSON*
of Lime& S-901 87 Umeb, Sweden
1981, revised version
received June 1981
The consequences of imbalances in markets for goods and factors are the prime issues in many project evaluations. This is so in evaluations of measures like migration, manpower training programmes and plant locations. In the present paper a general disequilibrium model of a small open economy is used to generate evaluation rules to be used when markets do not clear through price adjustments.
1. Introduction The traditional way to generate the rules the government should use when evaluating the efficiency gains from public sector projects is by using a general equilibrium model of the economy.’ This approach is reasonable if one wants to derive rules to be used in a tax-distorted economy since the markets still can be assumed to clear via price adjustments. However, in many empirical evaluations the consequences of imbalances in markets for goods and factors are the prime issues. Apparently, such distortions cannot be handled within the framework of a general equilibrium model. The purpose of this paper is to use a general disequilibrium model to generate project evaluation rules to be used in the presence of market imbalances. Regardless of how one feels about disequilibrium models as sensible explanations of the real world, the fact is that a great deal of the literature on cost-benefit analysis applications implicitly presupposes some form of disequilibrium fixed-price behavior. This is particularly true of project evaluation in developing countries; but, it is more generally true of the manner in which unemployed labor gets treated in cost-benefit analysis. *This paper is a result of research begun in 1979 as a part of a study at the University of Stockholm addressed to the question of what rules the government should use when evaluating energy projects. The study is supported by the Namnden f6r energiproduktionsforskning. I am indebted to Associate Professor Roland Andersson and Professor Peter Bohm for stimulation and assistance given during the period that I was engaged in the study. I would also like to thank two referees for helpful comments. Financial support has been provided by the Swedish Council for Social Science Research. ‘Compare, for example, Boadway (1975, 1978) and Lesourne (1975).
0047-2727/82/000&0000/$02.75
0
1982 North-Holland
122
P-O. Johansson, Costpbenefit rules in general disequilibrium
However, the existing literature is basically partial equilibrium,2 while this paper uses a general (dis-)equilibrium approach which takes account of all of the interactions between markets as well as the functioning of the individual markets. This approach also makes it possible to combine imbalances in several markets - as the practitioner often has to do - and to treat them in a consistent way. Moreover, cost-benefit analysts have always had difficulty in dealing with macroeconomic issues. This difficulty arose because of the lack of a satisfactory link between Keynesian macroeconomics and Walrasian or, for more practical cost-benefit analysts, Marshallian ~ microeconomics. Models of the kind developed in this paper provide such a link and can be used to derive a project evaluation model with a Keynesian textbook multiplier that has an explicitly choice-theoretic basis. The basic model used considers the case of the small open economy which takes the world market prices as given. The model can be viewed as composed of live ‘commodities’: an export good, an import good, a nontraded home good, homogeneous labor, and money.3 There is one ‘representative’ household (to concentrate on efficiency issues), three privately-owned firms and one state-owned firm in the model. The household demands all three goods and money and supplies labor. The privately-owned (profit-maximizing) firms produce export goods, imports and home goods. respectively, with homogeneous labor as the only variable input. Since the purpose is to study the impacts of market imbalances, and these typically will appear in the home goods market - the country is a price-taker in markets for traded goods - and in the labor market, the state-owned firm is assumed to produce home goods with labor as the only variable input. The government budget constraint is assumed to be of the simplest form for the purposes of the study any profits (deficits) in the public sector are disposed of (financed) by lump-sum payments. Of course, it would be interesting to have distorting taxes and subsidies in the model, but in order to concentrate solely on the market imbalance issue this generalization is not included in this paper; moreover, it demands so much space that it must be the subject of a separate paper. In section 2 I present a general equilibrium version of the model and derive shadow pricing rules for a (small) public project when all relative prices are flexible so that markets clear via price adjustments. The relative prices will then be fixed one by one in the following sections. Since the model contains live ‘commodities’, it is however possible to ‘See any standard text book in cost-benefit analysis. Niklasson (1971, 1976) has of disequilibrium model to generate project evaluation rules, and his ‘manual’ is an important contribution (but unfortunately not translated into English). However, diffuse as regards the properties of the model. This also constitutes the background Compare also Lesourne (1975). ‘A similar model is used in Johansson and Lofgren (1980).
used a kind original and Niklasson is to my paper.
P-O. Johansson, Cost-benefit rules in general disequilibrium
123
construct a lot of disequilibrium variations (24), but to keep the problem tractable I will only consider the following cases. First, the wage rate is fixed and in section 3 this is assumed to result in effective excess supply of labor (unemployment) while all other markets still clear through price adjustments. This fundamental disequilibrium situation is called orthodox Keynesian unemployment. Then, in addition, the price of home goods is fixed, while the exchange still is flexible, and this is assumed to generate one or the other of the following three fundamental disequilibrium situations.. In section 4 the wage rate and the price of home goods are assumed to be fixed and such that both the labor market and the market for home goods are in effective excess supply (Keynesian unemployment). In section 5 these two prices are fixed in such a way that the home goods market is in effective excess demand while the labor market still is in effective excess supply (classical unemployment). Then, in section 6, the price of home goods and the wage rate are fixed so that there is effective excess demand in both these markets (repressed inflation). Finally, in section 7 even the exchange rate is fixed, and I have a complete fixed-price model. In section 8 it is briefly indicated what rules are generated if the stateowned firm instead produces exportables or importables, and section 9 contains some concluding remarks. 2. The general equilibrium model The behavior of the representative privately-owned firm under a free market, i.e. when the firm perceives no restriction on trade, is described by the demand and supply functions:
L;:=Ijf(p,w-l),
(+) 1”= lgpi ew ‘),
(+I
yq= y;(& ew - ‘), (+)
(1)
where P,, = the price of home goods, pi = the exogenously given world market price in foreign currency of export goods (i=x) and imports (i=m), e= the exchange rate, w = the wage rate, Id=demand for labor, y”=supply of goods, and the algebraic signs beneath the functions indicate the assumed signs of the partial derivatives.
P-O.Johansson, Cost-benefit rules in general disequilibrium
124
These equations are derived from the maximization of profit, rc,(i = h, x, m), subject to the strictly concave and twice continuously differentiable production function FA$). The demand and supply functions are both notional and effective in this context since it is assumed that the firms perceive no restrictions on trade.4 The state-owned firm produces home goods, GE, with labor as the only variable input. It is generally assumed that the state-owned firm perceives that it can purchase all labor which it needs and sell all the output which it supplies at the existing levels of prices. On the other hand, it is not necessarily true that the state-owned firm maximizes the profit, rcg, i.e. p,, Fb(li)$ w, where F,($) is the production function, since the level of production of the firm may be used as an economic political mean. The behavior of the household is derived from the maximization of the strictly quasi-concave and twice continuously differentiable utility function u = U(Y;t,Y5 AL Ad, 1”) subject
to the budget
(2)
constraint
(3) where yy = the demand for the jth good (i= h, x,m), Md= the stock demand for cash, K a price index, tid = Md rep I, iii,= the initial stock of cash, and 1” = the supply of labor. The resulting behavior functions can be written as:
Ad = Ad(Mo + 71, ph,
p,
e, pme, w, K):
where rc=rcn,+rrh+nn,+n,. The first argument of these demand and supply functions is an income term, while the remaining arguments contain both a substitution and an income effect. In section 3 I return to the properties of the behavior functions and make explicit assumptions concerning the arguments. Observe, however, that the real balance effect is assumed to be so small that it is not explicitly included hereafter. i.e. it is assumed that K= 1 in the initial situation and that 4For the concept Barro and Grossman
of notional (1971).
demand
compared
to effective
demand,
see Clower
(1965) or
P-O. Johansson, Cost-benefit rules in general disequilibrium
125
the small project which will be considered leaves the general price level approximately unchanged.5 Another possible interpretation is that the individual real balance effects ~ in the case of several individuals - sum to zero. If I substitute for the profit argument in the budget constraint of the household I obtain:
md+ e(p, yi - fi, yk + P, y: - Px y:) + Ph(Y;t- Yi - GtJ + w( l;t+ 1;+ If + 1; - I”) = 0,
(5)
excess where md= Md-MO, which states that the value of all (notional) Hence, if the foreign demands sum to zero identically in prices (Walras’ law). exchange market, the market for home goods and the labor market are in equilibrium, then the flow demand for cash or savings, md, will be zero. Thus, there are three independent equilibrium conditions and three flexible relative prices; ph, e and w while the price of money = 1. This completes the description of the model. Now, I will use the model to obtain a monetary measure, dW, of the welfare change from increased production of home goods by the state-owned firm. Therefore, I totally differentiate the utility function, divide through by the marginal utility of money, A, and substitute the first-order utility maximization conditions, to obtain: d W= dull. = C(uj/l.)dy;
+ (u,,,Ji)dMd + (u,/i)dl”
= ph dyi + pXedy: + p, edyi + dMd - wdl”.
(6)
Next, substitute for the profit terms in the differentiated budget constraint of the household (observe that, for example, ph dy:- wdlz =0 from the profitmaximization conditions, and that all markets are in equilibrium). Substitutions into (6) give d W= du/3. = ph dG; - w dl;.
(7)
Eq. (7) can be used as a monetary measure of the welfare change if the changes in each of the variables concerned are sufficiently small for the marginal utility of money to be treated as a constant throughout the ‘For an analysis of the effects on the homogeneity properties of the demand functions that follow from the lack of a real balance effect, compare Howitt and Patinkin (1980a, 1980b). A real balance effect, however small, preserves the lack of money illusion.
126
P-O. Johansson,
Cost-henejit
rules in general disequilibrium
movement.6 In this case an appropriate rule for project evaluation could be to evaluate all outputs and inputs at their domestic market prices. This monetary measure of the welfare change which represents the has been arrived at in the change in profits in the public sector ‘traditional’ way, and is ‘in spirit’ identical to those obtained by Boadway (1975, 1978) Lesourne (1975) and Niklasson (1971, 1976) just to mention a few.7
3. Orthodox Keynesian unemployment When there is orthodox Keynesian unemployment the price vector (pi,, e, W,&?,) is assumed to be such that all markets except the labor market clear in the Walrasian sense, i.e. through price adjustments, while there is effective excess supply of labor - observe that the relative price between labor and money is c0nstant.s Hence, the household perceives a binding constraint upon its sales of labor, I: (A bar above a price indicates that it is exogenously given, while a bar above a quantity (with the exception of M,) indicates a perceived restriction.) Now, substitute l”=lin the utility function and in the budget constraint of the household; compare eqs. (2) and (3) in section 2 above. Maximization of utility then results in the following effective demand functions:
(8) The visible differences between (4) and (8) are that (8) contains the wage income, i.e. both price and quantity, as a term in the income effect argument of all four equations, while the wage rate does not appear as a separate argument. This is so because both the wage rate and the perceived restriction in the labor market are now decision parameters for the household. Also, as a rule T should enter as a separate argument because a change in T normally affects the marginal utilities of goods and money. To simplify the exposition I have eliminated the separate influence of r by assuming that there exists a monotone transformation such that u = H’(y;f, y:, y;,
Md)+ P(1”).
(9)
61f the changes are discrete, A W is a line integral and the value of W is path-dependent, i.e. depends upon the particular path of integration. Compare, for example, Silberberg (1972). ‘Of course, the measures may differ in the sense that different authors include different terms, as they consider different problems, but this does not alter the basic similarities. sThis seems to be the disequilibrium situation considered by Niklasson (1976, pp. 72274) and by Lesourne (1975, pp. 3S36).
The behavior equations of the privately-owned firms are the same as under general equilibrium, i.e. notional and effective demand and supply functions coincide. Regarding the market equilibrium conditions under this regime, it should be observed that although there is unemployment (underemployment) due to the constant relative price between labor and money, it holds that T=r,+& +c+r,=$+ I:+ I$+$; the household does the best possible and sells the amount of labor that is demanded. Thus, equilibrium in the balance of trade and in the market for home goods implies that the flow demand for cash is zero; compare eq. (5) in section 2 above. The monetary measure of the welfare change following from a small increase in the level of production of the state-owned firm can now be derived in an analogous way to eq. (7) in section 2. After some calculations 1 obtain: d W= du/A =
ph
dG; - Wdc + (p/1) d;
(10)
where p/A = W+ u,/i. The only visible difference between this measure and the one obtained in the general equilibrium case is that a term containing the wage rate, W, and the marginal disutility of effort in terms of the numeraire, u,/l, has been added. This is so because the household now perceives a binding constraint upon its sales of labor.’ In itself (10) could be used as a basis for shadow pricing rules under orthodox Keynesian unemployment. That is: (a) evaluate all outputs and inputs at domestic market prices, and (b) add a term representing the net change in the level of employment (dT) evaluated by the difference between the marginal disutility of effort and the wage rate. Eq. (10) is actually a straightforward application of Harberger’s general formula for applied welfare economics [Harberger (1971)]. Consider the last term. W is the demand price for labor while u,/1 is the supply price. Therefore, ti+u,/l is the marginal distortion on the labor market. Owing to lack of data many Swedish cost-benefit practitioners like Dahlberg (1972) and Johansson (1978) assume that the supply price of otherwise unemployed is equal to zero. This means that the supply price is put equal to the real opportunity cost in the absence of non-market production. In this case eq. (10) reduces to eq. (3.19) in Lesourne’s (1975) famous book in cost-benefit analysis. Finally, let us compare the partial equilibrium rules with those that can be obtained from eq. (10). The partial equilibrium view is to treat all labor ‘The household maximizes utility constraint F=r Thus, p is the Lagrange eqs. (20) below.
subject to the budget constraint and the additional multiplier associated with the constraint (T-I’). Compare
128
P-O. Johansson,
Cost-benefit
rules in general disequilibrium
employed in a marginal project as coming from the unemployed, which means that 1: is put equal to l To check this rule I can use the two independent equilibrium conditions, e.g. the market for home goods and the flow demand for cash, to derive the signs of 6p J&T~ and Se/6Gi. It turns out that pi,, the price of home goods, will decrease, while the sign of the effect on e, the exchange rate, is ambiguous. lo This means that an increased production by the state-owned firm will displace the private supply of home goods, while the sign of the effect on production and employment in the export sector and in the import sector is ambiguous. Thus, the sign of the effect on total employment is ambiguous. This means that even if the ‘individual’ supply price of unemployed resources is assumed to be zero and equal to the ‘individual’ real opportunity cost (in the absence of non-market production), the aggregate real opportunity cost might very well exceed the market wage; this is the case if d/CO in eq. (10) above. This result is rather different from the partial equilibrium one. The same result is possible to obtain if the assumption of a representative household is abandoned. If there are several households in the model there will be a redistribution of employment and unemployment between different households. However, when evaluating the welfare effects of policy changes in which several persons are affected, things are considerably complicated. The disputes over this issue are well known in the literature, and I have nothing to add to it. The reader is referred to Arrow (19.51) Boadway (1974) and Smith and Stephen (1975).
4. Keynesian unemployment Keynesian unemployment is defined as a situation where the priceewage constellation is such that there is effective excess supply in both the market for home goods and in the labor market. The market for foreign exchange is assumed to clear in the Walrasian sense due to a flexible exchange rate, while both the price of home goods and the nominal wage rate are fixed. Consistent trading is created through quantity adjustments in the market for home goods and in the labor market, and through a flexible exchange rate. If the privately-owned firm in the home goods sector perceives that demand is deficient, it will maximize profit by solving the program
(11) subject
to
Fh(c3= Y, “‘To sign the determinant of the equation substitutes. Compare Johansson (1980).
(12) system
I have to assume
normal
goods,
and net
P-O. Johansson, Cost-benefit rules in general disequilibrium
and the resulting
effective demand
129
for labor is given by
1;= FL l(Y).
(13)
The behavior equations of the household and of the firm producing exportables and importables, respectively, are the same as under orthodox Keynesian unemployment. The monetary welfare measure of an increased production of home goods by the state-owned firm contains, in addition to the terms obtained under orthodox Keynesian unemployment, the net change in the profit of the privately-owned firm producing home goods since MR > MC: d W= du/A = Pi, dGs, - Wdl; + Pi, dys, - % dl; + (p/A) dl: Note that P, dyi- Wdl;f has a Harberger interpretation. P, dyi- Wdlz is the marginal benefit less the marginal cost of an additional unit of labor and so is like a marginal distortion. To further examine this expression I have to solve, for example, the following equation system (I have substituted for rr in the effective demand functions of the household):
-
F,(l;t)- Gs,= 0; md(M,+ j&G; + Ph F,(l;)
+ Cpi eyf(p, eti I), & j, e, p, e) = 0. Observe that the labor market is in ‘equilibrium’ in the sense discussed section 3 above. Thus, by virtue of Walras’ law it follows that the balance trade is in equilibrium if the two markets in (14) are in equilibrium. Simple calculations reveal that Sl$SG; = - 1/F6;
Fb=6F,(1~)/61~;6e/6Gs,=0,
(14) in of
(15)
so employment
in the privately-owned part of the home goods sector will decrease, while the exchange rate, and as a consequence also employment in firms producing traded goods, wiil be left unchanged. Thus, the impact on monetary welfare of a small change in CL is: d W= du/A = P,, dGs, - Wdl; + &, dy; - Wdl; +(/L/A) dT = &, dGs, - Wdl; + &,(6F~c%;)(d@dGs,)
dGs, - Wdl; + (p/A) dT
=j& dG; - Wdl; - & dGs, - Wdl; + (p/A) di = - w(dl; + dl:) +(/~/l/1) di= [ - W+ (p/i)] dT= @,/A) dl
(16)
130
P-O. Johunsson,
Cost-benefit
rules in general disequilibrium
That is to say, since there is effective excess supply of home goods there will only be a redistribution of the production of such goods between the privately-owned and the state-owned firm. This result is a counterpart to the famous balanced-budget theorem, i.e. there is a one-to-one relationship between the decrease in the level of production of the privately-owned firm and the increase in the level of production of the state-owned firm. This is so hccause the initial change is on the supply side while the problem is that demand is deficient. The conventional supplyydemand figure in which the price of the good is fixed above the market clearing level gives an idea of this, So, the change in W is due to the difference, if any, in the marginal productivities between the two firms: dlf = - l/Fh and dli = l/F;. The effect on welfare is in a sense similar to the one obtained under full employment. That is, if the supply price of labor is positive (-u,>O), then welfare will increase only if the state-owned firm produces at a point where the marginal productivity, FL, exceeds the marginal productivity, Fh, of the privatelyowned firm, so that total employment decreases. In the general equilibrium case welfare will increase if the state-owned firm produces at a point where the value of the marginal product exceeds the wage, i.e. where pi, Fi> w. But this implies that Fb>Fh since the privately-owned firms produce where p,,Fh = w. To sum up, an appropriate rule for project evaluation under Keynesian unemployment could be: use the marginal productivities to calculate how a redistribution of the production of home goods between the private and the public sector affects employment. Next, multiply by ihe negative of the supply price of unemployed labor. This simple calculation gives the net change in the welfare. 5. Classical unemployment Classical unemployment is present when there is effective excess demand in the market for home goods, and effective excess supply in the labor market. In a situation like this, the firms perceive no restrictions on trade, while the household perceives restrictions on trade in both the market for home goods and in the labor market: Y;t=Yh
and
l’=l
(17)
The labor market is in excess supply while the market for home goods is in excess demand. It can be shown that the resulting effective demand functions can be written (after substitutions for n): YP= yWo
+
Px4 + P,
eyfP, e,&A
Md = Md(A, + pxey",+ p, ey",, p, e, pme).
i=x,m,
(18)
P-O. Johansson,
CostFbenej?t
rules in yeneral
disequilibrium
131
Since the home goods market is in excess demand, the household buys all that is produced. This and ‘additive separability’ in the sense used in section 3 above explains the absence of (J&Y:+ &, Gi) in the income argument and the absence of j& as a separate argument. The behavior functions of the privately-owned firms are the same as in sections 2 and 3 since firms perceive that they can buy and sell ‘unlimited’ quantities at the ruling prices [yE(p, W- l), yf(pi eF ‘)I. Since I retain the assumption of a flexible exchange rate and fixed prices of both labor and home goods, short-run equilibrium is now determined from the balance of payments condition:
Pxey”,k e*- ‘I+ P, &Xi% ew
‘)
- Cpi eyf (it?, + Qi eyf, & e, p, e) = 0.
(19)
However, it is obvious that 6e/6Gi = 0, i.e. a change in the level of production of the state-owned firm leaves the exchange rate constant. To obtain a welfare measure of the change in GL I have to replace both the price of home goods and the wage rate by shadow prices, say (Pi, +a/A) and (W-p/A), respectively. Too see this let the household maximize the utility function (2) subject to the budget constraint (3) and the two additional binding constraints yt = y,, and 1”= 1 The relevant first-order conditions are
Ui =
Apie,
i=x,m,
(20)
where a,1 and ,u are Lagrange multipliers. Thus, a/A may be understood to be the difference between the willingness to pay for an additonal unit of home goods and the market price; compare the conventional supply-demand figure in which the price of the good is fixed below the market clearing level. Regarding the interpretation of the imbalance in the labor market, the reader is referred to section 3 above. Next, after some substitutions I obtain: d W= du/;l =&
+ U/A)(dG”, + dy”,) - w(dl; + did,)
and an appropriate rule for project evaluation under classical unemployment could be: (a) use the marginal willingness to pay to evaluate rationed goods,
132
P-O. Johansson,
Cost-bent@
rules in general disequilibrium
and (b) use the supply price [ - u,/jL = W--/A] to obtain the cost of employing an otherwise unemployed (underemployed). In terms of compensating variations, (a/A) and (p/A) may be understood to be lump sums that leave an individual on the initial level of satisfaction after a change - evaluated at market prices - in the consumption of rationed goods and a reduction of underemployment, respectively. The labor market appears in eq. (21) in a way which is consistent with the partial equilibrium view discussed in section 3 above. That is, all labor employed in the marginal project is treated as coming from the unemployed. To point out another similarity, assume that the short-run equilibrium on the home goods market is close to the unconstrained one so that x z 0. Then, eq. (21) generates a rule which coincides with the partial equilibrium rule that the production of a marginal project can be treated as a net addition to the quantity traded so that the prevailing price measures the benefit. In other words, if there is classical unemployment it is possible to use the general disequilibrium model to generate a cost-benefit rule which is consistent with the partial equilibrium view. Finally, observe that there will be no income-induced multiplier effects under classical unemployment. The reason for this is that the levels of production of privately-owned firms are governed by the relative prices, and not by effective demand.’ 1 Moreover, since the household is rationed in the home goods market, it will use the new income (drcg+Wd$) to buy all that is produced of the good by the state-owned firm, and this implies in fact that there will be no multiplier effects even if I add a goods market where there is effective excess supply (‘Keynesian unemployment’). In a technical sense this is due to the fact that GE and 1: do not appear in the income term in the effective demand functions - compare eqs. (18) -~ since (pi, J$, + Pi, GT,- p,, J+,) and (~ 1: - lt - 1:- L”,+ 1) always arc equal to zero.
6. Repressed inflation Repressed inflation is present when the price vector is such that there is effective excess demand in both the market for home goods and in the labor market. This means that firms are constrained in the labor market and the household is constrained in the market for home goods. Consistent trading is created through quantity adjustments in the market for home goods and in the labor market, and through a flexible exchange rate. There are, however, four firms and I have to specify how the insufficient supply of labor is allocated between the firms. It is easily understood that the properties of this regime may depend on the kind of rationing scheme that is “Also observe that all relative prices are left unchanged. The exchange rate may change if the assumption of a ‘sufficient degree of additive separability’ is relaxed.
however
P-O. Johansson, Cost-benefit rules in general disequilibrium
133
postulated, and whether agents have the possibility to manipulate the scheme by emitting signals that are inconsistent with their budget restrictions. The behavior of the household under this regime is determined from the decision problem: maximize the utility function (2) subject to the budget constraint (3) and the additional binding constraint in the home goods market (yi = j&J. The resulting effective behavior functions can be written:
Md=Md(MO+ 71--ijhjh,&e,p,e,KQ, 1”= I”(R, +
7c-
jh y,,p, e, p, e, W).
(24
To derive the behavior of the firms, I have to specify a rationing scheme. Under this regime the total effective demand for labor [Id= l~+Z&‘& W-l) + Ii@, eW ‘) + lk&, eti- ‘)I exceeds the total effective supply of labor and the firms perceive a constraint on their demands in the labor market. A simple way to aliocate the insufficient supply of labor is to assume that the stateowned firm, whose level of production is exogenously given, gets all the labor which it needs, while each privately-owned firm gets a given share, bj, of the remaining supply of labor. This simple scheme, which will be used to illustrate the shadow pricing rules, yields y; = y$) = F,Q = F,(b, 0,
j=h,x,m,
(23)
where J= the perceived restriction on the amount of labor available for production of the jth good, ctj=cbjT=T= I”- Ii, i.e. the private sector uses up all the labor that is available. The monetary measure of the welfare change is d W= du/A = (Pi, + CC/~)(dGs, + dy;) - ti(dZ; + d&) +
c(p, edy: - WdS;).
(24)
All privately-owned firms now appear in the formula. This is so because they are rationed in the labor market and hence produce where the value of the marginal product exceeds the wage rate. Thus, one possible formulation of the project evaluation rules could be: (a) evaluate the net change in the level of production of rationed home goods at the marginal willingness to pay; (b) evaluate the net change in labor used in this sector at the market wage; and (c) evaluate changes in other sectors with the difference between the value of the marginal product and the market wage.
134
P-O. Johansson,
Costpbeneft
rules in yeneral disequilibrium
Finally, given the rationing scheme used it can be shown that SqSGi
not be equilibrium
(25) Thus, there will be an accumulation or a decumulation of cash. The assumption of a fixed exchange rate adds, however, nothing to the shadow pricing results obtained in the previous sections. Of course, an accumulation or a decumulation of cash generates long-run effects, but these are not discussed in this paper. Note, however, that the rules which can be derived support Boadway’s (1978) view that the Little-Mirrlees (1968) and the Dasgupta-Sen-Marglin (1972) measures do not correspond to the net social gain of a project under a fixed exchange rate regime. The Little-Mirrlees approach takes as a measure of welfare change the net contribution of the project to foreign exchange earnings. It follows straightforwardly from eq. (27) in section 8 below that such a measure does not correspond to the change in welfare. Consider also eqs. (16) and (21) in sections 4 and 5, respectively. In these cases increased public production of home goods leaves the exchange rate unchanged even under flexible exchange rates. This means that the net contribution to foreign exchange earnings is zero if the exchange rate is fixed. Nevertheless, the project affects welfare. According to the Dasgupta-Sen-Marglin approach non-traded goods are evaluated at domestic willingness to pay while traded goods are evaluated by a shadow exchange rate. This shadow exchange rate takes account of direct project purchases of traded goods. Therefore, this approach is partial equilibrium and will, as a rule, not generate the same measures as a general (dis-) equilibrium approach. This is most easily seen by introducing a tariff on imports, but I will not tire the reader by going through the calculations; the reader is referred to Boadway (1978) for a discussion. 8. Government production of traded goods So far I have
assumed
that
the state-owned
firm
produces
non-traded
P-O. Johansson, Cost-benefit rules in general disequilibrium
135
home goods. It is of course also possible to use the model to examine the case when the firm in question supplies (demands) exportables and/or importables. However, in the present context, where interest is focussed on market imbalances, the only interesting case seems to appear when there is some kind of ‘distortion’ in the foreign market, i.e. a tariff, a quota or some other distortion which makes the domestic and world prices differ. But, as far as I can see, this case adds little, if anything, to the results obtained above. There is, however, an exceptional case. Under Keynesian unemployment and fixed exchange rates, increased government production of traded goods will generate real income-induced multiplier effects in the home goods sector. Formally, by differentiating the equilibrium condition for the home goods market in eqs. (14) in section 4, remembering that the exchange rate, e, now is fixed, one obtains: sl~sGl=piey;f,/(l--p,y~,)Fh,
i=x,m,
(26)
so that
where P,, y& is the marginal propensity to consume home goods Fh = 6ysJ@. The following monetary welfare change measure is obtained:
(2 0), and
dW=du/l=j&edG;-(w-p/L)dl$+ji,dys,-(w-p/A)dl;
= pi FdG;/( 1 - P,, y;,) + (uJn) df.
(27)
So, the domestic market value of the direct change in output has to be multiplied by the ‘textbook’ Keynesian multiplier. At the same time, there will appear an employment multiplier. Hence, to obtain the welfare cost, the direct and the income-induced changes in employment should be multiplied by the supply price of otherwise unemployed. had difficulty in dealing with Cost-benefit analysts have always macroecomic issues. This difficulty arose because of the lack of a satisfactory link between microeconomics and Keynesian macroeconomics. To illustrate, empirical studies often include real multiplier effects.12 But these studies are based on the traditional microeconimic model which does not generate real multiplier effects. Eq. (27) shows that models of the kind used here provide “See
Bohm (1972), Dahlberg
(1972) and Johansson
(1978)
136
P-O. Johansson, Cost-benefit rules in general disequilibrium
‘the’ missing link between microeconomics and Keynesian macroeconomics. Eq. (27), moreover, provides a rationale for the mentioned practice to include real multiplier effects. Observe, however, that if the exchange rate is flexible, then the sign of the effect on production and employment in the home goods sector will be ambiguous. This is so because the exchange rate appreciates when the stateowned firm increases its production of traded goods, and this displaces private domestic supply of such goods and lowers the price of traded goods relative to the fixed price of non-traded home goods. 9. Concluding remarks In order to concentrate solely on the market disequilibrium issue I have ignored several other problems raised in both manuals and elsewhere: the small open economy assumption, the qualitative distinction between traded and non-traded goods, imbalances or other distortions on the capital market, and the shadow price of labor in a dual economy. Another problem which has been left out is how the cost-benefit practitioner should decide which regime is being experienced; the paper clearly shows that the kind of regime prevailing is essential for the kind of benefits and costs to be included. I have no simple recommendations to make, but in many instances already an examination of the sectors directly affected by a project should give valuable information. For example, in countries like Great Britain and Sweden the textile industry, the steel industry and the ship yards seem to be working under conditions which reminds of Keynesian unemployment; there is an ‘excess production’ and a high level of unemployment in these sectors. However, regardless of these practical problems I believe that the cost-benefit analysts could arrive at theoretically more well-founded and also empirically more realistic studies by making use of disequilibrium models. References Arrow, Kenneth J., 1951, Social choice and individual values (John Wiley & Sons, New York). Barro. Robert J. and Herschel I. Grossman, 1971. A general disequilibrium model of income and employment, American Economic Review 61, 82-63. Boadway, Robin W., 1974, The welfare foundations of cost-benelit analysis, Economic Journal 35, 926939. Boadway, Robin W., 1974, The welfare foundations of cost-benefit analysis, Economic Journal Studies XLII, 361-373. Boadwav. Robin W., 1978, A note on the treatment of foreign exchange in project evaluation, Economica 45, 391-399. Bohm, Peter, 1972, A cost-benefit analysis of mining at Stekenjokk (SOU, Stockholm) [in Swedish]. Clower, Robert, 1965, The Keynesian counterrevolution: A theoretical approach, in: Frank H. Hahn and Frank P. R. Brechling, eds., The theory of interest rates (Macmillan, London) 103125.
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Cost-benefit
rules in general disequilibrium
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Dahlberg, Ake, 1972, Manpower training programmes ~ consequences for the individual and society, Umed Economic Studies 3 [in Swedish]. Dasgupta, Partha, Amartya Sen and Stephen A. Marglin, 1972, Guidelines for project evaluation (United Nations Industrial Organisation, Vienna). Harberger, Arnold C., 1971, Three basic postulates for applied welfare economics: An interpretive essay, Journal of Economic Literature 9, 785-797. Howitt, Peter W. and Don Patinkin, 1980a, Utility function transformations and money illusion: Comment, American Economic Review 70, 819-822. Howitt, Peter W. and Don Patinkin, 1980b, Utility function transformations and money illusion: A further comment, American Economic Review 70, 826828. Johansson, Per-Olov, 1978, Economics of employment a study of the effects of a plant location, Umea Economic Studies 53 [in Swedish]. Johansson, Per-Olov, 1980, On potential employment effects of firm oriented labor market measures in a small open economy, Umel Economic Studies 74. Johansson, Per-Olov and Karl-Gustaf Lofgren, 1980, The effects of tariffs and real wages on employment in a Barro-Grossman model of an open economy, Scandinavian Journal of Economics 82, 1677183. Lesourne, Jacques, 1975, Cost-benefit analysis and economic theory (North-Holland, Amsterdam). Little, Ian M.D. and James A. Mirrlees, 1968, Manual of industrial project analysis in developing countries, vol 1 (Organisation for Economic Cooperation and Development, Paris). Niklasson, Harald, 1971,‘An examination of the theoretical foundations for the use of costbenefit analysis in evaluations of e.g. labor market policy measures; Lund,’ mimeo [[in Swedish].\ Niklasson, Harald, 1976, Costtbenefit analysis in theory and practice,l Samhallsvetenskapliga Studier fran Vaxjo,iSerie B, no. 1, Vaxjij [in Swedish], Silberberg, Eugene, 1972, Duality and the many consumer’s surpluses, American Economic Review 62, 942-952. Smith, Ben and Frank H. Stephen, 1975, Cost-benefit analysis and compensation criteria: A note, Economic Journal 85,902-905.