Public Pector Pricing and Poduction


Public Pector Pricing and Poduction


15-1 INTRODUCTION

Countries differ in the extent to which output is supplied by the public or nationalized sector, but in nearly all Western countries there has been a great deal of interest in policies and the behavior of private sector. Many managers of state enterprises seem to act in an identical way to the managers of private firms. In such behavior socially desirable? We focus in particular on pricing and production decisions. Should public sector prices be equal to marginal cost, and, if not, how should the deviate? In choosing the technique of production, should state enterprises use market prices or should the instructed to use shadow prices? If the latter, how should these shadow prices be calculated? What rate of discount should be employed investment decisions?
This Lecture is addressed to the questions. We begin with pricing policy, taking as reference point the first-best marginal cost pricing principle (described in more detail below). In section 15-2 we examine some of the arguments that may be advanced to justify departures from marginal cost pricing. These include the problem of financing public enterprise deficits, the implication of monopoly elements in the private sector, and the impact on the distribution of income. In part, this treatment exploits the parallel between public sector prices and commodity taxation, and we do not repeat the earlier analysis (e.g., of the distributional effect). However, we also draw attention to the differences that may arise, and the fact that a simple translation of algebraic result may obscure significant features of the problem. We then consider the production decisions of the public enterprise, and in particular whether the shadow prices inputs should be identical within the public sector and whether they should be equal to the market prices of inputs. Should British Steel use the same discount rate (for a given degree of risk) as British Gas, and should these rates be equal to an appropriate market rate? The question of the desirability of “production efficiency”, and the particular application to the social rate of discount, are the subjects of sections 15-3 and 15-4.

Marginal Cost Pricing Principle
The principles of public enterprise pricing have long been the subject of discussion, and particular reference should be made to the tradition in France, dating back the work of the Ecole des Ponts et Chaussees in the early nineteenth century (Ekelund, 1973). Much of this literature has argued in favour of marginal cost pricing, a case thet was put forcefully by Hotelling in his classic paper.

the optimum of the general welfare corresponds to the sale of everything at marginal cost. This means that toll bridges … are inefficient reversions: [that taxes on incomes, inheritances, and site values] might well be applied to cover the fixed cost are products of these industries. The common assumption … that “every tub must stand on its own bottom” [is thus] inconsistent with the maximum of social efficiency. [Hotelling, 1938, p. 242]

In short, Hotelling argued that prices should be set at marginal cost and that any resulting deficit, in decreasing cost industries, should be financed by taxation would be lump-sum. This case for marginal cost pricing must be seen therefore as a first-best argument. As we have emphasized in earlier Lectures, there are reasons why there may be limits on the use of lump-sum taxes and why government may have to rely on distortionary taxation (indeed, the taxes actually referred to by Hotelling are likely to finance deficits has to be raised in a way that is distortionary - so that for those economic activities price does not equal marginal cost – then there is no presumption that optimal pricing within the state enterprise will entail marginal cost pricing. If the desired distribution cannot be achieved by lump-sum means, the public sector pieces may have to be used as an instrument for this purpose.
The departures from marginal cost pricing in a second-best world have been subject of much of the postwar literature. Thus, the behavior of the public enterprise subject to revenue constraint (e.g., that they should break even) was investigated by Boiteux (1956). He derived a formula for optimal pricing (see Section 15-2) that appears to be virtually identical to those derived in the treatment of Ramsey tax problem (Lecture 12), e.g., that there should be equi-proportionate reduction in consumption, a long  price had been equal to marginal cost. That the formula appear similar by Hotelling. There is however the significant difference that each state enterprise may face a budget constraint that in separate from that faced by others.[1] So while we can exploit the earlier result, we need to bear in mind the special features of the position of the state enterprise and its relations with government – to which we now turn.

Control over Public Enterprises

The formal similarity between the problem of taxation and public enterprise pricing provides considerable insight, but fail to do justice to the complexity of the questions arising from the relationship between the state enterprise and government, and before presenting the analysis we should comment briefly on some of the relevant issues.
            On central question is the nature of the “directives” from the government to the “state enterprises”, and degree of autonomy of the latter. There are varying degrees of autonomy. At one extreme, the enterprise may be run like a government department (as is sometimes the case with the Post Office); at the other, the enterprise may be an autonomous corporation like IBM or ICI, with the state receiving the profits like the other shareholder (this is true, for example, of certain joint ventures). More common is the intermediate case, where the public enterprise has independent management, but is set objectives by the state and is subject to specified constraints. The formal organizational relations may, however, imperfectly reflect the degree of autonomy. The civil servants running the Post Office may have freedom of action that the head of the sate steel corporation, who may directed to locate his plans in certain areas, to build certain types of plants. Etc.
            In the design of the administrative structure, a key role is played by the incentive informational problem to which we have alluded on several occasions. The interesting questions raised are not ones that we have space to explore here, and simply assume that the structure is of the following two-stage nature. First, the government specifies the objectives of the enterprise and constraint. For example, it decides on the target rate of return on capital that has to be achieved by the industry and magnitude of the state subsidy (if any). Second, the enterprise determines its pricing policy so as maximize is objective function subject to the constraints. For example, given that the National Electricity Corporation has to make profits of $x, how should it determine the relative prices to households and industrial user? The efficiency and equity properties of the outcome depend on decisions made at both stages.
            With this kind of structure, there is no direct link between public enterprises. The National Electricity Corporation does not take account of the effect of its policy on the National Coal Corporation. Such interaction – affecting such matters as transfer pricing – need to be taken into consideration by the government in laying down the guidelines for individual enterprises, e.g., to allow for the fact that they may be producing closely competing products (e.g., long distance train and air service).
The two-level structure is one reason why the parallel with the literature on optimum taxation is not complete. In that case, there was a one-stage problem. If we apply the Ramsey result directly at the level of the individual enterprise, then we are ignoring the fact that the constraints are themselves the subject of choice. The profit targets for different industries are set by the government, and take account of the interdependencies. On the other hand, if we collapse the problem into one stage, and treat all public enterprises as a single entity, we are ignoring the fact that the design of decentralized guidelines is an important institutional feature of the public sector.
In that follows, we pay particular attention to the relationship between the two levels of decision-making, and the wider question of the links with other instruments of policy. Do we, for example, want to keep public sector prices low for redistributional reasons, and how does that depend on the scope for redistribution via tax system? In the analysis we of necessity have to leave on one side many important issues. We do not discuss the definition of marginal costs and related questions such as peak load pricing. We do not, in the models investigated, allow for uncertainty. The issue of incentives for the managers of public enterprises is not addressed directly, nor do we consider the information that is available to the government in its decision-making. We assume, for instance, that it knows the technologies and demand curves facing various industries, if this information is relevant for the directives it gives to different firms. (Of course, if it had this information, there would be bless need for decentralization.)
The analysis here is concerned with the publicly owned enterprise. In a number of countries, particularly the United States, industries such as telephones, electricity, and railroads, are typically privately owned but publicly regulated. There is again a two-tier structure. The government imposes constrains on the operation of the industry, and the enterprise maximizes subject to the constraints. The different is, however, that the enterprise objectives are now private objectives, such as the maximization of profits or sales. There has been an extensive literature on regulation (see, for example Baumol and Klevorick, 1970; Bailey, 1973). We make no attempt review this literature; we do however note an passant some of the main results. Finally, we shall have nothing to say about the relative merits of private ownership, with and without regulation, and direct public ownership and control. These issues are critical, but a full discussion is beyond the scope of these Lectures.

15-2 DEPATURES FROM MARGINAL COST PRICING

this section we consider the departures from marginal cost pricing that may be implied by need to finance deficits, by the existence of monopoly sewhere, and by redistributive goals. We assume initially that all individuals are identical, relaxing this when we refer to redistribution.

The Enterprise with a Profit Target

The simplest situation is that of a single public enterprise, producing one nal product in quality Z (in per capita terms), in a otherwise competitive economy (where the vector of per capita private outputs is denoted by X). all individuals have identical utility functions U(X,Z,L), where L is the quantity of  labour supplied per person. Labour is taken to be the numeraire. We denote by q the vector of prices of private sector product and by p the price of the public output. The production constraint assumed to be of the form

Ω ≡ F(X) + C(Z) – L = 0                            (15-1)

There F(X) gives the labour requirement in the private sector, and (Z) that in the public sector. It is assumed that the production set is convex, the condition for profit.
∏ ≡ q . X – F(X)                                         (15-2)

(e.g., value of the net output minus labour cost), is maximized a necessary condition for which is that qi=Fi, where the latter denotes the derivative of with respect to Xi. it is assumed at this stage that there are constant return to scale in the private sector. So ∏ ≡ 0. The implication of pure profits are discussed later.
            The public enterprise is assumed to determine its price, p, to maximize social welfare, as measured by the indirect utility function of the representative consumer, denoted by V(q.p). the enterprise is constrained by the profit condition (per person)

                                               pZ – C(Z) + T >0                                     (15-3)

where T denotes the subsidy by the government and  0  the profit target. The subsidy is assumed to the financed by lump-sum taxation, so that T enters the indirect utility function. There are assumed to be no other taxes at this stage. The solution to the pricing problem may be seen by forming the Lagrangean.

                               L    = V(q.p.T) + λ [pZ – C(Z) + T -∏0                        (15-4)

The first-other condition with respect to p may be written using the properties of the indirect utility function (the assumption of constant return to scale in the private  sector means that the only change in V is that arising directly from p):

                                        - αZ + λ [Z + (p-C1) ]                                      (15-5)

where α is the private marginal utility of income.
            Suppose first that T is freely variable, so that lump-sum taxation can be employed to finance any deficit. The first-other condition with respect to T (using the fact that (provided   is that
                                               - α + λ = 0                                                    (15-6)

From (15-5) it follows that (provided   /  a necessary condition for optimally is that
                                               p = C1 (Z)                                                    (15-7)

i.e., price equals marginal cost. This is an illustration of the standard argument for marginal cost pricing.
            Where there are constraints on the use of T, and the enterprise has an effective profit target, then the pricing rule must be modified. Suppose that T0, and that with marginal cost pricing this is not sufficient to allow the enterprise to satisfy (15-3). This situation is illustrated in Fig. 15-1, where the profit target is taken to be that of breaking even. At the level of output where the price equals marginal cost, there is the deficit indicated by the hatched area. In other to meet the profit target, the firm has to reduce output to ZB, where price exceeds marginal cost. As drawn in that diagram, the instruction to the enterprise to break even completely determines its pricing policy. It sets price above marginal cost to the extent necessary to avoid a deficit.
            In practice, public enterprises produce more than one produce, and this introduces degrees of freedom in to the choice of pricing policy. Suppose that there are two products Z1, Z2. There are typically many combinations of the prices p1,p2 that will satisfy the profit constraint. How should the
 















Figure 15-1  Public enterprise with break-even constraint.

m depart from the marginal cost principle? There have been two main phools of thought. One view is that the mark-up over marginal should very according to “that the market will bear”, i.e., inversely with the pasticity of demand. Opposed is the position that prices should be oportional to marginal cost, advanced by, among others, Frisch (1939) and Allais (1948).
In other to consider the merits of these rival views, we may modify the earlier analysis, so that maximization problem is now represented by the Lagrangean :

          L    = V(q, p1, p2, T) + λ [p1Z1 + p2Z2 + C(Z1, Z2) + T - ∏0]            (15-8)

The are assuming that the level of the lump-sum tax fixed. The first-other conditions with respect to the prices, p1 and p2, are:

                                                                                                                  (15-9)


There C1 denotes . The parallel with the Ramsey problem should at this point be clear, if we write p1-C1 ≡ t1, then these conditions are the same as those in Lecture 12 (see Eq. (12-13)).
            If we consider the special case where demands are independent and there are no income effects, then, re-arranging, we have for good 1


(15-10)
 
 

                         
This familiar Ramsey result, that the “tax” should be inversely related to the elastic of demand, supports therefore the “what the market will bear” view rather than the Frisch-Allais proportionality rule. This, and other implications, were brought out by Boiteux (1956). The extent of devinations from marginal cost pricing depends on the budget constraint. Where this is not binding (e.g., because lump-sum taxes can be employed) λ = α and p1 = C1. At the other extreme, as the required profit approaches the maximum possible, λ à α, and the right-hand side of (15-10) tends to unity. This yields as limiting case the price-discriminating monopolist, since marginal revenue equals marginal cost implies
                                                (                                       (15-11)

Where  is the elasticity of demand.
            As in the optimum tax literature, the analysis can be extended to interdependent demands. This is left as an exercise.
            Exercise 15-1 Examine the optimal pricing policy where there are two goods with interdependent demands. How does the excess of price over marginal cost depend on the cross-elasticities? Are there circumstances in which a price less than marginal cost may be justified?

Profits in the Private Sector
The assumptions made to date (both here and in the earlier treatment of optimum taxation) do not allow for pure profits in the private sector, which arise in the competitive case where there are decreasing returns to scale. We now consider the implications of the existence of profit for public enterprise pricing and the relation to the optimum tax formulae.
            With the introduction of pure profits, we have be careful about the normalization of producer and consumer prices (Munk, 1978). We write q for consumer prices, s for producer prices, and for the present treat labour as commodity zero. Since the supply functions are homogeneous of degree as commodity zero. Since the supply functions are homogeneous of degree zero in producer price (including those of the public sector), in the absence of lump-sum income, one can in that case normalize by fixing one producer price and one consumer price. There is no loss of generality in fuming one good to be untaxed (as in analysis to this point). This is however no longer true where the consumer receives profit income, since multiplying all producer prices by 5 implies that the profit income is also multiplied by 5. The effect can be offset only by multiplying all consumer prices by 5. The assumption of an untaxed good is not in this case inocouos. We can normalize one producer price ore one consumer price.
            The restrictions on commodity taxation are particularly important then considered in conjunction with restrictions on the tax rate on pure profit. Suppose that we set at unity one producer price (that of labour). The profit of the private sector is (per capita)

                                           ∏ = s X – F (X)                                           (15-2’)

This is assumed to be taxed at a rate t, so that the lump-sum income received by households is (1 –) ∏( ≡ I). Such a tax can be seen to be quivalent to a rise in all consumer prices by a factor of 1/(1 -), since demand function are homogeneous of degree zero in consumer prices and
            [2]The taxing of pure profit can therefore be achieved by uniform tax on all goods (and labour)[3]. As a result, any restriction on profits taxation must summit both  and the ability to tax all goods at uniform rate. In what follows we assume that   is fixed, and that there is one untaxed goods, which is taken to be labour. The individual budget constrain is then, with a sector Z of public sector outputs.
                                    q X + p Z = L + (1-                                     (15-12)

Combined with the production constraint and (15-2’), this yields the public sector budget constraint (per capita):
                                    p ZC (Z) + (q-s) X +  = 0                         (15-13)

or
                            p ZC (Z) + q X – F (X) – (1 –  = 0                (15-3’)

Let us now consider the position of the public sector as a whole determining the prices to be charged were all goods can be taxed except labour (and there is no poll tax or subsidy). The maximization problem may then be formulated in terms of the lagrangean:

  L    = V(q, p1, p2, I) + λ [ p ZC (Z) + q X – F (X) – (1 – ]      (15-14)

The first-other condition for the choice of pk is

          +   + λ [ ]          (15-15)

The effect on profit given by (from (15-2’))

                            (15-16)

From competitive profit maximization, si = Fi, so that the first term on the right-hand side is zero, and we can replace qj - Fj by qj – sj in (15-15). We may also make use of the properties of the indirect utility function, and observe that.
                                             (15-17)

Where  denotes the compensated demand derivative and  the income term (and there is a corresponding expression for private good demand). The first-other condition (15-15) can then be re-written as

(+ (
                
 -           (15-18)

                                             ≡ 0 (                                          (15-19)

This formula allows one to see the role played by profits. The percentage reduction of consumption along compensated demand schedule is no longer proportional for all commodities; there is a an additional term representing the effect of profit. If raising the price of the kth commodity the kth commodity should be reduced by less than the lth commodity[4].
            The derivation of the first-order conditions follows in the same way for the choice of indirect taxes (i.e., differentiating with respect to qk). there is however a significant difference between the case of public sector pricing and that optimal taxation. This is brought out clearly in the special case where there independent demands, no income effect, and no joint production. In case, a change in the public sector price has no effect on private sector profit, and the first-order condition reduces to the familiar inverse elasticity from (as in (15-10)). In contrast, in the case of a tax on a private sector product, the effect on profits is given by (where = 0 for m  k)

                                        (15-20)

Substituting into (15-19), with the cross-price and income derivatives set to zero (and replacing Zk by Xk).

          (     (15-21)

This differs from the result given earlier for the no-profit case in the appearance of the term in (1 - ). Defining  as before for the elasticity of demand, and

                                                                                        (15-22)

As the elasticity of supply, we can re-arrange (15-21) as (where 0 = 1 – α/ λ):
                                                                                                                                       (15-23)

This is a generation of the result originally derived by Ramsey (1927). In which the elasticity of supply now enters the termination of the optimum tax rate (he implicitly assumed that  = 0), and the other things being equal the tax rate should be higher on goods supplied inelastically[5]. The difference between public sector pricing and optimal tax case is brought out by the term (1 - ). in the public sector,  is in effect unity, all profits being returned to the government, so that no supply considerations enter. In the case of private goods, where the rate of tax on profit is less than 100 per cent, the supply side has to be taken into account.
            A natural question at this juncture is why governments do not impose 100 per cent profit taxes. Earlier we provided some explanation as to why lump-sum taxes should not be the only source of revenue. But, if profits taxes are non-distortionary, surely they should be set at 100 per cent, and thus the questions with which we have been concerned cease to be relevant? In practice, governments have not followed this Henry George-like policy. Although in wartime a few countries have impose 100 per cent surtax rates, they typically do not levy on a regular basis 100 per cent taxes on profits and the incomes of fixed factors. The reason for this goes back to the lack of information at the disposal of the government. Most importantly, it finds considerable difficulty in distinguishing pure profits from the return to capital, or the return to entrepreneurship. This is seen most clearly in the case of unincorporated enterprise. If there were a 100 per cent profit tax, no such enterprise would ever declare a profit it would always distribute the “pure profit” as wages to entrepreneurs.

Monopoly and Second-Best
The existence of monopoly profits in the private sector gives rise to effects similar to those just discussed, but also raises other significant issues. Should the attempt to offset in effect? Do departures from marginal cost pricing in the private sector provider grounds for deviating from marginal cost in the public sector? In order to concentrate on this kind of question, we abstract from the effects of profits by assuming a 100 per cent profits tax ( = 1). For the reasons just outlined this is not realistic; it does however help separate the issues.
            Suppose that we consider the choice of public enterprise pricing policy where there are no indirect taxes and the private sector monopolists have fixed prices, sj, where sj > Fj. since  = 1, we can write down the first-order condition by analogy with earlier result:
                 (= (          (15-24)

            The pricing rule in the absence of monopoly is now augmented by the term underlined in (15-24). This may be seen as the change in revenue from the profit tax arising from changes in private sector outputs include by a rise in pk (holding sj and Fj constant for all j). To see the implications, let us take the case of a single public enterprise, producing goods whose demands are independent and where there are no income effects. The usual elasticity formula is augmented by the underlined term. If private firms price above marginal cost, and if their output is an increasing function of the public price, then we shall on this account want to raise the price above marginal cost. Conversely, if their output is a declining function of the public price, then the underlined term is negative. If  = 0, which is effectively the case taken by Green (1961), the deviation from marginal cost depends solely on the divergences (sj – Fj) in the private sector. Intuitively, it might be felt that the corrective pi/Ci ratio for the public sector should lie somewhere between the maximum and minimum values of sj/Fj in the private sector, but Lipsey and Lancaster showed that this was not necessarily so. It is indeed possible that price could be below marginal cost if “monopolistic pricing of commodities complementary to it produce negative terms in the above sum [greater] an all the positive terms arising from … monopolistic pricing of substitutes” (Farrell, 1968, p, 48).
            The treatment just given is rather special, relating to an “irreducible” astortion, were indirect taxes cannot be employed to correct the deviant behavior. The analysis needs be extended to allow for the use of indirect taxation – see Guesnerie (1975, 1978). We should also note that lying behind this treatment of the implications of market imperfections for public factor pricing is a view of the behavior of the monopolist that is clearly not appropriate to oligopolistic markets, where strategic elements are likely to be important. Further development of the pricing rules requires a more boundly based general equibrium theory of imperfect competition, and – as he have seen in Lecture 7 – this at present at a rather early stage.

Exercise 15-2 in the context of the model of imperfect competition described in Lecture 7 (pages 208-217), we examine the optimal policy of state enterprise supplying the output Y. Should it charge more than marginal cost, on account of the mark-up in the private sector? Suppose that one of the firms in the X sector is taken oover by the public enterprise. What should be its pricing policy?

Redistribution and Public Sector Prices
There has been considerable debate about the role of the public sector prices in redistribution, as illustrated by the following quotations:
By far the simplest way of securing the distribution … desire is through the price system … the only price a public enterprise or nationalized industry can be expected to set is what we may as well call a just price-a price which is set with some regard for its effect on the distribution of wealth as well as for its effect on the allocation of resource. [Graaff, 1957, p. 155, his italics]
and:
subsidies to the consumption of commodities are a particularly inefficient way of redistributing income … the best way of making a particular individual better off is to give him a appropriate sum money… It is thus unlikely that consideration of the distribution of income should lead to an optimal price below marginal cost. [Farrell, 1958, pp, 113-14, our italic]
           
In attempting to assess the merits of these views, we can apply the same analysis as in earlier Lecture, introducing the distributional characteristic of the public sector good,  (Feldstein, 1972a). the application is straightforward, following the same lines as in Lecture 12, and is left to the reader as exercise:

Exercise 15-3 Suppose that individuals differ, being identified by superscript h. there is a single public enterprise producing two products subject to a break-even constraint. Derive the first-order conditions corresponding to (15-9). For the special case where demands are independent and there are no income effect, show that the first-order conditions corresponding to (15-9). For the special case where demands are independent and there are no income effect, show that the first-order condition reduce to

                                                           (15-25)

Where b is the social marginal valuation of income and ? the covariance between bh/ and k.
From the example just given, it is clear that there may be situations where redistributional reason dominate in the determination of the structure of price (e.g., where  and  ). In this sense, Graaff is correct. But the result depend critically on the extent to which orther vary the poll tax/subsidy element of direct taxation in such a way that  = 1, then prices is set below marginal cost only where  > 0. If the social marginal valuation of income falls with income, then  is positive only for an inferior good. If the good is normal Farrell’s conclusion is borne out.
We can go on to consider the range of pricing schedules open to the public enterprise. One commonly employed is the two-part tariff, involving a fixed payment coupled with a price per unit. This departs from a single price plus poll tax in that consumer choosing zero consumption are not liable for the fixed element[6]. More generally, the marginal price may vary with the quantity. The feasibility of such a nonlinear schedule depends on total consumption being observable. If resale or repeat purchasing is possible, then quantity discounts or premia can, respectively, be undone, For discussion of the optimal nonlinear schedule, the reader is referred to Spence (1977), Willig (1978), Goldman, Leland, and Sibley (1977), Roberts (1979), and Seade (1979).

15-3 CHOICE OF TECHNIQUE AND PRODUCTION EFFICIENCY

In this section we consider the choice of inputs for the public sector. The questions discussed may be posed- in a somewhat over-simplified from-in the mix of inputs used by each enterprise? Put another way, suppose that the government set shadows prices for input and tells enterprise to minimize costs at those prices. Should these prices be the same for all enterprises and all sub-units of enterprises? What should be the relationship between these shadow prices and market prices?



Production Efficiency

The questions just described are equivalent to asking whether there should the production efficiency. Should the public sector be productively efficient in the sense that the marginal rate of technical substitution between any two outputs should be the same in different enterprises? Or should the railways make the choice between coal and oil on a different basis from that faced by the electricity industry? Should the economy as a whole productively efficient in that marginal rates of substitution are the same in both public and private sector?[7]
            Intuitively, it seems plausible that production efficiency is desirable. The Literature on the second-best has however led one to be suspicious of such intuitive argument. Does, for example the need to meet a profit target lead to input choice being different from those made on the basis of market prices? Are distributional considerations relevant in the choice of technique?
Our discussions indicate that the most one can hope for is a result of the “separation” kind obtained in the Lectures on optimal taxation. Where there are departures from first-best in one area, can we none the less continue to hold? In the second-best conditions should have been analyzing, if the government can impose 100 per cent profits taxation and tax all commodities and factors, if the budget constraint of enterprises are optimally chosen, then we will want to have production efficiency in the economy as a whole. On the other hand, where these conditions do not hold, the presumption in favour of production efficiency no longer obtains. Where, for example, the budget constraints are arbitrarily fixed, we will want to have efficiency within each enterprise, but there may be different shadow prices in different enterprises.
            The issue of production efficiency was originally addressed in the classic paper by Boiteux (1956); he established basic efficiency theorem for an arbitrarily given constraint. Diamond and Mirrlees (1971) examined the question, using more general techniques, for the case of unrestricted taxation and no pure profits and established the desirability of production efficiency under fairly weak conditions. They require only that the social welfare function be individualistic and that there exist some good (with positive price) that is a “good” for all individuals.[8] The argument runs broadly as follows. If the optimum were in the interior of the production set, small changes in prices would still result in technically feasible demands. On the other hand, lowering the prices of the good consumed by everyone (strictly, a good that no consumer supplies), or raising the price of the good supplied by everyone (strictly, non-satiation and positive response of the welfare function). Therefore, given this condition, at an optimum production must occur on the production frontier.
            The extension of the analysis to economies where there are pure profits and restriction on the set of admissible taxes is studied in Stiglitz and Dasgupta (1971), Dasgupta and Stiglitz (1972), Mirrless (1972a) and Hahn (1973). The result show that, if there are enough instrument at the government’s disposal, and in particular if the government is free to set any rates of tax (including 100 per cent) on the pure profit of different producers, then production efficiency is desirable even with decreasing return to scale in the private sector (giving rise to pure profit). On the other hand, restrictions on the taxing possibilities of the government, for example, limits on taxing pure profit or when a tax cannot be levied on certain commodities or factors, may mean that production efficiency is not desirable.[9]
            We do not attempt to provide a rigorous account of production efficiency; instead, we take an example that brings out the role of several factors. We consider a set of public enterprises, identified by an index j, each of which produces an output Zj using inputs of two primary factors (types of labour)  and  according to the production function:

                               Zj =                                              (15-26)

Where Qj is assumed to be a differentiable, well-behaved production function exhibiting constant returns to scale (an assumption that can be relaxed). For each enterprise there is a profit constraint

                      =                                 (15-27)

Where wi denote the producer input prices. On the assumption that the government can vary freely the taxes on all goods and factors, but is restricted to taxing pure profits at rate, we may derive the following first-order condition for the choice of input in the jth enterprise (where  is the multiplier associated with the constraint (15-27) and µ a multiplier (associated with an overall revenue constraint) - see Stiglitz and Dasgupta (1971):

                                             (15-28)

            From this result we can see at once that there are several sufficient conditions for production efficiency. If there are no profit in the private sector, then the marginal rate substitution between L1 and L2 in the jth public enterprises and equal to the private sector rate of substitution. Profit taxes ( = 1), or if there lump-sum taxation such that µ = α. Thus, of the government’s revenue requirements can met by a partial profit tax, we may have  < 1 but µ= α. Where these do not hold, but the profit targets ∏? are set optimally, then  will be equated for all j. in this case, the marginal rates of substitution are equal within the public sector (since  is equal for all j). all public enterprises use the same shadow prices.
It should be noted that analysis assumes that the government can levy a full set of commodity taxes and that it can tax all factors in all uses. Otherwise, we may want to use distortionary factor taxes, in some industry, as partial substitute for the absent commodity taxes. Similarly, the fact thet we cannot tax labour in one use (e.g., household production) does not mean that we do not want to tax it in other uses. Many of the important instances of distortionary factor taxes can be related to these condition; for incorporated sector may arise for the impossibility within the unicorporated sector, the two factors must be treated the same.
            Finally, we have taken no account so far of distributional consideration, but they may also provide a reason for productive inefficiency.[10]

Implication of Production Efficiency/Inefficiency

The efficiency/inefficiency result has several important implications; and it serves to integrate the discussion of a number of different policy problems.
            First, production efficiency within the public sector implies that the transfer prices used by public enterprises for sales within the public sector should be marginal prices, and hence not necessarily equal to those charged to final consumers. The profit target should be met on sales outside the public sector; to charge a mark-up on the transfer of electricity to the public steel industry would lead to production inefficiency (as would taxes on any intermediate transactions).
The second implication of the efficiency result concerns the setting of enterprise objective related to input use. This applies particularly to the requirement of a minimum rate of return (as with regulated industries). Such rate of return constraints may be compared with the absolute profit target considered above. It can be shown that for a given output a cost-minimizing firm subject to a binding minimum rate of return constraint produces, its output using a more (less) capital-intensive method of production that with an absolute constraint if the minimum rate is less than (exceeds) the market rate (Gravelle, 1976). By contrast, a regulated private firm subject to a chooses a more capital-intensive technique than the one that minimizes cost for the output level produced (Baumol and Klevorick, 1970, Proposition 3). Where the conditions for production efficiency do not hold, then it may well be desirable for the shadow price of capital to differ from the market rate of interest, but such departures need to be derived from an explicit analysis of the kind described above, with full account taken of the instrument that government has at its disposal.
In a open economy, the possibilities for international trade can be treated as private sector industries, and the efficiency result implies that in evaluating public sector decisions the international prices should be employed. This result holds not only when commodity taxes are chosen optimally, but also when they are fixed arbitrarily (Dasgupta and Stiglitz, 1974).
Finally, in the context of intertemporal decisions, the efficiency result implies that the correct shadow price of capital (the social rate of discount) is the producer rate inters. This is in contrast to a substantial literature arguing that the social discount rate should be rate of time preference, or some weighted average of this rate and the private rate of return on capital. In the next section we take up this application in more detail.


15-4 COST-BENEFIT ANALYSIS AND SOCIAL RATE OF DISCOUNT

The choice of the social rate of discount plays a critical role in cost-benefit analysis, and we begin with a more general review of the issues involved.[11]

Cost-Benefit Analysis

In principle, cost-benefit analysis is straightforward. Any investment project can be viewed as representing a perturbation of the economy from what it would have been had the project not been undertaken. To evaluate whether the project should be undertaken, we need to look at the levels of consumption of all individuals of all commodities at all dates, under the two different situations. If all individuals are better of with the project than without it then it should be adopted (if there is an individualistic social welfare function); if all individuals are worse off, then it should be rejected. If some individuals are better off, and some worse off, whether we should adopt it depends on how we weight the gains and losses of different individuals.
            Although this is obviously the “correct” procedure to follow in evaluating projects, it is not a particular one; the problem of cost-benefit  analysis is simply whether we can find reasonable short cuts. In particular, we are presumed to have good information concerning the direct costs and benefits of a project (its inputs and its outputs);[12]  the question is whether there is any simple way of relating the total effect. Thus, in the case of the choice of discount rate, there is trivial sense. use the social rate of time preference for evaluating benefits and costs accruing in different periods. This however applies to total effects, and there is no reason to believe that these are simply proportional to direct effects that are observed. If the ratio of total effect to direct effects changes systematically over time, then we would not wish to use the social rate of time discount in evaluating a project when looking only at direct costs and benefits.
            In a first-best word, with no distortions and full scope for lump-sum redistributive taxation, if a project is “profitable” on the basis of its direct effect using market prices, then-with an individualistic social welfare function – it is socially desirable. The problem of finding the correct shadow princes for cost- benefit analysis arises from the existence of market imperfections and failures; it is concerned with situations where one cannot necessary infer social desirability on the basis of the profitability of the project. In the case of the social rate of discount, the difficulties stem from differences between the private rate return and rate at which society can transfer resources between periods. The former is equal, in a competitive model, to the marginal physical rate of transformation of output in one period into output in the next. The latter is the rate at which to the government can make the transfer, or what we refer to as the marginal economic rate of transformation.
            In the applying this approach, we need to begin with the reasons why a first-best cannot be attained. This depends on the initial sources of market failure, and on the extent to which government policy instruments can be employed to approach the first-best. As we have seen in earlier Lectures, there is no more reason to believe that the intervention is socially optimal at any point in time. Savings may be too low, e.g., where individuals give less weight to succeeding generations that they would collectively. Savings may be to high, e.g., because people can give as much to their descendants as they would like but are constrained in giving to antecedents. The direction of the misallocation may not therefore be clear, but there is certainly no presumption that the market solution is socially optimal.



Social Discount Rate in the Overlapping Generations Model

In order to explore this in more depth, we make use of the overlapping generations model described in earlier Lectures and which provided the basis for the treatment of the optimum taxation of savings in Lecture 14. The main modification is that there is now assumed to be a government capital good-the social discount rate being the return on public capital. Total government capital is denoted by G and enters the determination of aggregate output in period u, which is assumed to be given by a constant returns to scale production function:

                                        Yu = F (Ku Gu Lu Pu)                                      (15-29)

where Pu denotes the total population, Lu hours of work, and Ku the private capital stock, at time u. the return to public capital accrues to the government. output can be used interchangeably as either a consumption or capital good:

                                 Yu = Cu + (Ku+1 - Ku)+ (Gu+1 - Gu)                        (15-30)

where there is assumed to be no depreciation and no current government spending, and where Cu denotes total consumption.
            Initially we assume that all individuals are identical; this is a crictical assumption which is later relaxed. They live for two periods, working in the first, and the lifetime utility of a representative member of the generation born at u is U (, , Lu), where  denotes consumption in period i. total consumption at date u is therefore, per worker,

                                                                                             (15-31)

where n is the rate of growth of the population. The wage rate is wu before tax and u after tax, and price of second period consumption to generation u is pu. The indirect utility function, Vu, is a function of u and pu there are assumed to be no lump-sum taxes-although see below).
            The level of capital goods at date u is related to the savings of the order generation, and the capital market equation may be written (see e.q., (14-53))

                                        (1 + n) ku+1 = Au - Bu                                      (15-32)

Where ku+1 is capital per worker, Au savings per worker, and Bu the level of government bonds per worker. It is assumed that there are only two assets in the economy-bonds and real capital. In particular, there is at this stage no equity investment in firms. There are no “pure” profits. Again, this is a critical assumption. From the individual budget constraint,

                                        (1 + n) ku+1 = u Lu - - Bu                        (15-33)

Finally, the production constraint may be expressed in per worker terms and re-arranged to yield

     (1 + n) ku+1 = u + Lu (,  ) - -  - (1 + n) gu+1+ gu     (15-34)

Where gu is government capital per worker (and we revert to f for the production function).
            As in the previous Lecture, we assume that the government maximizes the sum of lifetime utilities over generations discounted by factor  (where  1);

                                                                                               (15-35)

We introduce as before the state valuation function:
            Ku, gu, Lu Pu)
           
                     = max          
                                                                                                              
                                               - (1 + n) gu+1+ gu]+           (15-36)

Where we have eliminated ku+1 between (15-33) and (15-34), and ku+1 in Γ(u+1) is given by the latter. The analysis of the optimal wage tax () and interest tax (p) follows the same lines as in Lecture 14. Here we concentrate on the effects of government capital. The first-order condition for the choice of gu+1 is:

                                        (15-37)

The difference equations governing Γi are:

                                        (15-38)
                             (15-39)

Where fk, fg denote the marginal products of private and government capital respectively. As before, we assume that an optimal policy exist.


Implications Results

we begin by considering the case where the government has full control over debt policy. since Bu does not effect Vu, the first-order condition is that ? = 0. It then follows from (15-37) – (15-39) that

                                         = )                                             

and
                                         =  = 1                                         (15-40)


It is immediate therefore that in this model, with identical individuals, all pure profits taxed away and a completely flexible debt policy, the rate of return on public capital must equal that on private capital. The social rate of discount is private rate of return. The intuitive reason for this is that, with optimally chosen taxes and debt policies, aggregate savings are fixed, and unit of public capital displaces precisely one unit of private capital. And has no further repercussions. From the difference equation (15-38), with  = 0, the steady-state value of fk is given by (in steady sate k, g, L and all other per capita variables are constant):[13]

                                             =                                              (15-41)


If the objective function is the total sum of utilities (so  = (1+n)/(1+)), the steady-state rate of return is equal to the social rate of time preference. This does not in general hold outside the steady state.[14]
            These results give a straightforward answer to the question of the choice of social discount rate. The assumptions required are, however, of subious validity. The government does not use debt policy primarily for purpose of intertemporal redistribution, as the above analysis requires, and does not impose 100 per cent pure profits taxes; finally, individuals are not identical.
            The assumption about debt policy can be interpreted more generally as applying to monetary policy: the issue of money can have the same effect as Bu. moreover, as noted is earlier Lectures, the use of differential lump-sum taxes on the two generations equivalent to the use of debt. Thus a combination of lump-sum taxes with zero present value has no impact on allow the government to shift resources through time. Nevertheless, even when debt policy is viewed more broadly, there may be situations where the government does not have complete freedom to achieve intertemporal redistribution. Where this is so   0, and we have 9from (15-37) – (15-39)):

                          =  = 1+ (                    (15-42)


If the government cannot use debt/monetary policy freely for purposes of intertemporal redistribution, the social rate of discount is not necessarily equal to the producer rate of interest. In steady state, it is still true that  the social rate of time preference, but   < 0 implies fk > . the value of  depends on the choice of tax instrument – see the following exercise (based on Pestieau, 1974).

Exercise 15-4  Derive the first-order conditions for the choice of pu and u (by differentiating (15-36)), and examine their interpretation in steady state (making use of the difference equations in Γp and ). use these result to exspress the rate of return on public capital as a function of the private return and consumer rate of interest. Show that where there is no tax on wages fg is a weighted average of the two rates, but that the weights need not lie in the interval [0,1].

The model discussed above is parallel to that of Diamond and Mirrless (1971), assuming constant returns to scale and hence no pure profits.[15] We have seen however that production efficiency may not be desirable: the social discount rate is not necessary equal to the private rate of return. The reason for this is that the government cannot-because of the assumption of restricted debt/monetary policy-transfer resources at will between priods. It cannot in effect trade freely on all markets. Where this restriction is present, and ? > 0, then the marginal physical rate of transformation of output in one period into output in the next (1+fk) is not necessarily equal to the rate of transformation thet can be achieved by the government using a restricted class of instrument (the marginal economic rate of transformation). This can be viewed another way. Private savings are only channeled into private capital accumulation; the government on the other hand is constrained in its ability to influence private capital formation (where Bu is fixed).
Finally, we allow for difference between people. If the government is constrained in its ability too levy differential taxation, then, even with 100 per cent profits taxes and complete control of debt policy, the social rate of discount may deviate from the private rate of return for distributional reasons. Moreover, if the distributional impact differs across capital goods, then different social rates of discount ought to be employed for different types investment.



15-5 CONCLUDING COMMENTS

Our treatment of decisions in the public sector has been highly selective, and there are many issues that have been left on one side. We have not considered peak load pricing or the interrelation between pricing and investment decisions. No account has been taken of uncertainty or of rationed demand. the discussion of cost-benefit has been very circumscribed. There is however one shortcoming that we should emphasize: the lark of any explicit analysis of the information available to the government and of the process by which it is obtained.
            This may be illustration by reference to the setting of guidelines to public enterprises. Suppose that the government possesses the same information as the producers (in both private and public sector). Then clearly, it could solve the problem of optimal pricing for the entire each enterprise and then face the enterprise with constraint of not exceeding this level. The government does not however possess this quantity of information: and if it did, it would hardly need to decentralize. We have therefore to consider the mechanisms available to the government that enable it to elicit the necessary information, and the motives of those in charge of the enterprises. They latter will depend on the kind of considerations discussed in Lecture 10 and on the specific incentive structures that are in effect. Thus, there is an extensive literature in the field of economic planning concerned with the effect of different incentive structures that are discussed in Lecture 10 and on the specific incentive structures that area in effect. Thus, there is an extensive literature in the field of economic planning concerned with the effect of different incentive schemes (for a recent review, see Johansen, 1978, Ch 5). The problem is moreover close to that of the revelation of preferences for public goods. In the next Lecture, we examine some of the procedures that have been proposed.









READING

The Anglo-Saxon literature on marginal cost pricing is usefully collected in Turvey (1968); for an extensive account of the French literature, see Dreze (1972), Kolm (1968, 1971) Rees (1968, 1976) and Turvey (1971). For discussion of the distributional aspect, see Feldstein (1972a, 1972b). On the regulation of public utilities, see the original article by Averch and Jonson (1962), and surveys of the fields by Baumol and Klevorick (1970) and Bailey (1973). The results on production efficiency are discussed in a Diamond and Mirrlees (1971); for a broader treatment of the results in relation to second-best theory, see Guesnerie (1978). There is an extensive literature on cost-benrfit and the choice of the social discount rate-see Little and Mirrless (1974) and the references given there.

























[1] Informal sense (as we note below) the taxation problem discussed earlier is special case of the more general problem that Boiteux analysed: wehere there is a single budget constraint for the whole public sector, and the level of revenue raised for the public sector is set optimally (not, as in the Boiteux formulation, arbitrarily)
[2] That is the demand of the household for goods  i is
   

[3] This means in the optimal tax problem that, where the value of private profit exceeds the government requirement, denominated appropriately, afirst-best solution can be attained either by a pure profits tax or by taxing all goods and factors, See Munk (1978).
[4] in the formulation given, it has been ssumed that all goods (expect labour) can be taxed: the effect of the lagrangean (15-14)
[5] Most textbooks refer to the formula that taxes should be proportional to ? (e.g., Pigou, 1947, p. 108). For further discussion, see Stiglitz and Dasgupta (1971, p. 170)
[6] This feature of the two-part tariff is not captured in the empirical application by Feldstein (1972b), who treats the fixed payment as a uniform lump-sum tax
[7]There is a closely related question, which we should more properly have asked in the Lectures on optimal taxation. Should the government impose differential factor taxes on different industries within the private sector; i.e., should the private sector economy be production efficient? See Stiglitz and Dasgupta (1971)
[8] Even this restriction may be dropped if we allow trade taxes, i.e., taxes that are differentiated on the basis of whether the individual is buying or selling a commodity.
[9] Also, we noted in Lecture 12 that, in the course of a process of tax reform, where only limitied steps may be made, there may be situations in which temporary inefficiencies are desirable even when the full optimum is characterized by production efficiency – Guesnerie (1977)
[10] See Dasgupad and Stiglitz (1972) and Mirrlees (1972a).
[11] The situation with which we are concerned in this section are those where the government directly undertakes the project; there are other circumstances in which the government is called upon to license some private project (particularly in less developed countries) or to provide some critical input (capital). Although many of the same considerations arise, one must bear in mind the distinction between these circumstances. In general, the criteria for evaluating project in these two situations will be different.
[12] In practice
[13] As in Lecture 14, we assume that an optimum, if it exists, converges to a steady state.
[14] It should be noted that the discount factor relates to utilities, not consumption-see Presteau (1974).
[15] The assumption of constant return to scale encures that in steady state all relevant variables are constant in  per capita terms. It also means that, if the return to the public capital (? Per worker in steady state) accrues to the government, it is sufficient the new public capital formation (ng per worker). Since the assumption ? 1 implies that ? n. if therefore the return to public capital is appropriated by the state or 100 per cent pure profits taxation is possible, then there is no need for distotionary taxes to finance public capital formation.

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