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Pareto Efficiency

1. Introduction

Consider an economy facing the allocation of n goods, involving M firms and N households.

The i-th household has preferences represented by a utility function ui(xi), where xi = (xi1, , xin)

Rn is a vector of the goods consumed by the i-th household, xi 0, i = 1, , N. We assumethroughout that ui(xi) is a quasi-concave function of x i, i = 1, , N. Denote by x = (x1, , xN) thevector of all consumption decisions.

Note: This assumes that the consumer goods are private goods. In other words, x i is a vector ofgoods consumed only by the i-th household. This means of the benefits generated by thegoods xi are captured entirely by the i-th household. Thus, at this point, we rule out theexistence of externalities and/or public goods for the consumption goods x.

The j-th firm chooses netputs yj = (yj1, , yjn) Rn, where outputs are positive and inputsnegative, j = 1, , M. Denote by y = (y1, , yM) the vector of all production decisions. The

technological feasibility of producing y is denoted by the feasible set Y, with y Y RMn.

Note: This allows the production goods (denoted by the netputs y Y) to include private goods,public goods, as well as externalities. Indeed, depending on the nature of the feasible setY, the netput yj associated with the j-th firm can affect the production possibility of anyor all of the M firms. See below.

Definition: A feasible allocation is an allocation z = (x, y) satisfying

xi 0, i = 1, , N, (1a)y Y, (1b)

and

i xi j yj. (1c)

Expression (1c) is simply a quantity balance: it states that aggregate consumption i xi cannotexceed aggregate production j yj.

Throughout, we will make the following two assumptions.

Assumption A1: A feasible allocation exists.

Assumption A2: There exists a feasible allocation such that j yj > 0.Assumptions A1 and A2 are intuitive. Without A1, no allocation would be feasible and askingwhich allocation to choose would be irrelevant. And assumption A2 states that it must be feasible

to produce a positive aggregate quantity of all n commodities.

Among all the feasible allocations, we would like to identify the ones that seem more desirablefrom a social viewpoint.

2. Pareto efficiency

Definition: An allocation is Pareto efficient if it is feasible and there is no other feasibleallocation that can make one household better off without making any other worse off.

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Let z* = (x1*, , xN*; y1*, , yM*) = (x*, y*) be a Pareto efficient allocation. It follows that

there does not exist any other feasible allocation z = (x, y) such that ui(xi) ui(xi*) for allhouseholds i = 1, , N, and ui'(xi') > ui(xi'*) for some household i'.

Alternatively, an allocation z# is not Pareto efficient (or is Pareto inefficient) if there exists some

other feasible allocation z = (x, y) such that ui(xi) ui(xi#) for all households i = 1, , N, andui'(xi') > ui(xi'

#) for some household i'. Pareto inefficiency means that it is possible to make at leastone household better off without making any other worth off. Intuitively, Pareto inefficiencyseems quite undesirable from a social viewpoint. It states that some resources are being wasted inthe sense that they could be used better so as to improve the welfare of households in general.

Note: Using the Pareto criterion, welfare levels are expressed entirely in terms of householdwelfare. This does not mean that firm welfare is irrelevant. Rather, it means that that firmwelfare is relevant, but only to the extent that production activities contribute toincreasing consumer welfare. In this context, production activities are not an end; ratherthey are a means of generating goods that will eventually be consumed by households.

It would be very useful to develop insights into the identification of Pareto efficient/inefficientallocations. This would help address two important issues:

Identifying Pareto inefficient allocations can help discover which existing decision rulesare inefficient.

By identifying Pareto efficient allocations, we can gain some insights into improveddecision rules that can help enhance household and social welfare.

Note: The efficiency of decision rules apply at all levels: the micro level (e.g., firm or household)as well as the aggregate level (e.g., government policy, trade policy).

3. Identifying efficient allocations

To identify Pareto efficient allocations, we need to rely on some household welfare measurement.

It will convenient to rely on the benefit function. As we have seen earlier, the benefit function forthe i-th household is

bi(xi, Ui) = max { : ui(xi - g) Ui, (xi - g) 0}, (2)if there is a satisfying ui(xi - g) Ui and (xi - g) 0},

= - otherwise.

where g Rn is reference bundle of private goods satisfying g 0, g 0. bi(xi, Ui) measures thequantity of the bundle g the i-th household is willing to give up to reach xi starting with the levelof utility Ui. It is a measure of the i-th household's benefit associated with consuming xi. If thevalue of one unit of the bundle g is $1, then bi(xi, Ui) is the i-th household willingness to pay toobtain xi starting at the utility level Ui. (To simplify the notation, we drop the argument g in thebenefit function; keep in mind that g is taken to be the same for all consumers in the discussionbelow).

We have shown earlier that, under the assumption that ui(xi) is a quasi-concave function, thebenefit function bi(xi, Ui) is concave in xi, and non-increasing in Ui.

We will focus our attention on the aggregate benefit function defined as the sum of the individualbenefit functions across all households:

B(x, U) = i bi(xi, Ui), (3)

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where x = (x1, , xN) and U = (U1, , UN). From the properties of bi(xi, Ui), it follows that theaggregate benefit function B(x, U) is concave in x, and non-increasing in U.

Intuitively, we expect that efficient allocations will maximize aggregate benefit. This suggestconsidering the following allocations.

Definition: An allocation z is maximal if it maximizes the aggregate benefit function B(x, U) in(3) subject to the feasibility conditions (1a)-(1c). Thus a maximal allocation is a solutionz*(U) to following optimization problem

W(U) = Maxz {B(x, U): equations (1a), (1b), (1c)}, (4)where W(U) is the indirect objective function.

The indirect objective function W(U) = B(x*(U), U) is the largest feasible aggregate benefit thatcan be obtained to reach utility levels U = (U1, , UN). There are three possibilities.

If the largest feasible aggregate benefit W(U) is negative, this means that reaching utilityU requires a negative aggregate quantity of the bundle g. This implies that reachingutility level U = (U1, , UN) is not feasible.

If the largest feasible aggregate benefit W(U) is positive, this means that reaching utilitylevel U can be attained using a positive aggregate quantity of the bundle g. This impliesthat reaching utility level U = (U1, , UN) is feasible. In this context, W(U) can beinterpreted as a measure of aggregate surplus (expressed in terms of quantity of thebundle g). When W(U) > 0, this surplus can in general be redistributed among the Nhouseholds.

If the largest feasible aggregate benefit W(U) is equal to zero, this corresponds to afeasible allocation where aggregate benefit have been maximized but there is noaggregate surplus to redistribute.

This suggests considering the following allocations.

Definition: An allocation z is zero-maximal if it is maximal and if U is chosen such that

W(U) = 0. (5)

Thus, a zero-maximal allocation is feasible, maximizes aggregate benefit, and corresponds to asituation where there is no aggregate surplus to redistribute. We show next that, under someregularity conditions, a zero-maximal allocation identifies a Pareto efficient allocation.

Proposition 1: Assume that there is at least one household that is non-satiated in g (with u i(xi + g) being strictly increasing in for some i). If the allocation z* is Pareto efficient, then itis zero maximal.

Proof: The allocation z* is feasible. It follows from (2) that bi(xi, Ui) 0 for all i = 1, , n.Assume that bi' > 0 for some household i'. This implies that B > 0. But the aggregatesurplus B can be redistributed to the household that is non-satiated in g. This would makethis household better off without making any other worse off, thus contradicting Paretoefficiency. It follows that Pareto efficiency implies zero maximality.

Proposition 2: If z* is zero maximal, then it is Pareto efficient compared to all feasible allocationssatisfying xi > 0.

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Proof: Assume that there is a feasible allocation z satisfying xi > 0, where ui(xi) ui(xi*) for all i,but with ui'(xi') > ui'(xi'*) for some household i'. This means that z* is not Pareto efficient.

It follows from (2) that bi(xi, ui(xi*)) 0 for all i, and bi'(xi', ui'(xi'*)) > 0. This implies thatz cannot be zero-maximal. Thus, zero maximality implies Pareto efficiency.

Propositions 1 and 2 establish close relationships between Pareto efficiency and zero-maximality.

They state the equivalence of two concepts when the following two conditions hold1. Non-satiation in g for at least one household.2. xi = (xi1, , xin) > 0.

These two conditions appear reasonable. Condition 1 is intuitive. Condition 2 states that, underPareto efficiency, consumption must be positive. In the discussion presented below, we willassume that both conditions are satisfied. Thus, we proceed with our analysis assuming thatPareto efficiency and zero-maximality are equivalent. Note that such equivalence holds withoutimposing any restriction on the feasible set Y.

Under zero-maximality, the distributable aggregate surplus W(U) is zero. This provides a simpleand intuitive interpretation of Pareto efficiency. An allocation is Pareto efficient when:

First, resource allocation z = (x, y) is chosen such as to maximize aggregate benefit B(x, U),conditional on U = (U1, , UN) (as stated in (4)).

Second, the level of utilities U = (U1, , UN) is chosen such that the associated distributablesurplus W(U) is entirely redistributed to the N households.

We have seen above that a positive distributable surplus W(U) 0 corresponds to a feasibleallocation. In other words, the set of feasible utilities U = (U1, , UN) is given by {U: W(U) 0}. The boundary of this feasible set is of special interest.

Definition: The Pareto utility frontier is given by the set of utilities U = (U1, , UN) satisfyingW(U) = 0.

Note that W(U) = 0 is an equation involving N variables: U1, , UN. This equation typically hasan infinite number of solutions. The set of its solutions constitutes the Pareto utility frontier. Itmeans that an allocation is efficient if and only if it is associated with a point on the Pareto utilityfrontier. Alternatively, an allocation is inefficient if it generates utilities U that are below thePareto utility frontier. This is illustrated in figure 1 in the context of a two-household economy (N= 2).

Figure 1U1 A

Feasible utilities:

W(U1, U2) 0.C

B

U2

4

Pareto utility frontier:W(U

1, U

2) = 0.


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This allows the identification of feasible points that are inefficient. The feasible point C in figure1 illustrates this. Point C is Pareto inefficient since there exist feasible points to the north-east ofC that can make both households better off (i.e., with higher utilities U1 and U2). As such, theallocation associated with point C is not socially desirable.

Figure 1 also shows that any point along the line between A and B is Pareto efficient. This

illustrates that the Pareto efficiency criterion says little about distribution issues. Indeed, thereexist efficient allocations that are not equitable. For example, point A corresponds to an efficientpoint that benefits greatly household 1 at the expense of household 2. Alternatively, point B is anefficient point that benefits greatly household 2 at the expense of household 1. The Paretocriterion provides no information about whether point A is better (or worse) than point B.

Again, it is worth stressing that all these results are obtained without imposing any restriction onthe production technology, as represented by the feasible set Y (e.g. it holds even under a non-convex technology, and in the presence of externalities across firms). And they hold whether ornot markets are present.

Note: Define the shortage function

S(y) = min {j j: (y1 - 1 g, y2 - 2 g, ) Y},if there are 's satisfying (y1 - 1 g, y2 - 2 g, ) Y,

= otherwise.The shortage function S(y) has the following properties:

y Y implies that S(y) 0. Under free disposal, S(y) 0 implies y Y. Under free disposal, the shortage function S(Y) is non-decreasing in y. S(y1 + 1 g, y2 + 2 g, ) = i i + S(y1, y2, ). S(y) is convex in y if the set Y is convex.

Proof: Consider any y = (y1, y2, ) and y' = (y1', y2', ). Let *(y) = ( 1*(y), 2*(y), )denote the solution for = ( 1, 2, ..) in the definition of the shortage function.

Assuming that S(y) and S(y') are finite, it follows that (y1 - 1*(y) g, y2 - 2*(y) g, ) Y and (y1' - 1*(y') g, y2' - 2*(y') g, ) Y. Let y" = y + (1- ) y', for any [0, 1]. Under the convexity of the set Y, this implies (y1" - [ 1*(y) + (1- ) 1*(y')]g, y2" - [ 2*(y) + (1- ) 2*(y')] g, ) Y. By definition of the shortage function,this gives

S(y") S( y1 + (1- ) y1', y2 + (1- ) y2', ) j [ j*(y) + (1- ) j*(y')]= S(y) + (1- ) S(y'),

implying that the function S(y) is convex in y.

Thus, under free disposal, {y: S(y) 0} = Y, and S(y) = 0 can be interpreted as an implicitmulti-input multi-output aggregate production function. And under a convex technology (i.e.

a technology exhibiting decreasing marginal productivity), S(y) is a convex functionreflecting increasing marginal cost.

Finally, when the reference bundle g is the same one as used in the definition of the benefitfunctions, then the maximal allocation (4) can be alternatively written as

W(U) = Maxz {B(x, U) - S(y): equations (1a) and (1c)}, (4')

where (1a) is: xi 0, i = 1, , N; (1b) is: y Y; and (1c) is: i xi j yj.

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Proof: First, start with equation (4): W(U) Maxz {B(x, U): equations (1a), (1b), (1c)}. Notethat any feasible z (x, y) satisfies S(y) 0. It follows that

W(U) Maxz {B(x, U): equations (1a), (1b), (1c)}, Maxz {B(x, U) S(y): equations (1a), (1b), (1c)}, Maxz {B(x, U) S(y): equations (1a) and (1c)}.

where (1a) corresponds to {xi 0, i = 1, , N}, (1b) corresponds to {y Y}, and (1c)corresponds to {i xi j yj}.Thus, to prove (4'), we now need to show that W(U) Maxz {B(x, U) - S(y): equations(1a) and (1c)}. This inequality clearly holds if S(y) = . Consider a point y (notnecessarily feasible) satisfying S(y) < . Denote * = ( 1*, 2*, ..) the solution for= ( 1, 2, ..) in the definition of the shortage function, where S(y) = j j*(y). Let y" =(y1 - 1*(y) g, y2 - 2*(y) g, ), where *(y) = ( 1*(y), 2*(y), ) It follows that y" Y, i.e. that y" satisfies equation (1b). Define x" = (x1 - 1 g, x2 - 2 g, ...), where the 's are chosen such that i i = j j*(y), and (x, y) are chosen to satisfy x 0, i xi j yj, and (xi - i g) 0 for i = 1, 2, .... It follows that i xi = i xi - i i g j yj -j j*(y) g = j yj. Thus, (x", y") is a feasible point satisfying equations (1a), (1b) and(1c). In addition, note that B(x", U) = i bi(xi", Ui) = i bi(xi - i g, Ui) = -i i + B(x,U) = B(x, U) - S(y). It follows that

W(U) Maxz {B(x, U): equations (1a), (1b), (1c)}

B(x", U), since (x", y") is a feasible point,= i bi(xi, Ui)= i bi(xi - i g, Ui)= i bi(xi, Ui) - i i= i bi(xi, Ui) - S(y)= B(x, U) - S(y).

This gives

W(U) Maxz {B(x, U) - S(y): equations (1a) and (1c)}.

By being subtracted from aggregate benefit in (4), this makes it clear that the shortagefunction S(y) measures the social cost of producing y. Again, this holds without imposingany restriction on the production technology Y (e.g. it holds even under a non-convextechnology and in the presence of externalities across firms). And this holds whether or notmarkets are present.

Note that (4') differs from (4) in two ways: 1/ the technology constraint (1b), y Y, ispresent in (4) but not in (4'); 2/ it has been replaced by subtracting the shortage function S(y)from the objective function in (4'). As such, equation (4') is easier to use empirically.

Note: The aggregate net benefit function B(x, U) - S(y) in (4) has a useful alternativeinterpretation. To see that, consider the case where there are no externalities across firms, i.e.

where S(y) = j Sj(yj), where Sj(yj) is the shortage function for the j-th firm. Then, theaggregate net benefit function can be written as: i bi(xi, Ui) - j Sj(yj). Underdifferentiability, applying the fundamental theorem of calculus gives bi(xi, Ui) = bi0 +

ikn

k=1 0

x

(bi/xik) dxikwhere (bi/xik) measures the marginal benefit of xik, and Sj(yj) = sj0

+jkyn

k=1 0 (Sj/yjk) dyjkwhere (Sj/yjk) measures the marginal cost of yjk. Consider the

case where p denotes the market prices satisfying the market clearing condition: j p yj = i pxi. It follows that the aggregate net benefit can be written as:

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i bi(xi, Ui) - j Sj(yj),= i bi(xi, Ui) - i p xi + j p yj - j Sj(yj),= i [bi(xi, Ui) - p xi] + j [p yj - Sj(yj)],

= i [bi0 +ikn

k=1 0

x

(bi/xik) dxik- p xi] + j [p yj - sj0 -jkyn

k=1 0 (Sj/yjk) dyjk],

= K + i CSi + j PSj,

where K i bi0 + j sj0, CSi [ikn

k=1 0

x

(bi/xik) dxik- p xi] can be interpreted as the i-th

consumer surplus, and PSj [p yj -jkyn

k=1 0 (Sj/yjk) dyjk] can be interpreted as the j-th

producer surplus. In this case, except for a constant of integration K, the aggregate netbenefit can be expressed as the sum of producer surplus and consumer surplus across allagents. This result can be very useful in the empirical evaluation of both efficiency anddistribution issues. See homework #4.

4. Welfare measurements

The aggregate benefit B(x, U) can provide a convenient way to measure aggregate welfare. Ingeneral, B(x, U) measures the aggregate quantity of the bundle g that the N households are

willing to give up starting with utility levels U to reach consumption levels x. And in the casewhere the price of the bundle g is one unit, then B(x, U) is the aggregate amount of money the Nhouseholds are willing to pay starting with utility levels U to reach consumption levels x.

Given the maximal allocations defined in (4), aggregate benefit B(x, U) can be compared withW(U). From (4), we have

W(U) B(x, U) for all feasible z = (x, y). (6)

And from (5), a zero maximal/Pareto efficient allocation satisfies W(U) = 0. This suggestsmeasuring aggregate welfare loss by the amount

W(U) - B(x, U) 0

When U is chosen such that W(U) = 0, this gives

-B(x, U) 0.

This suggests that, with U chosen such that W(U) = 0, -B(x, U) 0 is a measure of the welfareloss associated with inefficient resource allocation. This is illustrated in figure 2, where point A isPareto inefficient (being below the Pareto utility frontier), and where -B(x, U) provides a welfaremeasurement of the distance between point A and the Pareto utility frontier. And in the casewhere the price of the bundle g is one unit, then -B(x, U) is an aggregate monetary amount of thewelfare loss associated with Pareto inefficiency.

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Figure 2

U1

A

U2

Again, these results are very general in the sense that they apply without imposing any restrictionon the production technology, as represented by the feasible set Y. And they hold whether or notmarkets are present.

5. Shadow pricing

The zero-maximal allocations just identified provide useful insights in to the valuation ofresources. To see that, consider the Lagrangean associated with the maximization problem in (4)

L(z, U) = B(x, U) + T [j yj - i xi], (7)

where = ( 1, , n) is a (n 1) vector of Lagrange multipliers associated with constraint(1c). Note that the maximization problem (4) can be equivalently written in terms of the saddlepoint of the Lagrangean (7) under some regularity conditions. These regularity conditions are

1. there is a feasible point where the constraint (1c) is not binding (Slater's condition)2. The objective function and constraint functions are concave3. The feasible set is convex.

Here, condition 1 (Slater's condition) corresponds to assumption A2. Condition 2 is satisfied since

the objective function B(x, ) is concave, and the constraint (1c) is linear hence concave. Finally,since the feasible set for x 0 is convex, condition 3 is satisfied if the following assumptionholds.

Assumption A3: The feasible set Y is convex.

Assumption 3 imposes convexity restrictions on the production technology. Intuitively, it imposesdiminishing marginal productivity in production activities. This is important. While the resultspresented above applied for any production technology, the assumption of diminishing marginalproductivity is needed to obtain the results presented below.

Using the saddle point characterization under assumptions A2 and A3, a maximal allocation

(defined in (4)) can be equivalently expressed in terms of the saddle point of the Lagrangean (7):W(U) = Min 0 Maxz {L(z, U): x 0, y Y}, (8a)

which has for solution z*(U), *(U). Note that (8a) can be written asW(U) = Min 0 Maxz {B(x, U) + T [j yj - i xi]: x 0, y Y},

= Min 0 {Maxx 0 {B(x, U) - Ti xi} + Maxy { Tj yj: y Y}},= Min 0 {Maxx 0 {i bi(xi, Ui) - Ti xi} + Maxy { Tj yj: y Y}},= Min 0 {i [Maxxi 0 {bi(xi, Ui) - T xi}] + Maxy { Tj yj: y Y}}, (8b)

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1, U

2) = 0.

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The Lagrange multipliers *(U) have the standard interpretation: they measure the shadow valueof the constraint (1c). In the case where the bundle g has a unit price (i.e., where T g = 1), theyare the shadow prices of the n commodities .

In addition, note that (8b) can be decomposed into the following sub-problems

1. ( ) = Maxy { Tj yj: y Y}, (8c1)2. -Ei( , Ui) = Maxxi 0 {bi(xi, Ui) - T xi}, (by duality, assuming that T g = 1), (8c2)3. E( , U) = i Ei( , Ui), (8c3)4. W(U) = Min 0 { ( ) - E( , U)}. (8c4)

Expression (8c1) implies profit maximization for production activities, using the Lagrange

multipliers as prices for the n commodities. It defines the aggregate indirect profit function ( ).Expression (8c2) defines the expenditure function for the i-th household, again using the

Lagrange multipliers as prices.Expression (8c3) defines the aggregate expenditure function as the sum of the individualexpenditure functions across all households.

Finally, expression (8c4) defines the Lagrange multipliers/ shadow prices * as the solution of aminimization problem involving aggregate profit net of aggregate expenditures E. Equation(8c4) also defines the distributable surplus W(U). It provides an alternative intuitive interpretationof the distributable surplus: it is the value of aggregate profit net of consumer expenditures,

evaluated at *.

It follows that, under assumptions A2 and A3, z*(U) obtained from (8c1)-(8c3) provides a

characterization of a maximal allocation. In this context, the feasibility condition W(U) 0 canbe interpreted intuitively in terms of the aggregate budget constraint: ( ) - E( , U) 0(evaluated at *), stating that consumers cannot spend more than aggregate income .

To obtain a zero-maximal allocation, from (5), it remains to choose U such that the distributable

surplus is completely redistributed: W(U) = 0. Then, from (8c4), W(U) = 0 is equivalent to stating

that the aggregate budget constraint must be binding: ( ) - E( , U) = 0 (evaluated at *), asthe distributable surplus is completely redistributed among the N households.

It should be emphasized that, so far, we have not assumed the existence of markets. Thus, the role

of shadow prices * is relevant with or without markets. This stresses that the concept of Paretoefficiency is relevant whether or not resource allocation is supported by a market economy.

6. Competitive market allocations

But how can we attain efficient allocations? There are many possibilities. In this section, weexplore the role of markets in attaining Pareto efficiency.

Assumption A4: The production technology is Y = (Y1 Y2 YM), where yi Yi, i =1, , M.

Assumption A4 means that the production technology for y = (y1, , yM) is non-joint acrossfirms in the sense that the feasible set for the j-th firm, Yj, is independent of the other firms. Itfollows that the production goods yj are private goods for the j-th firm, implying the absence of

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external effect of the decisions of each firm on any other firm. In this context, assumption A4implies a situation ofwell-defined property rights and no externalities.

Note: All the results presented above applied without assumption A4. It means that all our earlierresults allowed forexternal effects across firms , as represented by thejoint technology

Y, where y = (y1, , yM) Y allows for the decision of each firm to affect other firms.See below.

Assumption A4 implies some modifications to the analysis presented in the previous section.From (8a), under assumptions A2 and A3, a maximal allocation (defined in (4)) can beequivalently expressed in terms of the saddle point of the Lagrangean (7)

W(U) = Min 0 Maxz {L(z, U): x 0, yj Yj, j = i, , M}, (9a)

which has for solution z*(U), *(U). Note that (9a) can be written asW(U) = Min 0 Maxz {B(x, U) + T [j yj - i xi]: x 0, yj Yj, j = 1, , M},

= Min 0 {Maxx 0 {i Bi(xi, Ui) - Ti xi} + Maxy { Tj yj: yj Yj, j = 1, ,M }},

= Min 0 {i [Maxxi 0 {Bi(xi, Ui) -

T

xi}] + j Maxyj {

T

yj: yj Yj}}. (9b)

Now, let be actual market prices for the n commodities, and assume that the bundle g has aunit price where T g = 1. Then, in a way similar to (8b), note that (9b) can be decomposed intothe following sub-problems

1. j( ) = Maxyj { T yj: yj Yj}, (9c1)2. ( ) = j j( ), (9c2)3. -Ei( , Ui) = Maxxi 0 {Bi(xi, Ui) - T xi}, (assuming that T g = 1), (9c3)4. E( , U) = i Ei( , Ui), (9c4)5. W(U) = Min 0 { ( ) - E( , U)}. (9c5)

Expression (9c1) implies profit maximization for the j-th firm. Since this optimization problem

takes prices as given, this corresponds to the behavior of a competitive firm that takes marketprices as being exogenous. Equation (9c2) defines the aggregate profit function ( ) as thesum of the firm profit functions across all competitive firms.

Expression (9c3) defines the expenditure function for the i-th household, given market prices .Expression (9c4) defines the aggregate expenditure function as the sum of the individualexpenditure functions across all households.

Finally, expression (9c5) defines the competitive market prices * as the solution of aminimization problem involving aggregate profit net of aggregate expenditures E. Equation(9c5) also defines the distributable surplus W(U). Again, in the context of competitive markets, itprovides another intuitive interpretation of the distributable surplus: it is the value of aggregate

profit net of consumer expenditures, evaluated at *.

Finally, to obtain a zero-maximal allocation, from (5), it remains to choose U such that thedistributable surplus is completely redistributed: W(U) = 0. Then, from (9c5), W(U) = 0 is

equivalent to stating that the aggregate budget constraint must be binding: ( ) - E( , U) = 0(evaluated at *), as the distributable surplus is completely redistributed among the Nhouseholds.

Definition: A competitive equilibrium is an allocation satisfying

market prices * 0, normalized such that *T g = 1,

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z* = (x*, y*) satisfying (9c1)-(9c5), and where ( *) = E( *, U).

This means that a competitive equilibrium (z*, *) is characterized by Allocations satisfying the feasibility conditions x 0, yj Yj, j = 1, , N.

M competitive firms maximizing profit (as stated in (9c1). N households minimizing expenditures (as expressed through the dual relationship (9c3)) Allocations satisfying the aggregate budget constraint ( *) = E( *, U). Under differentiability, the Kuhn-Tucker conditions associated with the minimization

problem (9c5) are

/ - E/ 0, * 0, *T [ / - E/ ] = 0.Using the envelope theorem, this gives that

j yj* - i xi* 0, * 0, *T [j yj* - i xi*] = 0,which implies the feasibility constraint (1c).

We obtain the following important results.

First welfare theorem: Assume that assumption A4 holds, and that household preferences arenon-satiated. Then, a competitive equilibrium (z*, *) is Pareto efficient.

Proof: By definition, a competitive equilibrium is equivalent to (9c1)-(9c5). But this is equivalentto the saddle point of the Lagrangean in (9a). From the saddle point theorem, this implies

that z* is a maximal equilibrium. In addition, ( *) = E( *, U) implies that z* is azero maximal equilibrium. Thus, under non-satiation, z* = (x*, y*) is Pareto efficient.

Second welfare theorem: Assume that assumptions A2, A3 and A4 hold, and that householdpreferences are non-satiated. Assume that z* is a Pareto efficient allocation. Then, there

exists prices * 0 such that (x*, *) is a competitive equilibrium .

Proof: The Pareto efficient z* allocation is zero maximal. Under assumptions A2, A3 and A4, amaximal equilibrium z* implies that (z*, *) is a saddle point of the Lagrangean in (9a).But this implies (9c). And zero maximality implies that W(U) = ( *) - E( *, U) = 0.It follows that (z*, *) is a competitive equilibrium.

The two welfare theorems state that, under some regularity conditions, competitive markets cansupport a Pareto efficient allocation. This is a remarkable result in that it requires no centralplanning. It shows that competitive market prices can reflect the social value of commodities andprovide appropriate incentives for efficient decentralized decision making by firms andhouseholds. This is sometimes called Adam Smith's invisible hand. However, this result holdsonly under some regularity conditions. They are:

Assumption A2, stating the existence of a feasible allocation such that j yj > 0. Assumption A3: the feasible set for production activities, Y, is convex. Assumption A4: The production technology is non-joint: Y = (Y1 Y2 YM).

While assumption A2 appears non-controversial, assumptions A3 and A4 can be moreproblematic. There are situations where the feasible set Y is non-convex (thus violating A3). Forexample, this includes cases where the production technology exhibits increasing returns to scale(e.g., national defense, public utilities). Also, there are situations where the productiontechnology Y is joint. This includes the case where external effects exist (violating assumption

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A4). This includes the case of positive externalities (e.g., bees producing honey while they helpfertilize flowers) as well as negative externalities (e.g., pollution). In such situations, competitivemarkets typically fail to generate efficient allocations.

7. Gains from trade

Assume that the economy involves N locations (e.g., N countries), where each location is

associated with a representative consumer and a representative firm. Then, M = N, and eachlocation exhibits production as well as consumption activities and has the option to trade the ngoods with any other location. This raises the question: what are the benefits from trade?

To answer that question, consider the case where there are trade restrictions. Write these traderestrictions as

xi = yi, for some locations i = 1, , N. (10)

The constraint (10) implies that the restricted locations do not trade: they do not export and do notimport. As a result, they are in autarky: they consume all of their domestic production. Given(10), define the autarky-constrained maximal allocation

Wa(U) = Maxz {B(x, U): equ. (1a)-(1c)}, xi = yi for the locations under autarky}. (11)

It is clear that (11) is a restricted version of (4). Since the restrictions (10) can only reduce thefeasible set in (4), it follows that the maximum value attained Wa(U) cannot increase, implyingthat

Wa(U) W(U),or

W(U) - Wa(U) 0. (12)

This suggests that W(U) - Wa(U) 0 is a welfare measure of the gains from trade.

We have seen that the sets of U = (U1, , UN) satisfying W(U) = 0 identifies the (unconstrained)Pareto utility frontier. This suggests that the set of utilities U = (U1, , UN) satisfying W

a(U) = 0

can be interpreted as the trade-constrained utility frontier. From (12) and using the fact that thatW and Wa are non-increasing in U, it follows that the trade-constrained Pareto utility frontiermust be below the unconstrained Pareto utility frontier. This is illustrated in Figure 3, showingthat trade restrictions imply an inward shift of the utility frontier.

Figure 3

U1

U2

In this context, [W(U) - Wa(U)] 0 measures the distance between the Pareto utility frontier andthe trade restricted frontier. And in the case where U is chosen such that W(U) = 0 and g is worth

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1, U

2) = 0.

Trade-restricted utility

frontier: Wa

(U1, U2) = 0


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$1, it follows that -Wa(U) 0 provides a monetary measure of the gains from trade . Alternativelystated, -Wa(U) 0 is measure of the economic losses from the trade restrictions (10).

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