Disequilibrium cost-benefit rules for natural resources

Resources

and Energy

6 (1984) 355-372.

North-Holland

DISEQUILIBRIUM COST-BENEFIT RULES NATURAL RESOURCES* Per-Olov

JOHANSSON

Swedish Uniuersity of Agricultural Received November

FOR

Sciences,

1983, final version

S-901 83 Umed, Sweden received

June 1984

This paper derives cost-benefit rules to be used when evaluating the efliciency gains from extraction of non-renewable resources. The interest is concentrated on consequences of market imbalances. Both excess supply situations and excess demand situations are considered. To derive the shadow pricing rules the paper develops an intertemporal multi-sector model of an open economy which is making use of natural resources. It turns out that the existing partial equilibrium rules or, for that matter, general disequilibrium rules derived for ordinary produced goods are not applicable. The intertemporal interactions, which may be (or at least are) ignored when deriving rules for ordinary produced goods, are of vital importance for the shadow prices to be used when evaluating the efficiency gains of increased extraction. Of particular interest seems to be the fact that the level of unemployment in many cases turns out to be of no importance for extraction decisions based on welfare considerations (even if full employment 1s expected to prevail in the future). Moreover, increased extraction in a period characterized by excess demand (or excess supply) in the market for a natural resource may be socially unprotitable.

1. Introduction The assumption of continuous market clearing widely employed in deriving cost-benefit rules is a strong one. For practical applications it is invaluable to determine how the shadow pricing rules are changed by different kinds of market imbalances. This is true, not the least, in open economies which are making use of natural resources. There are, indeed, important situations where one can reasonably expect quantity constraints in the markets for natural resources. Even if the economy is small in the sense that changes in domestic supply/demand have an imperceptible influence on world prices, domestic producers or consumers may face rationing if the world prices are slow to adjust to eliminate world excess supply or demand. A good example of such a situation would be a small oil-exporting nation facing a world price and sales constraint imposed by the OPEC oil cartel. Quantity constraints can also arise in situations where the country in question is not large if it imposes import quotas coupled with domestic price *I would like to thank K.G. Liifgren and an anonymous 0165~572/84/$3.00

0

1984, Elsevier Science Publishers

referee for helpful comments. B.V. (North-Holland)

356

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J~hansson,

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controls. This may result in domestic rationing, yet the country may be smali in the sense that it would, in the absence of such distortionary policies, perceive perfectly elastic world supply or demand curves for these products at prevailing world prices. Even in the absence of import quotas agents may face rationing. The domestic wage may be fixed at a level which, given other prices, results in excess supply or excess demand for labor, This suggests a priori that the rules for the optimal management of natural resources in situations with rationing in the labor market are very different from the rules to be used under fullemployment conditions (although the outcome of the analysis in this paper will turn out to be quite surprising). Cost-benefit analysts have always had di~culty in dealing with market imbalances. It is straightforward to analyse a market imbalance in a partiai setting; i.e., to examine a single market assuming that the price is fixed above or below the market clearing level. The problems appear when the analyst tries to take account of ail of the interactions as well as the functioning of the individual markets. These problems arose because of the lack of a satisfactory link between Keynesian macroeconomics and Walrasian microeconomics. So-called ‘disequilibrium’ models or temporary equilibrium models provide such a link and have recently been used to derive general disequilibrium cost-benefit rules. See Bell and Devarajan (1983), Blitzer, Dasgupta and Stiglitz (1981), Cuddington, Johansson and Lijfgren (1984), J.H. Drize (1982), J.P. Dreze (1982), Johansson (1982), and Roberts (1982). The general disequilibrium approach used by these authors represents a marked improvement over the standard partial equilibrium approach, which ignores that a disequilibrium in one market will affect decisions in other markets. Unfortunately, the cost-benefit rules derived by the above mentioned authors cannot be applied in an economy which is making use of natural resources. This is so because the rules are derived for ordinary produced goods in si~gZe-period models, while natural resources require an explicit intertemporal treatment. With a resource that is extracted over an extremely long period one may doubt that temporary market imbalances will be of critical importance. Nevertheless, this is precisely what intertemporal extensions of traditional cost-benefit rules for ordinary produced goods suggest; production should be stimulated in times of quantity rationing in markets for goods or factors. To avoid possible confusion it should be stressed that such intertemporal extensions, like the model employed in this paper, assume perfect foresight or rational expectations; see, e.g., Dreze and Stern (1983). Hence, the reason for the fact that market imbalances do matter, even if the time horizon approaches infinity, is basically the same as in atemporal models. [See Neary and Stiglitz (1983) for an informed discussion of the role of expectations in

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fix-price models.] The main message of this paper is that these models do not capture the kind of intertemporal constraints that are faced by extractors of natural resources; e.g., that one cannot extract more than the initial stock. In turn, these constraints imply that the traditional set of cost-benefit rules will be misleading when applied to natural resources. In order to derive cost-benefit rules for an economy which is making use of natural resources, this paper develops an intertemporal model of an open economy. The country considered produces an ordinary traded good using an exhaustible resource and labor as variable inputs. The non-renewable resource, which is also traded internationally, is extracted using labor as the sole variable input. Households consume both kinds of goods and supply labor. To date there seems to be no other model of this kind. A two-period model of a single-sector, closed economy facing quantity constraints can be found in Neary and Stiglitz (1983). See Dasgupta and Heal (1979), Herlindahl and Kneese (1974) and Hoe1 (1977) for traditional extraction models. For empirical research on welfare issues in intertemporal models, see e.g., Bauman and Kalt (1983). The paper is structured in the following way. Section 2 develops the model for the case where there are no market imbalances and derives cost-benefit rules to be used in the unconstrained, or general equilibrium, case. Then, in section 3 excess supply cases are introduced. Two particular situations are considered. The first situation, labelled classical unemployment, is characterized by excess supply in the labor market due to excessive real wages. The other situation, called Keynesian unemployment, refers to a situation where there is unemployment due to deficient demand. In this situation firms extracting the natural resource may perceive a sales constraint on their final output. Section 4 concentrates on the reverse case, i.e., a situation where there is excess demand for natural resources. Section 5 contains a brief discussion of different tax schemes, while a concluding remark can be found in section 6. Behavior functions and equilibrium conditions can be found in the appendix at the end of the paper. 2. Cost-benefit

rules in the absence of quantity constraints

2.1. Analytical

preliminaries

This section considers a small open economy which can buy and sell without limit in each period at fixed foreign-currency prices. Given a fixed exchange rate and assuming the law of one price holds, the domesticcurrency prices of both produced goods and natural resources are also fixed. Moreover, assuming perfect capital mobility and a single international traded bond, the interest rate is exogenously fixed at the world level for the small open economy under consideration. The nominal wage rate is allowed to adjust so as to achieve full employment.

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2.1.1.

Production

natural resources

sectors

this environment, the (representative) firm producing traded goods the present value of profits given the production technology, the wage w,. the price of the natural resource pt, and the price of the traded Within

Yf maximizes

good e,,

where xyt denotes the present value of profits in period C (t = 1,2,. . . ,h), all prices are expressed as present values, F, denotes the strictly concave and twice continuously differentiable production function in period t, N$ is demand for the natural resource, and L$ is demand for labor; a superscript d (s) denotes demand (supply), and a subscript y refers to producers of traded goods. Maximization of (1) yields the well-known condition that the marginal revenue product of each factor of production should equal the price of the factor (in each period). Behavior functions can be found in the appendix at the end of the paper. Private firms extracting the non-renewable resource are assumed to use labor as the sole variable production factor. These firms maximize

where f, denotes the twice continuously differentiable production function in h), N; is supply (extraction) of the resource, Lt, is period t (t=1,2,..., demand for labor, the n subscript refers to private firms extracting the resource, and 1 is a Lagrange multiplier. Firms cannot extract more than the initial stock N, of the resource.’ This explains that there is a constraint on the maximization problem over and above the constraint (i.e., the production technology) faced by tradeables producers in (1). An interior solution to (2) requires that aG/aL:,=p,fj-w,-Afj=O,

(3)

where a prime denotes the derivative of the production function in period t with respect to L$. According to (3) there is in each period of extraction a positive difference between the value of the marginal product and the wage if the resource constraint is binding. In fact, the present value of the marginal profit is constant and equal to 1>0 in all periods of extraction (and not ‘Throughout this paper extract more of a resource

the term quantity constraint excludes than the initial stock available.

the constraint

that lirms cannot

P.-O Johansson, Cost-benefit rulesfor natural resources

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equal to zero as it was for producers of ordinary goods). The Lagrange multiplier 3, may be interpreted as the present value of future profits lost if extraction today is increased by one unit (or as the increase in the present value of profits if the stock of the resource increases by one unit). If production functions are strictly concave, extraction is throughout the paper assumed to take place in all periods regardless of the time path of prices. Note, however, if there are no costs of extraction, then we have Hotelling’s (1931) well-known rule that extraction will take place in periods in which the rate of change of the price in current terms equals the market rate of interest. If the price grows at a faster rate it pays to keep the stock intact and accumulate capital gains. A slower rate of change of the price (than the interest rate) induces firms to immediately extract the stock. Remember that firms are assumed to be able to sell unlimited quantities at the prevailing world market price. 2.1.2. The government To derive cost-benefit rules we will follow the tradition within and introduce a state-owned firm. This firm extracts the natural using labor as the sole variable input. The production function is

the field resource

where the g subscript denotes government output supply or labor demand. The state-owned firm hires labor L$ at the prevailing wage w, and sells output Nit at the prevailing price R,. Because the level of public-sector employment is an exogenously-determined policy variable, the marginal revenue product of government-employed labor may exceed or fall short of the wage. However, the state-owned firm cannot extract more than its initial stock N,, of the resource,

Any profits (or losses) incurred by the state-owned disposed of (financed) by lump-sum taxes (transfers) ment budget constraint takes the form

firm are assumed to be 7;. Hence, the govern-

T = c 7; = - 1 [p,N”,, - w,L&]. This simple specification allows us to concentrate on the market imbalance issue in later sections. See, however, section 5 for a discussion of different tax schemes.

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2.1.3. Households In order to focus on efficiency considerations while setting aside matters of equity and income distribution, the common-employed assumption of a ‘representative’ household is used. This household maximizes utility, u=u(Yi

,...)

Ylf,Ni

subject to the intertemporal

C Cnyt+ x,, -I- w,Lf

Ni,LS,

)...,

)...,

Li),

(7)

budget constraint

- q - e,

Yf- p,Nf] = 0,

(8)

where Yf =demand for traded goods, Nf =demand for the natural resource (e.g., as a fuel) and L: = supply of labor (in the t-th period: t = 1,. . . , h). Both borrowing and lending are allowed at the prevailing interest rate. Once the possibility of borrowing or lending at the prevailing interest rate is introduced, the issue of whether profits are distributed in the period in which they are generated or in the subsequent period becomes less important. In what follows it will be assumed that the sum of current profits rryt+ TC,~from both tradeables producers and resource extractors are distributed within the current period. 2.2. Cost-bene$t rules in situations without rationing To obtain a monetary welfare measure, totally differentiate the utility function (7) and substitute the traditional first-order conditions for utility maximization to obtain dU=du/p=

c [e, dYt +pr dNf - w, dL:J,

(9)

where p is the marginal utility of money (income). It is useful to rewrite (9) in terms of policy instruments. Using the household budget constraint, (9) simplifies to dU =du,fp =c [p, dN;,-

w, dL;,-J.

(10)

Given the assumption of a continuous market-clearing wage plus the small open economy assumption (so that de, = dp, = 0), changes in private sector output and employment net out. However, because public-sector employment is policy determined and need not reflect profit maximization, the sum of the terms in (10) need not equal zero.’ If the state-owned firm produced an ordinary (perishable) good welfare ‘The monetary For the problems

welfare measures considered of examining discrete changes.

in this paper are only valid for small see e.g., Starrett (1979).

changes.

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could be increased as long as the price of the good differs from the marginal cost of producing the good. This is the well-known general equilibrium rule found in the literature on cost-benefit analysis. See e.g., Boadway (1975), Harberger (1971), Lesourne (1975), and Starrett (1979). This simple criterion is not valid for (small) changes in the level of extraction of an exhaustible resource. Increased extraction in the current period requires that extraction is reduced in some future period (provided, of course, that the exhaustion constraint is binding). Hence, the loss in future marginal profit must be accounted for. For welfare to be maximized (so that dU = 0) the state-owned firm should, like private sector firms, in each period select a level of extraction such that pt dN”,, -w, dL;, = A, dN”,, > 0,

(11)

where 1, may be interpreted as the Lagrange multiplier associated with the constraint (5) that the firm cannot extract more than the stock initially available. The Lagrange multiplier Ag reflects the loss in future profits that will occur if extraction in period t increases by one unit. [Recall the discussion following eq. (3) above.] Hence, in each period of extraction there should be a positive difference (equal to A.,) between the market price of the resource and the marginal cost of extracting the resource. If the production function of the state-owned firm is strictly concave it (is assumed that it) should extract in all periods if it aims at maximizing welfare. (Remember that the firm can sell whatever amount it chooses to extract in the world market due to the small open economy assumption.) The level of extraction in the t-th period can be obtained by inserting the production function into (11). On the other hand, if extraction costs happen to be independent of the level of extraction, the time parth of prices is of critical importance. Consider a simple two-period example, dU=p,dN;,+P,dN;,=(P,-P,)dNS,l.

(12)

Clearly, if the rate of growth of the price falls short of the interest rate (so that p2 pJ it pays to delay extraction to the second period. In summary, if a cost-benefit analysis is used to decide whether increased extraction in a given year is socially profitable it would be highly misleading to compare only revenues and costs of that year. Yet empirical works seem to be based on the assumption that an exhaustible resource may be treated as an ordinary produced good. See Bohm (1974) and Lesourne (1975, pp. 233-243). The foregoing analysis demonstrates that such an assumption may be misleading when evaluating the efficiency gains from increased extraction of a natural resource.

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3. Cost-benefit

for natural

resources

rules in classical and Keynesian unemployment

situations

The assumption of no market imbalances (quantity rationing) in deriving cost-benefit rules is a strong one. In this section the consequences of excess supplies will be considered. Two important cases, well-known from modern disequilibrium macroeconomics, can be distilled. In the first case the wage is sticky at a level which, given the levels of all other prices, results in unemployment. This is classicaf ~ne~~Z~y~e~~ caused by excessive real wages. The second situation is known as Keynesian unemployment. In this case unemployment is caused by the deficiency of aggregate demand for output rather than excessive real wages. 3.1. Classical unemployment Assume that the wage is fixed at such a level that there is an excess supply of labor. The fact that there is unempIoyment, but no other market imbalances, means that firms still are unconstrained in all markets. Hence, the maximization problems of firms are those described in section 2.1 above. On the other hand, the household maximizes its utility function subject to the budget constraint plus the empIoym~nt constraint Li=L,. The first-order condition for utility maximization with respect to Lf is now a@!.; = - pw, + e,,

(13)

where 0, is the Lagrange multiplier associated with the constraint in the labor market (in period t). Using this new first-order condition, eq. (10) is replaced by

dU = C EP~dN”,,- w,dL:z+(%‘P)dL:J

(14)

Observe, that there will be no ‘crowding-out’ effects in the private sector, i.e., dY;=dN;=O. The reason for this result is that the levels of production of private firms still are governed by the relative prices, and these are fixed. The welfare measure in (14) differs from the one obtained in the full employment case in that a term t3,/@ reflecting the difference between the market wage and the marginal disutility of effort has been added. This difference (i.e., 6,/,u) may be interpreted as the amount of compensation that can be taken from the household while leaving it just as well off as it was before the change in employment. Due to lack of data cost-benefit practitioners often assume that the marginal disutility of effort au/aL’ is equal to zero (which is a reasonable assumption if unemployment is ‘chronic’). In effect, this means that e,/,u==w, in eq. (13) above. The monetary welfare measure (14) then reduces to

(15)

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resources

This is the Hotelling rule discussed in section 2.1 above. Hence, in order to maximize welfare, the state-owned firm should extract the resource in period t provided the rate of growth of the output price equals the interest rate. The firm should pay no attention to its own marginal costs of extraction, nor to The latter means that increased the prevailing level of unemployment. extraction ‘today’ cannot be justified unemployment. In fact, the firm should

by

‘today’s’

relatively

high

level

of

keep the stock intact if (and as long as) the rate of change of the output price exceeds the interest rate. This is indeed quite a strong result in an economy suffering from unemployment. Next, suppose that there is full employment in some periods. To highlight the main point consider the two-period case, with unemployment in the first period (0, = pw, >0) and full employment in the second period (0, =O). Period 1 may be interpreted as representing ‘today’ and period 2 as in period 1 by the representing the (expected) ‘future’. Increased extraction state-owned firm then causes the following change in monetary welfare: ~U=(P,

-PA

dNS,,- w,(aL$IaNS,,) dNS,,

+ WI c(aLB,iaw,)(aw,iaNs,,)1 dx,.

(16)

The first term in (16) has already been interpreted above. The second term represents saved (social = producer) extraction costs of the state-owned firm in the second, full employment, period. Recall that increased extraction in one period has to be followed by decreased extraction (and employment) in the other period. In turn, this causes the wage w2 to adjust so as to restore full employment in the second period. The change in w2 affects extraction decisions of private firms in both periods provided agents have perfect foresight or at least rational expectations, as is assumed throughout. An examination of the short-run equilibrium condition (AS) in the appendix suggests that both w2 and extraction by private firms in the first period decreases. Hence, the third term in (16) reflects the social loss (equal to 0,/p= wi) of the accompanying reduction in employment by private extractors in the first, unemployment, period. These effects suggest a priori that a reallocation of extraction by the state-owned firm from full-employment periods to unemployment periods may cause a rise or fall in monetary welfare. Recall that the first

term in (16) may be positive or negative, the second term is positive and the third term is negative. If saved extraction costs in period 2 (the second term) equals the value of production foregone due to crowding-out effects in period 1 (the third term), however, the Hotelling rule once again applies. Hence, one obtains the ‘perverse’ result that extraction by the public sector firm should be decreased in the period with unemployment if the present value of the output price increases over time (so that pz > pl). See the discussion following eq. (15) above.

R.E-B

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As the foregoing analysis has demonstrated, it is far from self-evident that the presence of unemployment changes the extraction rules for an exhaustible resource, or affects the rules in a way predictable from the traditional partial equilibrium approach. According to the partial equilibrium rule production of a good should be increased as long as the price exceeds the social marginal cost of production. Clearly, such a rule of thumb may be quite misleading when applied on natural resources. 3.2. Keynesian unemployment If prices are fixed and such that there is excess supplies of both goods and labor, one speaks of Keynesian unemployment. In this section, however, the analysis is restricted to a situation where there is unemployment and resource extractors face a sales constraint for their output. Firms producing the ordinary traded goods are throughout this section assumed to never face quantity constraints. Private producers extracting the resource face a sales constraint mi on their output in period i if domestic plus foreign demand falls short of their profit-maximizing output. Assuming that the price path over time is such that the resource is completely exhausted (i.e., that firms do not face a sales constraint in all periods), firms maximize (17) In periods where firms face no constraint on the level of sales, the first-order conditions for profit maximization are still those derived in eq. (3); y is equal to zero in such periods. If resource extractors face a sales constraint, on the other hand, the Lagrange multipliCr y is positive and reflects the loss in marginal profit due to this constraint,

aG/aL;i=pJj

-wt

-a& -yJ-;

=o,

(18)

where a prime denotes a partial derivative of the production function with respect to labor L$. Changes in the level of extraction by the state-owned firm will now affect private sector extractors as their levels of extraction are demand determined. Hence, the monetary welfare measure includes, in addition to the terms in (14), the loss in marginal profits of private extractors in sales constrained periods,3 dU=C[p, where

dN”,,-W,

dz, = dLi, + dLi,.

dL$+(O,/p)dL,]

A close examination

‘It is assumed that the state-owned

firm gets priority

+Cyi

dmi,

of the behavior in the market.

(19) functions

and

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short-run equilibrium conditions, however, reveals that increased extraction by the state-owned firm in a sales constrained period crowds-out private sector supply, i.e., dNS,, = -dNf. This result means that aggregate demand in the ith period is unaffected by increased extraction by the state-owned firm. See the appendix at the end of the paper. Consider once again the twoperiod case with a sales constraint in the first period and the marginal disutility of effort equal to zero so that 0,/p = w,,

+[pl-;1-(w,/f;)]dN;=p,dNS,, = -[p2-A-(wllf;]dN;=O,

+w,dL$-AdN”, (20)

where the fact that dN;, = - dNS, = - dNiz = dN”, has been used, and the last equality equals zero from the first-order conditions for profit maximization in a period without a sales constraint. Hence, monetary welfare is left unchanged by increased extraction by the state-owned firm in one period (followed by decreased extraction in a second period due to the exhaustion condition). This result is identical to the result for ordinary produced goods derived in Johansson (1982) and may be given the following interpretation: The government should increase extraction in a certain period only if it can do it more profitable, calculated at producer prices, than private sector firms.4 This is a rather restrictive condition in an economy where there is unemployment.

4. Cost-benefit

rules in excess demand situations

There are also important situations where one can reasonably expect excess demand for a natural resource. The experiences from the market for crude oil support this supposition. In this section world market prices are assumed to be fixed and such that there is excess demand for the nonrenewable resource. Recall the discussion in the introduction. Both tradeables producers (using natural resources as an input) and domestic consumers are assumed to face a quantity constraint in at least one period. Firms extracting the natural resource, on the other hand, never face a sales constraint. When faced with a constraint iVYl on demand for natural resources in the period, tradeables producers maximize (1) subject to N$ =myr. The resulting first-order conditions for profit maximization are 4This is most easily seen by assuming that the marginal disutility of effort (au/aLS) is negative [and not equal to zero as in (20)]. Then we have dU=C[(au/aL*)(dL~~+dL~,)/~] so that the sign of the change in welfare depends on the marginal productivities of firms.

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for all t, (21) p,(dF,/cYN$)

-p, - a, = 0

for all t.

The Lagrange multiplier CC,represents the positive difference between the value of the marginal product of a rationed natural resource and its price. If the constraint does not bite in a certain period, then ct, =O. The imbalance in the resource market is assumed to be so severe that also households are rationed: NP=N,. If households also face a quantity constraint in the labor market, the first-order conditions for utility maximization equal (&@Y:‘)-pe,=O

for all t, for all t,

(au/aL;) + pw, -

ez= 0

(22)

for all t,

where 8, is the Lagrange multiplier associated with the constraint in the resource market in the t-th period. Using (21) and (22), the monetary welfare measure can be shown to equal dU=C(p,

*

dNi,-W,

dL:t) +C[cLi dm,i +(Bi/P)

I

dmil (23)

where the i subscript refers to periods where the constraint market is binding. To highlight the main points assume to (world) price of natural resources grows at a rate equal to the that pl=pZ=... =p,,, and that the marginal disutility of effort 8, =pw,. Assuming that tradeables producers demand for the priority, (23) simplifies to dU=&dN;i+xwidL;i. I I

in the resource begin that the interest rate so is zero so that resource takes

(24)

Increased extraction by the state-owned firm increases marginal profit of rationed tradeables producers by an amount equal to ai. Moreover, the increase in the available quantity of natural resources affects employment in the tradeables sector. Employment increases if natural resources and labor are technical complements (so that the marginal productivity of labor increases). In this case both terms in (24) are positive; the second term represents the social value of increased employment (when the marginal

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disutility of effort equals zero). If the production factors are technical substitutes, on the other hand, tradeables sector employment decreases implying that the second term in (24) is negative. Hence, increased extraction by the state-owned Jirm in a period ~har~cter~~ed by excess demand, may, quite surprisingly, cause monetary welfare to rise or fall. A sufficient condition for welfare to fall is that factors are technical substitutes and the increase in profits tli d& falls short of the loss of labor income wi dL$. This outcome is likely to occur if the rationing in the resource market is not too severe so that cli is close to zero. Moreover, if the resource price varies over time welfare would decrease regardless of the size of cli if there exists at least one future resource price pj such that pi zpi +~r:; in this case the term (pi -pj) dtvii < 0 is added to eq. (24). Finally, if domestic consumers get priority in the natural resources market, (24) is replaced by (25) The term /Ii/p reflects the positive difference between the marginal willingness to pay and the market price of a rationed resource in the ith period. Clearly, the social profitability of a marginal increase in extraction by the state-owned firm exceeds its private profitability, provided the resource is consumed but not used as a factor of production.

Note, however, that this result assumes that the present value of the price of the natural resource is constant over time. If pj >pi +(flJp) for at least one pj #pi, it would clearly be socially unprofitable to increase extraction in the ith period where consumers are rationed. As indicated above, a similar argument applies in the case where rationed tradeables producers (and not consumers) get priority in the natural resource market. 5. Observations on optimal taxation

The assumption of lump-sum taxation employed in this paper is a restrictive one. Policy rules assuming lump-sum transactions often are questioned as lump-sum transactions seldom, if ever, are possibie to use. It is beyond the scope of this paper to derive detailed rules for different tax schemes; the purpose is to highlight the consequences of market imbalances. There are, however, a few important observations to report. First of all, if households face rationing in the labor market a proportional or progressive tax on income works like a lump-sum tax. This is because a change in the wage rate only has an income effect (but no substitution effect) when a household is unemployed; see the behavior functions of an unemployed household in the appendix. Hence, the results derived in sections 3 and 4 are valid even if the government

taxes wage (and/or profit) incomes and

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changes the tax rate to balance its budget. This is an important generalization of the results derived in this and other papers on disequilibrium cost-benefit rules. Recall that rules assuming lump-sum transactions often are questioned. Second, a wage subsidy to firms extracting the natural resource leaves their levels of extraction unchanged if there is Keynesian unemployment. Production is governed by aggregate demand and a wage subsidy leaves the consumer prices unaffected. Instead the government should subsidize the consumer price of the natural resource; it can be shown (although the proof is omitted) that the optimal tax on consumers of the resource is negative in periods with Keynesian unemployment, which seems reasonable as demand is deficient. On the other hand if there is classical unemployment, one obtains the reverse results. A subsidy to producers in a period with unemployment works while a subsidy on consumption leaves extraction unchanged. The result that a wage subsidy to firms extracting the natural resources increases welfare, however, presumes that unemployment is not persistent. See the discussion following eq. (15) in section 3 above. In summary, the results reported in this section in many cases deviate from those found in textbooks on cost-benefit rules. For example, a standard result seems to be that a subsidy to producers is ‘Iirst-best’ in unemployment situations. Clearly, this is not true in a situation characterized by Keynesian unemployment. 6. A con~~~djng remark

The optimal management of natural resources is basically a long-run issue. At the same time futures markets are quite rare in the real world. This means that real world (governments’ attitudes toward) extraction decisions probably are strongly affected by ‘temporary’ phenomena like world excess demand for a certain resource, increasing or decreasing unemployment rates in depressed areas of a country, etc. This paper has been devoted to an examination of the difference between the expected private pro~tability and the social profitability of extraction decisions in such situations. Of course, such comparisons do not solve some of the basic problems, e.g., how to make appropriate forecasts of the price path (in an infinite-horizon setting). Nevertheless, if it is admitted that temporary phenomena affect extraction decisions it is important to have a set of appropriate project evaluation rules that obviates the need for ad hoc evaluations based on output, employment or external balance effects. In any case, the results of this paper indicate that the case for government intervention is much weaker in markets for natural resources than in markets for ordinary goods: under rather reasonable assumptions government interventions in a rationed market for a natural resource would decrease societal welfare.

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Appendix In this appendix we derive behavior functions and short-run equilibrium conditions for the different regimes. To keep the paper to manageable proportions only the two-period case is considered; the number of possible regimes drastically increases as the number of periods increases. Period 1 may be interpreted as representing ‘today’ and period 2 as representing the ‘future’. Throughout agents are assumed to have rational (constraint) expectations. See Neary and Stightz (1983) for details. A.I. The unconstrained case

Firms producing the ordinary traded goods never perceive quantity constraints. The first-order conditions for profit maximization give rise to the usual supply and demand functions,

If private firms extracting the non-renewable resource perceive no quantity constraints in either the labor or output markets, their behavior functions wifl depend on prices in both periods and the initial stock of the resource,

The household’s behavior functions are

V= YP(el,e2,pl,wI,wz,I), (A-3)

where I is the present value of non-labor income plus initial wealth minus taxes. A short-run equilibrium requires that wages adjust so as to simultaneously achieve fuil employment in both periods. Recall that all other prices are fixed. A.2. Classical unemp~o~ent

If the household with subjective certainty expects to perceive constraints L, and i,, on its supply of labor, it will maximize (7) subject to (8) and the quantity constraints L; =L1 and LS,=L,. Assuming weak separability to eliminate the separate influence of a rationed variable on the marginal

370

utilities

P.-O. Johansson,

of unrationed

Cost-benefit

commodities.

rules for natural

the behavior

resource6

functions

can be written

The levels of production and employment are now exogenously given as all prices are fixed and firms perceive no quantity constraints. Hence, demand for goods can be found by inserting profit and wage incomes into (A.4). However, if the wage w2 adjusts so as to achieve full employment in the second period a short-run equilibrium requires that

(A.9

+L~,(Pl,p,,wl,w,,N,)+L~,. This equation determines the equilibrium wage w2 in the second Observe that r=x[e,Y;+p,(N;+N”,,)] - w,E, -ww,(L$ + Li, + L$).

period.

A.3. Keynesian unemployment If firms extracting the natural resource face a sales constraint output in period 1, their behavior functions equal N;=No-ml,

t,“, =&(iif,),

N1 on final

L~2=L~,(N,-~l).

64.6)

As prices are assumed to be such that the resource is completely extracted, supply in the second period equals what is left over from the first period. The labor demand functions reflect the fact that it never pays to hire more labor than the minimum required to produce the output demanded in Ihe first period and the output possible to produce in the second period, respectively. If the household is unemployed in both periods, a short-run equilibrium requires that

where I=Ce,Y;+(p, --p&Vi +p2No-1 w,E, +xp,Ni,. Eq. (A.7) determines the equilibrium level of extraction N, of private extractors in period 1. The remaining variables (like demand for tradeables) are recursively determined. A.4. The excess demand case When firms producing the traded goods face a quantity constraint the market for natural resources their behavior functions are

m,,i in

P.-O. Johansson,

Cost-benefit

rules_/or natural

Households are now assumed to be rationed in periods) and in the market for natural resources sufficient degree of additive separability to eliminate a rationed variable on the marginal utility of an behavior functions are

resources

371

the labor market (both (period 1). Assuming a the separate influence of unrationed variable, the

(A-9)

A short-run equilibrium now requires that (A.lO) Domestic extraction by private firms is determined by the fixed prices and the stock of the resource. Production by the state-owned firm is policydetermined. Finally, due to world excess demand or domestic regulations, net import R, is fixed. Hence, the total available quantity is also fixed. This quantity, which is assumed to fall short of total domestic demand at prevailing prices, is allocated between tradeables producers and households according to some (exogenously determined) rationing rule. Then, the levels of supplies and demands in (A.8) and (A.9) are recursively determined.

References Bauman, Michael G. and Joseph P. Kalt, 1983, Intertemporal consumer surplus in laggedadjusted demand models, Discussion paper no. 956 (Harvard Institute of Economic Research, Cambridge, MA). 1983, Shadow prices for project evaluation under Bell, Clive and Shantayanan Devarajan, alternative macroeconomic specifications, Quarterly Journal of Economics XCVIII, 457-477. Blitzer, Charles, Partha S. Dasgupta and Joseph Stiglitz, 1981, Project appraisal and foreign exchange constraints, The Economic Journal 91,58874. Boadway, Robin, 197.5, Cost-benefit rules in general equilibrium, Review of Economic Studies 42. 361-373. Bohm; Peter, 1974, Social efficiency: A concise introduction (Macmillan, London). Cuddington, John T., Per-Olov Johansson and Karl Gustaf Lofgren, 1984, Disequilibrium macroeconomics in open economies (Basil BlackweB, Oxford). Dasgupta, Partha S. and Geoffrey M. Heal, 1979, Economic theory and exhaustible resources (Cambridge University Press, Cambridge). D&e, Jacques H., 1982, Second-best analysis with markets in disequilibrium: Public sector pricing in a Keynesian regime, CORE discussion paper no. 8216. Drize, Jean P., 1982, On the choice of shadow prices for project evaluation. Discussion paper no. 16 (Indian Statistical Institute, Calcutta).

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P.-O. Johansson. Cost-benefit rules

for natural

resources

Dreze, Jean P. and Nick Stern, 1983, Cost-benefit analysts, Mimeo., m: Arlan Auerbach and Martm Feldstem, eds., Handbook of pubhc economtcs (North-Holland, Amsterdam) forthcommg. Harberger, Arnold C., 1971. Three baste postulates for applied welfare economtcs: An mterpretattve essay, Journal of Economic Lrterature 9, 785-797 Hertindahl, Orrts C. and Allen V. Kneese, 1974, Economic theory of natural resources (Charles E. Merrill, Columbus, OH). Hoel, Michael, 1977, Five essays on the extraction of an exhaustible resource which has a nonexhaustible substttute. Memorandum. 17 Oct. 1977 (Instttute of Economics, Universitv of Oslo). Johansson, Per-Olov, 1982, Cost-benefit rules in general disequihbrium, Journal of Pubhc Economics 18, 121-137. and economic theory (North-Holland, Lesourne, Jacques, 1975, Cost-benefit analysis Amsterdam). Neary, Peter J. and Joseph Stiglitz, 1983, Toward a reconstruction of Keynesian economics: Expectations and constrained equilibria, Quarterly Journal of Economics 98, suppl., 199-228. Roberts, Kevin, 1982, Desirable fiscal policies under Keynesian unemployment, Oxford Economic Papers 34, 1-22. Starrett, David, 1979, Second-best welfare economics in the mixed economy, Journal of Public Economics 12. 329-349.