# Pareto Efficiency and Market Equilibrium Under Transaction Costs

Policy Analysis

1. Introduction

We want to investigate the economic and welfare implications of economic policies. We build on the analysis presented in the previous chapter. We consider a global economy consisting of commodities produced by a set Nsof production units and a set Ncof consumers. It will be convenient to start with the optimal allocation given in equation (12) of the previous chapter. It is

W(U) = MaxD,S {B(D, U) - C(S): D  S}, (1)

where D  (D1, …, Dm) = jNc xai Rmis the vector of aggregate demandform final goods, S  (S1, …, Sm) = iNs yai Rmis the vector of their corresponding aggregate supply, B(D, U) is the aggregate consumer benefitobtained from consuming D and holding consumer utilities constant at U = {Ui : i  Nc}, and C(S) is the aggregate cost of producing S. Equation (1) chooses aggregate production and consumption decisions so as to maximize aggregate net benefit [B(D, U) - C(S)] subject to the feasibility constraints [D  S]. As discussed earlier, this corresponds to a maximal allocation, W(U) being the maximized aggregate net benefit. In addition, the set of U satisfying W(U) = 0 identifies the Pareto utility frontier. It means that an efficient allocation is one that maximizes aggregate net benefit (as given in (1)) and then redistribute all net benefit (with W(U) = 0).

Equation (1) is a constrained optimization problem. It can be alternatively written in terms of the Lagrangean L = B(D, U) - C(S) + p [S - D], where p = (p1, …, pm)  Ris a vector of Lagrange multipliers for the constraints [D  S]. In a market economy, the vector p represents market prices reflecting resource scarcity in the economy. In this context, (1) can be alternatively written as

W(U) = Minp 0MaxD,S {B(D, U) - C(S) + p [S – D]},(2)

which has for solution D*, S* and p*. Under differentiability and assuming that p* > 0, the associated first-order necessary conditions are

B(D, U)/D = p,(3a)

C(S)/S = p, (3b)

S = D. (3c)

Equations (3a)-(3c) state that an efficient allocation is attained when marginal benefits equal prices (as given in (3a)), marginal costs equal prices (as given in (3b), and the market is in equilibrium (where aggregate supply equals aggregate demand, as given in (3c)). This characterizes the efficient aggregate supply and demand, S* = D*, along with the market equilibrium prices p*.

Below, we investigate situations where actual allocations are not necessarily efficient, i.e. where S  S* and/or D  D*. We examine a set of scenarios where this can happen. When the scenarios are associated with alternative economic policies, this amounts to conducting policy analysis. In each scenario, we ask three sets of questions: 1/ How does (S, D) differ from (S*, D*)? 2/ In a market economy, how do prices get affected? 3/What are the welfare implications of each scenario?

1. The Case of Aggregate Distortions

Equation (1) provides a good starting point for our investigation.We are interested in situations where the aggregate quantities Q differ from S* and D*. First, we examine the case where the aggregate supplies S = (S1, …, Sm) are constrained to be equal to QS (QS1, …, QSm), where QS are “aggregate production quotas”. This can be introduced into (1) to give

WS(U, QS) = MaxD,S {B(D, U) - C(S): D  S, S = QS}. (4)

Equation (4) is a constrained optimization problem which can be alternatively written in terms of the Lagrangean L = B(D, U) - C(S) + p [S - D] + qS [QS – S]}, where p = (p1, …, pm)  R is a vector of Lagrange multipliers for the constraints [D  S] and qS= (qS1, …, qSm)  Rmis a vector of Lagrange multipliers for the quota constraints [QS = S]. In this context, (4) can be alternatively written as

WS(U, QS) = Minp0,qS MaxD,S {B(D, U) - C(S) + p [S – D] + qS [QS – S]},(5)

which has for solution Dc, Sc, pc and qSc. Under differentiability and assuming that pc > 0, the associated first-order necessary conditions are

B(D, U)/D = p, (6a)

C(S)/S = p - qS, (6b)

S = D = QS. (6c)

Equations (6a)-(6c) state that the associated allocation is attained when marginal benefits equal prices p (as given in (6a)), marginal costs equal “distorted prices” (p – qS) (as given in (6b), and the aggregate supplies and the aggregate demands equal the aggregate production quota (as given in (6c)).

Expressions (6a) and (6b) indicate that consumers face prices p, while producers face prices (p – qS). This hasseveral interpretations. First, it shows that qS are price wedges that separate consumer prices from producer prices. Second, as Lagrange multipliers, qS can be interpreted as shadow prices of the quota constraints [QS = S]. Third, qS can be interpreted as unit quota rents, with [qS QS] measuring the aggregate quota rent. These quota rents get captured by market participants. As discussed below, who capture the quota rents affect the distribution of welfare.

The shadow prices qSc solving (5) depend on QS: qSc(QS). In general, these shadow prices can be shown to be decreasing in QS, where

qSic(QSi’) qSic(QSi) for any QSi’ > QSi, i = 1, …, m. (7)

Equation (7) has the following interpretation. When the production quota QSi is “very small”, then increasing this quota would make the quota constraint “less binding”, implying a lower shadow price qSic. Similarly, when the production quota QSi is “very large”, then increasing this quota would make the quota constraint “more binding”, implies a higher shadow price qSic. It means that theprice wedge qSic(QSi) starts positive for “small QSi”, declines to zero as QSi rises, and turns negative as QSi becomes “large”. See Figures 1 and 2 below for an illustration. In the context of model (4) or (5), it means that the absence of quota for some good i can be represented by setting the i-th quota QSi such that it is non-binding (at the point where qSic = 0). Below, we are interested in the case where the quotas Q = S are binding for at least one good.

2.1.Welfare Analysis

What the welfare implications of the quotas QS? This can be evaluated by comparing the aggregate benefit obtained with quotas (as given by W(U, QS) in (4) or (5)) and without quotas (as given by W(U) in (1) or (2)). Define the change in aggregate welfare (or aggregate benefit) due to the production quotas QSas WLS(QS)  W(U) – WS(U, QS). Note that adding a constraint in a maximization problem cannot increase the value of the objective function. Then, comparing (1) and (4) gives

WLS(QS)  W(U) – WS(U, QS)  0. (8)

Equation (8) states that imposing production quotas tends to decrease aggregate welfare and thus reduce efficiency. This result is obtained under general conditions. The lower bound for WS(QS) in (8) is 0; and it is attained when QS = Q*, i.e. when the quotas QS are set equal to the efficient aggregate quantities Q*. Otherwise, WLS(QS) in (8) will be positive (when QS Q*). Then, the term WS(QS) in (8) provides a convenient measure of theaggregate welfare loss (reflecting a reduction in aggregate benefit) associated with imposing the production quotas QS.

Is there a simple way to measure the aggregate welfare loss WLS(QS)? Under differentiability and using the fundamental theorem of calculus, note that WLS(QS) in (8) can be written as

WLS(QS) =,

= ,

=,(9)

where WS(U, Q)/Qi = qSic(Q) is obtained from applying the envelop theorem to (5). Equation (9) shows that the aggregate welfare loss WS(QS) can be conveniently measured from the unit quota rents qSc(Q).

2.2.Graphical Illustration

The above results are illustrated below in Figures 1 and 2. Consider the simple case where m = 1 and where the quota Qsrestricts aggregate production, with Qs< S*. Assume that producers capture the quota rent (this would occur if producers can sell their product at the same price paid by consumers).

Figure 1: Market Allocation under Aggregate Production Quota QS< S*

Price p

A

pc

qSc B C

pc - qSc

D

E

0QS S* Aggregate Quantity Q

Benefit = area (A + B + D + E)

Cost = area E

Quota rent = area B

Consumer surplus = area A

(= Benefit – Expenditure = area (A + B + D + E) – area (B+ D + E))

Producer surplus (including Quota rent) = area (B+ D)

(= Revenue – Cost + Quota rent = area (D + E) – area E + area B)

Net benefit = Benefit – Cost = area (A + B + D)

= Consumer surplus + Producer surplus (including Quota rent)= area (A + B + D)

Figure 2: Comparison with the efficient allocation

Price p

A

pc

B1 C1

p*

qSc B2 C2

pc - qSc

D

E1 E2

0QS S* Aggregate Quantity Q

Welfare Evaluation / (Gross) Benefit / Cost / Consumer surplus, CS / Quota rent / Producer surplus, PS (including quota rent) / Net Benefit
= Benefit – Cost
= CS + PS
Efficient
allocation / A+B1+C1
+B2+C2
+D+E1+E2 / E1+E2 / A+B1+C1 / none / B2+C2+D / A+B1+C1
+B2+C2+D
Allocation under production quota
Qs / A+B1+B2
+D+E1 / E1 / A / B1+B2 / D
+B1+B2 / A+B1+B2+D
Welfare Change (“efficient” - “quota”) / C1+C2+E2 / E2 / B1+C1 / -- / C2-B1 / C1+C2
WELFARE LOSS (WL) due to quota / -- / -- / B1+C1
(> 0) / -- / C2-B1
(< 0 if B1>C2)
(> 0 if C2>B1) / C1+C2
(> 0)

Figure 2 evaluates the effects of the quota QS compared to the efficient allocation. And the above table summarizes the welfare effects. The key results are:

• Restricting supply (QS < S*) increases consumer price (pc > p*).
• The price increase (pc – p*) is less than the price wedge qSc.
• The magnitude of the price effects depends on the slopes/elasticities of the supply and demand functions.
• Imposing an aggregate production quota QS is inefficient, the aggregate welfare loss being measured by the area (C1+C2).
• In a way consistent with (9), the area (C1+C2) can be approximated as

WLS(QS)  (1/2) [S* - QS] qSc.

• The magnitude of the welfare loss depends on the slopes/elasticities of the supply and demand functions.
• Consumers loose under quota QS, as consumer surplus CS declines by the area (B1+C1).
• Producers can gain from a production quota QS (e.g., producer surplus would increase when (area B1) > (area C2)).
• The magnitudes of the gains/losses depend on the slopes/elasticities of the supply and demand functions.
• Even if producers can gain under quota Qs, their gain can never compensate the welfare loss for consumers.
• Thus imposing quota QS can have significant distribution effects.

2.3.Extensions

Our analysis has focused on the case of aggregate production quotas QS. In policy analysis, this would correspond to a quantity policy, where QS is the policy instrument. An example is the “US biofuel mandate” that restricts the amount of corn used to produce food (and diverts it to produce fuel). In this context, the policy would induce market adjustments, including price change (from p* to pc) and the quota rents qSc.

But the above analysis can have an alternative interpretation. It can also cover a price policy, where policy makers decide to tax or subsidize producers. In this case, the policy makers would choose qSc, where qSic > 0 would correspond to a unit tax on the production of the i-th good, while qSic < 0 would correspond to a unit subsidy on the production of the i-th good. With qSas policy instruments, the associated market response would include price adjustments (from p* to pc) and supply adjustments (from S* to QS). This can be illustrated using equation (7). Equation (7) states that the unit quota rent qSicis decreasing in QSi: qSic ispositive when the quota QSi is small and restricts supply, but qSicis negative when the quota QSi is large and expands supply. When qS are treated as policy instruments, then (7) has the following alternative interpretation: taxes (qSic > 0) reduce the incentive to produce (thus lowering supply), while subsidies (qSic< 0) provide an incentive to expand supply. This indicates that there is a one-to-one mapping between QS and qSc: we can switch from aggregate supply QS to its implied qSc (under a quantity policy) to taxes/subsidies qScand their implied aggregate supply QS (under a pricing policy).

Importantly, the welfare effects and price effects of these two interpretations would be similar. It means that a pricing policy (choosing taxes/subsidies qS 0) would be inefficient as it contributes to reducing aggregate welfare. It also means that the “price wedges” qS can be used in the characterization of both quantity policy and price policy.

Yet, there are two key differencesbetween these two interpretations.

• The first key difference is: the quantity policy would treat QS as policy instruments and qSas endogenous market response, while the price policy would treat qS as policy instruments and QS as endogenous market response.
• The second key difference is: the issue of who captures the quota rent in a quota policy becomes the issue of who pays for the subsidies (typically the “taxpayers”) or “captures” the taxes paid in the implementation of a pricing policy. Of course, who pays the cost of a pricing policy has implications for welfare distribution…
1. The Case of Domestic and Trade Policies

Policies are often designed to affect a subset of economic agents. This is the case of domestic policies and trade policies. Analyzing such policies requires a more disaggregate analysis that can support an investigation of distribution issues. For that purpose, we consider the case of two regions: region A and region B. Denote the set of consumers in region r be Ncr, r  {A, B}, and by Nsrthe set of production units in region r, r  {A, B}. We focus our attention on policies implemented in region A (with the understanding that such policies can have economic and welfare implications for both regions A and B). In this context, region A can be interpreted as the country where the policies are implemented, while region B is the “rest of the world”.

3.1.Efficiency Analysis

We refine the analysis presented above in the context of two regions (A, B). Denote by Dr (D1r, …, Dmr) = jNcr xai Rm the vector of aggregate demandfor m final goods in region r (A, B),with D = DA + DB. And denote by Sr (S1r, …, Smr) = iNsr yai Rm the vector of corresponding aggregate supplyin region r (A, B), with S = SA + SB. Let U = (UA, UB), where UA = {Ui: i  NcA} and UB = {Ui: i  NcB}. Denote aggregate benefit by B(D, U) = BA(DA, UA) + BB(DB, UB), where Br(Dr, Ur) is the benefit function for all consumers in region r, r  (A, B). Anddenote aggregate cost by C(SA, SB) = CA(SA, SB) + CB(SA, SB), where Cr(SA, SB) is the production cost for all producers in region r, r  (A, B). Note that this allows for external effects across regions (as SA can affect CB() and SB can affect CA()).

In the context of two regions (A, B), the maximal allocation given in equation (1) can be written as

W(U) = MaxD,S {BA(DA, UA) + BB(DB, UB) - C(SA, SB): DA+ DB SA + SB}, (10)

where W(U) is the maximized aggregate net benefit. In addition, the set of U satisfying W(U) = 0 identifies the Pareto utility frontier. It means that an efficient allocation is one that maximizes aggregate net benefit (as given in (10)) and then redistribute all net benefit (with W(U) = 0).

Equation (10) is a constrained optimization problem. It can be alternatively written in terms of the Lagrangean L = BA(DA, UA) + BB(DB, UB) - C(SA, SB) + p [SA + SB- DA- DB], where p = (p1, …, pm)  R is a vector of Lagrange multipliers for the constraints [DA+ DB SA + SB]. In a market economy, the vector p represents market prices reflecting resource scarcity in the economy. In this context, (10) can be alternatively written as

W(U) = Minp 0MaxD,S {BA(DA, UA) + BB(DB, UB) - C(SA, SB)

+ p [SA + SB- DA- DB]},(11)

which has for solution DA*, DB*, SA*, SB*and p*.

Below, we investigate situations where actual allocations are not necessarily efficient. As noted above, we examine economic policies specific to region A. In particular, we study situations where SA SA* and DA DA*. We consider the case of a domestic policywhere supplies SAare constrained to be equal to QA (QA1, …, QAm), where QA are “production quotas” for region A. And we consider the case of trade policywhere net exports from region A, SA- DA, are restricted to be equal to TA, where TA = (TA1, …, TAm) are “trade quotas” for region A. The trade quotas takes the form of an export quota when (SAi – DAi) > 0, and an import quota when (SAi – DAi) < 0. Introducing thesepoliciesas additional constraints into (10) gives

WA(U, QA, TA) = MaxD,S {BA(DA, UA) + BB(DB, UB) - C(SA, SB):

DA+ DB SA + SB;SA = QA, SA - DA= TA}, (12)

Equation (12) is a constrained optimization problem which can be alternatively written in terms of the Lagrangean L = BA(DA, UA) + BB(DB, UB) - C(SA, SB) + p [SA + SB- DA- DB] + qA [QA – SA] + qT [TA - SA + DA], where p = (p1, …, pm)  R is a vector of Lagrange multipliers for the constraints [D  S], qA = (qA1, …, qAm)  Rm is a vector of Lagrange multipliers for the production quota constraints [SA = QA], and qT = (qT1, …, qTm)  Rm is a vector of Lagrange multipliers for the import quota constraints [SA - DA= TA]. In this context, (12) can be alternatively written as

WA(U, QA, TA) = Minp0,qA,qTMaxD,S {BA(DA, UA) + BB(DB, UB) - C(SA, SB):

+ p [SA + SB- DA- DB] + qA [QA- SA] + qT [TA - SA + DA]}, (13)

which has for solution DAo, DBo, SAo, SBo, po, qAoand qTo. Under differentiability and assuming that po > 0, the associated first-order necessary conditions are

BA/DA = p - qT, (14a)

BB/DB = p, (14b)

C/SA = p - qA - qT, (14c)

C/SB = p, (14d)

QA = SA = -SB+ DA+ DB, (14e)

TA = SA- DA. (14f)

Equations (14a)-(14f) characterize the market allocation under domestic and trade policies (QA, TA). Equation (14b) and (14d) identify p as the prices faced by producers as well as consumers in region B. They state that marginal benefit pricing and marginal cost pricing applies to region B (where there is no domestic or trade policy). Equation (14a) identifies (p - qT) as “distorted prices” facing consumers in region A. And (14c) identifies (p - qA - qT) as “distorted prices” facing producers in region A. Finally, (14e) and (14f) state that, in addition to market equilibrium, production quotas and trade quotas are imposed for region A.

Expression (14a) indicates that qT are price wedges that separate consumer prices in region A from producer/consumer prices in region B. As Lagrange multipliers, qT can be interpreted as shadow prices of the trade quota constraints [TA = SA- DA]. The vector qT can also be interpreted as unit trade quota rents, with [qTTA] measuring theaggregatetradequota rent. These quota rents get captured by market participants. Who capture these quota rents affects the distribution of welfare…

Similarly, expressions (14a) and (14c) indicate that qA are price wedges that separate consumer prices from producer prices in region A. As Lagrange multipliers, qA can be interpreted as shadow prices of the production quota constraints [QA = SA]. The vector qA can also be interpreted as unit production quota rents, with [qT QA] measuring the aggregate production quota rent. These quota rents get captured by market participants. Again, who capture these quota rents affect the distribution of welfare…

The shadow prices qAo solving (13) depend on QA: qAo(QA). In general, these shadow prices can be shown to be decreasing in QA, where

qAio(QAi’)  qAio(QAi) for any QAi’ > QAi, i = 1, …, m. (15a)

Equation (15a) means that the price wedge qAio(QA) starts positive for “small QAi”, declines to zero as QAi rises, and turns negative as QAi becomes “large”.

Similarly, the shadow prices qTo solving (13) depend on TA: qTo(TA). In general, these shadow prices can be shown to be decreasing in TA, where

qTio(TAi’)  qTio(TAi) for any TAi’ > TAi, i = 1, …, m. (15b)

Equation (15b) means that the price wedge qTio(TA) starts positive for some negative TAi, declines to zero as TAi rises, and turns negative as TAi becomes “large”.

The above discussion indicates that the shadow prices qAo and qTo can be positive or negative depending on the supply/demand characteristics of each region and the policy choice (QA, TA). In this context, equation (14c) is of special interest: it identifies the “distorted prices” facing producers in region A by [p - qA - qT]. Accordingly, any increase (decrease) in (qA + qT) implies a decrease (increase) in the distorted prices affecting producers in region A. It means that both domestic policy and trade policy have direct effects on supply incentives in region A. In addition, negative values taken by either qA or qTare associated with policies favoring producers in region A (as they provide incentives to increase supply SA). Andpositive values taken by either qA or qT are associated with policies against producers in region A (as they provide disincentives to produce in region A).

3.2.Welfare Analysis

What the welfare implications of the production quotas QA and trade quotas TA? This can be evaluated by comparing the aggregate benefit obtained with quotas (as given by WA(U, QA, TA) in (12) or (13)) and without quotas (as given by W(U) in (10) or (11)). Define the change in aggregate welfare (or aggregate benefit) due to (QA, TA) as WLA(QA, TA)  W(U) – WA(U, QA, TA). Note that adding constraints in a maximization problem cannot increase the value of the objective function. Then, comparing (10) and (12) gives