Lecture 3 Welfare Economics

Transcription

1 Lecture 3 Welfare Economics Welfare economics is essentially about udging the desirability of social outcomes. In this lecture we will introduce the normative concept of Pareto efficiency and two positive concepts; the core and general competitive equilibrium. Much of the lecture will be devoted to analyzing the relationships among these concepts. General equilibrium in pure echange To begin our discussion of welfare economics we will consider a general equilibrium model of a pure (barter) echange economy. In such an economy we will analyze n consumers trading in m markets. For now, there is no production, no prices, and no money. Define N = (,,..., n) -- the set of traders (consumers) in the economy. M = (,,..., m) -- the set of goods available in the economy. ssume that each good is of homogeneous quality. Two goods that are the same ecept for quality should be thought of as different goods. Furthermore, assume that each good is perfectly divisible. This last is not really necessary, but it makes the analysis easier. Endowments Instead of consumers purchasing a bundle of goods out of money income, consumers are endowed with a bundle of the goods in the economy. Let, w -- i's endowment of good. w i = (w i, w i,..., w im ) -- i's endowment bundle. W = (w, w,..., w n ) -- the economy's endowment. llocations -- consumers take their endowments to market and trade with others to obtain another bundle of goods which they consume. Let, -- i's final demand (consumption ) of good. i = ( i, i,..., im ) -- i's consumption bundle. X = (,,..., n ) -- an allocation. Feasibility Note that, w = w + w w, n is the aggregate amount of good available in the economy. Furthermore, 3.

2 is the aggregate consumption of good. = n Definition n allocation X is a feasible allocation if and only if w,. Note that a feasible allocation is one in which the final consumption of each good does not eceed the amount available at the start of trade. Preferences Each individual is fully described by their preferences and endowments. ssume that each individual trader's preferences are fully described by a utility function. For person i, u i ( i ) = u i ( i, i,..., im ). Most of the time we will assume that each utility function is strongly monotonic and strictly quasi-concave. Edgeworth o Let's restrict our economy to two individuals (, ) trading two goods (, ). To illustrate barter trade we construct what is known as an Edgeworth o. To construct this bo first consider our description (preferences and endowment points) of the two individuals. We do not need to draw budget lines since there are no prices in this economy. w w u u u u Α w Β w Now, take 's indifference map, rotate it 8 degree and place it on top of 's indifference map so the endowment points coincide. 3.

3 * w Β u * P * w + w w Α * W w u w w + w Remarks ) The economy's endowment is W = [w, w ] = [(w, w ), (w, w )]. It is also called the no-trade allocation. The indifference curves that pass through W gives us the utility levels in the absence of trade ( u, u). If these individuals are to trade with each other they have to do at least as well as this. ) The height of the bo is the amount of good available in the economy, while the width of the bo is the amount of good available. 3) ny point in the bo or on the boundary represents a feasible allocation. To see this take point P = [, ] = [(, ), (, )], and note that and + + = w + w (supply is equal to demand for good ) = w + w. (supply is equal to demand for good ) Trade in the Edgeworth o Consider the following: Β Q Α P W u u u u u u 3.3

4 W is the endowment point again. Trade between the two individuals will take place when both are made better off. For eample, they might trade to allocation P. However, they won't stop trading at P because there are still gains from trade to be had. t allocation Q, all gains from trade are ehausted. Notes about Q ) It is a feasible allocation. ) There is no other feasible allocation that will make one of them better off without harming the other. 3) n indifference curve of 's is tangent to an indifference curve of 's. 4) oth individuals prefer allocation Q to allocation W. Pareto Efficient llocations Definition : Pareto efficient allocation is a feasible allocation from which there is no way to make at least one individual better off without harming another. Definition : feasible allocation X = ( n ) if there is no other feasible allocation X = ( n ),,..., is Pareto efficient if and only,,..., such that u i (X ) u i (X ) i N [no one is harmed by moving to X ] and u i (X ) > u i (X ) for at least one i N. [at least one person is better off] Remarks ) The two definitions are equivalent if one accepts the assumption that preferences can be represented by a monotonic utility function. ) In the previous graph, allocation Q is Pareto efficient, while P and the endowment W are not. 3) These definitions require nothing about tangent indifference curves. Thus, tangent indifference curves is sufficient for Pareto efficiency, but not necessary. More on this later. 4) Notice that there is nothing about fairness or equity in these definitions. More later. Pareto Efficiency as a Constrained Optimization Problem Consider a pure-echange economy with two people (, ) and two goods (, ), and the constrained maimization problem of choosing an allocation X = [, ] = [(, ), (, )] to solve ma u (, ) s.t. u (, ) = u i) + = w + w ii) + = w + w. iii) ) 3.4

5 What we are doing here is trying to find a feasible allocation [constraints ii) and iii)] to make as well off as possible (the obective) without harming [constraint i)]. The solution to this problem will be a Pareto efficient allocation. The Lagrange equation for ) is L = u (, ) + µ[u (, ) - u ] + λ [w + w - - ] + λ [w + w - - ]. The first-order conditions for an interior solution to ) are L L L L = u λ = ) = u λ = 3) µ u = λ = 4) µ u = λ = 5) L L L = = = 6) µ λ λ The first-order conditions ) and 3) imply u u λ =. 7) λ The first-order conditions 4) and 5) imply u u λ =. 8) λ 7) and 8) imply u u u λ = =. 9) u λ Recall that u i is the marginal rate of substitution between goods and k for ui ik person i. It is the slope of an indifference curve of i's. In terms of valuation: The marginal rate of substitution is the subective marginal value i places on consumption of good in terms of good k. Equation 9) states that at an interior solution to the optimization problem ), the slope of an indifference curve for is equal to the slope of an indifference curve for. In terms of value 9) says that at an interior solution, 's subective marginal valuation of good in terms of good must be equal to 's subective marginal valuation of good in terms of good. 3.5

6 "Equal slopes" and feasibility require a tangency. One way to think about the solution to ) is to choose the highest indifference curve for that satisfies the constraint u. * * * * (,,, ) u u u u The Set of Efficient llocations Definition: contract curve is the set of Pareto efficient allocations. ctually, the "contract curve" may be a space or a point or something other than a continuous function. In the optimization problem above, the contract curve is found by varying the u constraint. u u u u u u 3.6

7 Core llocations Definition (blocking): coalition S N can block an allocation X if there is some other allocation X such that i S = w, M i) i S u i (X ) u i (X ) i S u i (X ) > u i (X ) for at least one i S. ii) iii) The first requirement for blocking i) is that the allocation X is feasible for the coalition. The requirements ii) and iii) are that at S. That is, the members of S can 'afford' ( i ) least one member of S is better off and no member is harmed. i S Definition (core): feasible allocation X is a core allocation if it cannot be blocked by any coalition S N. In a core allocation no subgroup can break away from the rest of the economy and trade only among themselves and be better off. Notice that the definition for blocking is like the definition for a Pareto efficient allocation. In fact, the similarity is very real. Proposition: Every core allocation is also a Pareto efficient allocation. Proof: Toward a contradiction assume that an allocation X is a core allocation but is not Pareto efficient. If X is not Pareto efficient there eists another feasible allocation X such that u i (X ) u i (X ) i N and u i (X ) > u i (X ) for at least one i N. ut then, X can be blocked by S = N. Hence, X could not have been a core allocation. This contradiction proves the proposition. The following graph illustrates the set of core allocations in our two-person, two-good economy. 3.7

8 Β P contract curve Q core allocations Α W u u Notice that the allocation P is efficient but is not in the core because it can be blocked by person (he can choose to consume his own endowment). Thus, the reverse of the proposition is not true. That is, not every Pareto efficient allocation is a core allocation. llocation Q is a core allocation because neither or, or both together can block it. would be worse off consuming her own endowment as would. Together they cannot block Q because they can't move from it without harming one or the other or both. Notice that core allocations depend on the initial endowments, while Pareto efficient allocations do not. Positive and normative concepts The concept of Pareto efficiency is a normative concept. That is, it is a notion of 'what ought to be'. This definition of efficiency is not derivable from some obective theory of economic behavior, so, like all normative concepts, there is a subective value system underlying its use. e aware of this. The core on the other hand is an equilibrium concept. It is reasonable to require that an economic equilibrium be stable in the sense that no coalition can block it using their own resources. s an equilibrium concept, the definition of the core is also not derivable from obective economic theorizing. However, the notion itself is devoid of value statements. 3.8

9 Competitive echange So far we have considered barter echange to define efficient and core allocations. Now we want to look at trade governed by competitive pricing. Once we have characterized the outcomes of competitive trading (i.e., competitive general equilibrium) we will analyze these outcomes in terms of efficiency and the core. ssume: ) general equilibrium model as before ) gain there is no production, no storage, and no money. 3) Each good has a price per unit which traders take as given (the basic assumption of competitive behavior). 4) Each trader has a strongly monotonic and strictly quasi-concave utility function. Let: N = (,,..., n) -- the set of traders (consumers) in the economy. M = (,,..., m) -- the set of goods available in the economy. p -- the competitive price of the th good The budget constraint for any trader i is p i + p i p m im p w i + p w i p m w im or p pw. Remarks ) p is the market value of i's consumption of good. Therefore, M p is the market value of i's consumption bundle, or i's ependiture on consumption. ) p w is the market value of i's endowment of good. Therefore, M p w is the market value of i's endowment bundle. 3) Since we assume that utility functions are strongly monotonic we can replace ' ' with '='. We will do this from now on. The assumption of competitive behavior also requires that traders maimize their utility subect to their budget constraint. Thus, each i N chooses a consumption bundle ( i, i,..., im ) to solve ma u i ( i, i,..., im ). s.t. p = pw ) 3.9

10 The Lagrange equation for ) is L i = u i ( i, i,..., im ) + λ i [ M p w - M p ]. ) The first-order conditions for an interior optimum are Li L i λ i ui = λ ip =, M ) =. 3) ) implies that for any two goods h and k from M, ui u i ih ik ph =, h and k M, h k. 4) p k This is the result that at an individual, interior optimum, the marginal rate of substitution between any two goods must be equal to the price ratio. Note that since i was chosen arbitrarily, 4) must be true for each i N. The optimal consumption bundle for i is illustrated for the case M = (, ) in the first graph below. The second graph illustrates the comparative static of an increase in the price of good. ui i u i i p = p slope = p ', p > p p i w i (, ) i = i i u i w i i u i i w i i w i 3.

11 How are prices determined? -- The Walrasian uctioneer To mimic actual market operations we add a player whose role is as follows: It calls out a set of prices. Each trader tells the auctioneer its optimal consumption bundle at those prices. If quantity demanded is not equal to the amount available for each good, the auctioneer adusts prices until all markets clear. When all the markets clear, the traders consume their final consumption bundles. That all markets clear is another requirement for a competitive equilibrium. That is, we require that final consumption in a competitive equilibrium be feasible: = w, M. 5) n illustration Consider the graph of a two-person (, ), two-good (, ) economy below. Suppose that the auctioneer calls out initial prices ( p, p ) which results in the allocation [(, ), (, )]. t ( p, p ) neither market clears. In fact, + < w + w (ecess supply of good ) and + > w + w (ecess demand for good ). In this situation, to move toward market clearing, the auctioneer should decrease the relative price of good and increase the relative price of good. That is, in the net round the auctioneer should call out ( p, p ) such that p p < p p. t prices (p *, p *) and allocation X*, each consumer is optimizing on their budget sets and both markets clear. This set of prices and allocation is a competitive equilibrium. u u Β X* Α slope = p p slope = p p u u W 3.

12 Definition: competitive (Walrasian) equilibrium in a pure echange economy is a set of prices p = (p, p,..., p m ) and an allocation X* = ( *, *,..., n *) such that ) For each i N, i * = ( i *, i *,..., im *) is the solution to ma u i ( i, i,..., im ). s.t. p = pw (utility maimization on a budget set) ) ) For each M, = w. (feasibility) 5) In the graph above, you noticed that the competitive equilibrium allocation X* is also a Pareto efficient allocation. It is also a core allocation. It turns out that these are general results. The First Theorem of Welfare Economics If [(,,..., n), ( p, p,..., pm)] [X*, p] is a competitive equilibrium, then X* is a core allocation. y implication it is also a Pareto efficient allocation. Proof: To prove the theorem we use the following facts: Fact Let [X*, p] be a competitive equilibrium. If u i ( i ') > u i ( i *) for some i ', then p ' > p w = p. That is, if i ' is strictly preferred to i * by i, it must not be affordable for i. To show this assume that i can afford i '. Then, since he prefers i ' to i *, he would have chosen i ' instead of i *. ut then, X* could not have been a competitive equilibrium allocation. This contradiction establishes the result. Fact Let [X*, p] be a competitive equilibrium. If u i ( i ') u i ( i *) for some i ', then p ' p w = p. That is, if i ' is weakly preferred to i * by i, it cannot cost less than i *. 3.

13 Toward a contradiction of the welfare theorem, suppose that [X*, p] is a competitive equilibrium but X* is not a core allocation. From the definition of the core, if X* is not a core allocation there eists another allocation X and a blocking coalition S N for which i) = w i S i S, M. -- the members of S must be able to achieve their part of X with their own resources. ii) u i ( i ) u i ( i *), i S. -- no member of S strictly prefers X* to X. iii) u i ( i ) > u i ( i *), for at least one i S. -- at least one person in S strictly prefers X to X*. Fact and ii) imply that iv) p pw, i S. Fact and iii) v) p > pw, for at least one i S. Summing iv) and v) over the members of S yields p > i S i S pw, which can be written as i i m im i i m i S i S i S i S i S i S p + p p > p w + p w p w. This can rewritten again as p w i S i S ut, since prices are positive, >. im i S w for some M. i S This implies that X cannot be a feasible allocation for S -- it violates i). We conclude that an allocation like X cannot eist. ut this contradicts our assertion that X* is not a 3.3

14 core allocation. Therefore, X* must be a core allocation. Furthermore, since all core allocations are Pareto efficient, X* must be Pareto efficient. Q.E.D. Remarks a) The theorem implies that competitive behavior will lead to a desirable (in the Pareto sense) social outcome. b) Unfortunately the theorem does not hold if the assumptions of competitive trading are not met (i.e, no government intervention, no eternalities, no market power, etc.). c) Still we haven't said anything about fairness. There should be no presumption that competitive trading will lead to a fair allocation. However, the Second Welfare Theorem reveals that we can induce a fair (by some criteria) and efficient allocation. The Second Welfare Theorem Suppose that all traders have strongly monotonic and strictly quasi-concave utility functions. Let X* be an efficient allocation such that * >, i N and M. Then there eists a set of prices p = (p, p,..., p m ) and an assignment of endowments W = (w, w,..., w n ) such that (X*, p) is a competitive equilibrium. Notes a) The assumption that * >, i N and M can be relaed. b) In a sense, the theorem says that if you let me choose prices and endowments I can guarantee that any efficient allocation of your choice will be a competitive equilibrium allocation. To illustrate the Second Welfare Theorem, consider trade in an Edgeworth bo. Β u u X* W u u Α slope = p p W 3.4

15 Suppose that W is the initial allocation of endowments. Suppose we think that trade between the two individuals will lead to an unfair allocation, and we prefer to see them trade to the 'fair' and efficient allocation X*. The theorem guarantees that if we pick the appropriate prices and rearrangement of endowments, X* will result from competitive trading. The appropriate prices here are (p, p ) so that p /p = MRS (X*) = MRS (X*). Now pick an endowment W so that the budget line in the Edgeworth bo is the common tangent line at X*. Now if and start at W and trade at prices (p, p ) they will trade to X*. Remarks a) The welfare theorems are important because they let us conclude that if we believe that people trade in competitive situations, any complaints about the price system can be reduced to issues of equity. Furthermore, issues of equity can be addressed by rearranging endowments. b) In the real-world, we can rearrange endowments by what are called lump-sum transfers. Lump-sum transfers are ta/subsidy policies that don't distort competitive prices. Unfortunately, there aren't many types of transfers that don't distort prices. c) Though the welfare theorems are quite powerful, they do depend heavily on the assumptions of competitive behavior. d) There is another problem that we can't address. What criteria will we use to determine what is and what is not fair? Furthermore, what rule do we use to choose among fairness criteria? Shadow prices and competitive prices The purpose of this section is to show that the Lagrange multipliers from the constrained optimization problem that characterizes efficient allocations coincide with competitive market prices. Proposition : Suppose all traders have strongly monotonic and strictly quasi-concave utility functions. Then, if (X*, p) is a competitive equilibrium with * >, i N and M, ui u i ih ik ph =, h and k M, and i N. p k Proof: Recall that if (X*, p) is a competitive equilibrium each i N will choose a consumption bundle ( i, i,..., im ) to solve ma u i ( i, i,..., im ). s.t. p = pw ) 3.5

16 Recall that the necessary conditions for an interior solution to this problem include ui u i ih ik ph =, h and M. 4) p k Note that 5) must be true for each i N. Q.E.D. Proposition : Continue to assume that all traders have strongly monotonic and strictly quasi-concave utility functions. Then, if X* is a Pareto efficient allocation with * > i N and M, ui u i ih λ h =, h and M, and i N, λ where λ k is the Lagrange multiplier for the feasibility constraint on the k th good. Proof: If X* is an efficient allocation, it solves the following optimization problem for each i N: ma (, M, i N) s.t. u i ( i, i,..., im ) u k ( k, k,..., km ) = u k, k N, k i = w, M. For an arbitrary i N, the Lagrange equation is L i = u i ( i, i,..., im ) + [ ( ) ] k N, k i µ k uk k uk + λ w. The first-order conditions are i) ii) iii) From i) and ii) Li L i k Li λ ui = λ =, M. µ uk = k λ =, M, k N, k i. k Li = =, M, k N, k i. µ k 3.6

17 ui u i ih λ h =, h and M and i N. Q.E.D. 6) λ Now, Propositions and imply that p p h h = λ, h and M. λ Thus, we can interpret the Lagrange multipliers from the problem of finding efficient allocations as the market prices that would emerge from competitive trading. Welfare maimization ssume the eistence of a social welfare function. This is a mapping U: R n R such that U(u, u,..., u n ) gives us the collective welfare of N = (,,..., n) for any distribution of private utility levels (u, u,..., u n ). Typically we assume that the social welfare function is increasing in each private utility, That is, U/ u i >, for all i N. Now suppose we have the worthwhile goal of maimizing social welfare. How does the solution to this optimization relate to Pareto efficiency? Proposition: If an allocation X* maimizes U, X* is efficient. Proof: Toward a contradiction of the proposition, suppose that X* maimizes social welfare but is not efficient. If X* is not efficient, there eists a feasible allocation X, such that i) u i ( i ) u i ( i *), i N and ii) u i ( i ) > u i ( i *), for at least one i N. ut, since U is monotonically increasing in each u i, i) and ii) imply U[u ( ), u ( ),..., u n ( n )] > U[u ( *), u ( *),..., u n ( n *)]. Therefore, X* could not have maimized U. This contradiction proves the proposition. Q.E.D. Now, consider the problem of maimizing social welfare subect to the feasibility constraints: 3.7

18 ma (, M, i N) U[u ( ), u ( ),..., u n ( n )] s.t. = w, M. The Lagrange equation for this problem is L = U[u ( ), u ( ),..., u n ( n )] + λ w. ssuming an interior solution, the first-order conditions are i) ii) L L λ U ui = λ =, M, and i N. u i =, M. From i) we have ui u i ih λ h =, h and M, and i N. λ Since this holds for every i, ui u i ih uk kh =, h and M, and i and k N. u k k These marginal conditions are the same as those for Pareto efficient allocations. Remarks a) Though an allocation that maimizes social welfare is efficient, an efficient allocation does not necessarily maimize a particular social welfare function. This implies that though we may have an efficient allocation, there might be another that gives us greater social welfare. In such a case, we would be able to make society better off in aggregate, but doing so would harm someone. b) However, under certain conditions, it can be shown that an efficient allocation always maimizes some social welfare function. c) There are real problems with assuming that a social welfare function eists. ut, at times they are convenient to use. 3.8

19 General equilibrium and the welfare theorems with production We have eamined the relationships among Pareto efficient allocations, core allocations, and competitive equilibria in pure echange economies. Now we introduce production into the economy. Let H = (,,..., h) -- the set of firms in the economy. M = (,,..., m) -- the set of goods available in the economy. p = (p, p,..., p m ) -- constant (competitive) prices. production plan for the k th firm is y k = (y k, y k,..., y km ). If y k >, firm k produces good as an output y k <, firm k uses good as an input Note that the goods set M includes outputs for consumption and inputs to production. n aggregate production plan for the entire economy is y = (y, y,..., y h ). production possibilities set for the k th firm is a collection of all production plans that are technically feasible. Denote the production possibilities set of the k th firm as Y k. ssume that firms are competitive, and that they choose a production plan (y k ) to maimize profit (π k ), taking the vector of prices (p) and the production possibilities set (Y k ) as given. Here, π k = p y k + p y k p m y km = py k Note that if good is an input p y k < (a cost to the firm), and if good is an output p y k > (a source of revenue for the firm). The k th firm's optimization problem is to choose a feasible production plan y k to solve ma π k = py k s.t. y k Y k 7) 3.9

20 The solution to 7) is a production plan ( yk, yk,..., ykm) = ( yk( ), yk( ),..., ykm( )) p p p. In vector notation y k = y k (p). Note that if y k (p) <, it is an input demand function for good, and if y k (p) >, it is a supply function for good. n aggregate production plan in which each firm chooses inputs and outputs to maimize profit is y(p) = [y (p), y (p),..., y h (p)]. Proposition: n aggregate production plan y(p) maimizes aggregate profit k H π k if and only if each firm's production plan y k (p) maimizes its individual profit π k. [For this proposition and its proof see Varian, pg. 339]. Note: For a competitive equilibrium we are going to require that each firm maimizes profit. Sometimes it is more convenient to maimize aggregate profit. The proposition tells us that there is no difference between the two operations. Consumers Recall that in a competitive echange economy we required that an equilibrium allocation X* = ( *, *,..., n *) satisfy ma u i ( i *) s.t. p = pw, i N. In an economy with production there is a complication. What do we do with profit? ssume that each firm is owned by consumers (not necessarily all consumers). Suppose that if i is an owner of firm k, she is entitled to a share s ik of its profit. ssume ) Each firm is completely owned by individuals so that =. s ik ) The shares s ik are fied, and hence, are not traded. In this model there is no stock market although we could have included one. 3.

21 Individual i's share of the profit from firm k is s ik π k = s p y ( p ). ik k Individual i's income from owning shares in a number of firms is k H s π = s p y ( p ). ik k ik k k H Thus, i's budget constraint in this economy with production is p = pw + s py ( p ). 8) ik k k H We will require that in a competitive equilibrium with production, each i maimizes utility subect to 8). Efficient allocations and competitive equilibria Recall: X denotes a consumption allocation. y denotes an aggregate production plan. The pair (X, y) will now be called an allocation. n allocation (X, y) is feasible if and only if = w + y, M. k H k n allocation (X, y) is Pareto efficient if and only if there is no other allocation (X, y ) such that i) = w + y,, M. [(X, y ) is feasible] k H ii) u i ( i ) u i ( i ), i N. [no one is harmed by moving to (X, y )] iii) u i ( i ) > u i ( i ), for some i N. [at least one person is better off at (X, y )] 3.

22 competitive equilibrium is a triple (X, y, p) such that i) Each production plan y k y = (y, y,..., y h ) is the solution to ma π k = py s.t. y k Y k, ii) Each consumption bundle i * X is the solution to ma u i ( i *) s.t. p = pw + s py ( p ) k ik k k H iii) The consumption allocation X is feasible, so that = w + y M. k H k The First Welfare Theorem If (X, y, p) is a competitive equilibrium, then (X, y) is a core allocation. It is also a Pareto efficient allocation. [For the proof, see Varian, pp ]. The Second Welfare Theorem Suppose that (X, y) is a Pareto efficient allocation with >, i N and M. ssume further that each consumer has a strongly monotonic and strictly quasi-concave utility function, and each firm has a closed and conve production possibilities set. Then with an appropriate choice of endowments and profit shares, there eists a set of prices p such that (X, y, p) is a competitive equilibrium. Note: For the second theorem to hold we need each firm's production possibilities set Y k to be closed and conve. set is conve if every point on a line segment oining two points in the set is also in the set. set is closed if the boundaries of the set are included in the set. concave production function will imply a closed and conve production possibilities set. However, a quasi-concave production function may not. If there is a region of increasing returns to scale, the production possibilities set will not be conve. 3.

23 General equilibrium and efficiency with production: The calculus approach Now we are going to characterize efficient allocations and competitive equilibria with the marginal conditions from a series of optimization problems. We will derive the marginal conditions for ) technical efficiency, ) Pareto efficiency, and 3) competitive equilibria. In order to keep things simple we will not fully specify the economy in as much detail as we did above ssume Two consumers, and. Two consumption goods, and. Two inputs into production, L and K. Technical efficiency ssume production functions that are strictly concave: = f(l, K ), = g(l, K ). The resource constraints are L = L + L, K = K + K, where L and K are the aggregate amounts available in the economy. We will ignore the question of where they come from and who owns them. To characterize technical efficiency we choose (L, L, K, K ) to solve the following: ma = f(l, K ) s.t. = g(l, K ) L = L + L K = K + K. The Lagrange equation is Φ = f(l, K ) + λ[g(l, K ) - ] + λ L [L - L - L ] + λ K [K - K - K ]. The necessary conditions are Φ L = f L λ L = 9) Φ K = f K λ K = ) Φ L = λg L λ L = ) Φ K = λg K λ K = ) 3.3

24 Φ λ = Φ λ L = Φ λ K = 3) The first-order conditions 9) through ) imply f f L K gl L = = λ g λ. 4) K K That is, the ratio of the marginal products must be equal for all goods. Recall that the ratio of marginal products is called the marginal rate of technical substitution. ny production plan (,, L, L, K, K ) that satisfies 4) and the resource constraints L = L + L and K = K + K, is technically efficient. Production possibilities frontier Note that the first-order conditions 9) through 3) implicitly define L = L (, L, K) L = L (, L, K) K = K (, L, K) K = K (, L, K) λ = λ (, L, K) λ L = λ L (, L, K) and λ K = λ K (, L, K). The indirect obective function is = f( L, K ) = (, L, K). This is the production possibilities frontier (PPF). It gives us the maimum possible production of for each level of, given the availability of L and K. From the Envelope Theorem, we have the slope of the PPF Φ = = λ. nd from the first-order conditions we have f L f K λ = = < g g L K. 3.4

25 Since = λ <, the PPF is downward sloping. The slope of the PPF is sometimes called the marginal rate of transformation. If you check the second derivative of (, L, K).you will realize that it is strictly concave if f(l, K ) and g(l, K ) are both strictly concave. f L = = g L f g K K PPF Remarks ) The production possibilities frontier collects all the combinations of the production of the two goods that are technically efficient. ) Technical efficiency is a necessary condition for Pareto efficiency Pareto efficiency s before, to find the set of Pareto efficient allocations we choose an allocation (,,, ) to solve ma u (, ) s.t. u (, ) = u + = + = (, L, K) = i) ii) iii) iv) Constraints ii), iii) and iv) are 'feasibility' constraints. Constraints ii) and iii) state the supply of each good must be equal to aggregate demand, and iv) states that a production plan (,, L, L, K, K ) must be technically efficient. Combine the last three constraints into one to make the problem a little simpler: + = ( +, L, K) v) 3.5

26 3.6 The Lagrange equation is then L = u (, ) + µ[u (, ) - u ] + θ [ ( +, L, K) - - ] The first-order conditions are θ θ L u L u u u = = = + = =, 5) µ θ µ θ L u L u u u = = = + = =, 6) and L θ = L µ =. Equations 5) and 6) imply u u u u = =. 7) Equation 7) states that at a Pareto efficient allocation, the marginal rates of substitution between any two goods is equal for every consumer, and in turn equal to the slope of the production possibilities frontier. PPF u u u u u u = slope =

27 Competitive behavior Profit maimization: Let prices in the economy be (p, p, p L, p K ). For the production of good we choose (L, K ) to solve The necessary conditions are ma p f(l, K ) - p L L - p K K. pf pf L K pl = pk = f f L K pl = 8) p K Similarly, the necessary conditions for a profit maimizing plan to produce good imply 8) and 9) imply g g f f L K L K pl = 9) p K gl pl = = 3) g p K K Compare 3) and 4) to note that competitive, profit maimizing behavior induces a technically efficient allocation of L and K to the production of and. Utility maimization: Each consumer chooses ( i, i ) to solve ma u i ( i, i ) s.t. a budget constraint, i =,. The necessary conditions imply u u u p = = 3) u p Compare 3) and 7) to verify that competitive behavior by consumers induces a Pareto efficient consumption allocation. lso note that the ratio of the goods prices is equal to the slope of the production possibilities frontier. Notes ) Obviously, all the marginal conditions are not enough to make the statements we have been making. We also need the resource constraints. ) These marginal relationships are not the only ones that can be inferred. See Silberberg for more. 3.7

28 The compensation criterion s a criterion for evaluating public policy proposals, the concept of Pareto efficiency is considered by most to be too restrictive. Specifically, the criterion is said to be incomplete in the sense that it does not allow us to rank every possible allocation (or, more generally, social outcome). For eample, consider the following graph. ccording to the Pareto criterion, a social planner will not be able to rank bundles R and T. Even worse, the Pareto criterion does not tell us that society prefers R and T to S even though S is inefficient. contract curve Β T Α R S u u In this lecture we will consider a modification of the Pareto criterion called the compensation criterion, which is commonly used in applied welfare economics. It is similar to the Pareto criterion but it allows us to compare more outcomes. That is, it is more complete than the Pareto criterion. Unfortunately, as you will see, it is also incomplete and it can provide inconsistent comparisons. The Pareto frontier It will be useful to use Pareto (sometimes called utility possibility) frontiers. Pareto frontier collects all bundles of utility levels that are generated by efficient allocations. Consider a society with two people (, ) and two goods (, ) and the problem of finding the Pareto efficient allocations: ma u (, ) s.t. u (, ) = u + = w + w + = w + w. 3) 3.8

29 The Lagrange equation for 3) is L = u (, ) + µ[u (, ) - u ] + λ [w + w - - ] + λ [w + w - - ]. The necessary conditions for an interior solution to 3) are L u = λ = 33) L u = λ = 34) L µ u = λ = 35) L µ u = λ = 36) L µ = L L λ = λ = 37) s usual, these first-order conditions imply that at an efficient allocation with > for i (, ) and (, ), u u = u u u (, ) = u + = w + w + = w + w 38) [Note that the collection of conditions 38) are identical to 33) through 37)]. ssuming that a solution to 3) eists and is unique, conditions 38) implicitly define (,, ) ( u, w w, w w ) (,, ) = u w + w w + w, i (, ) and (, ) λ = λ + +, (, ) µ = µ u w + w w + w. The indirect obective function is (,, ) u = u u w + w w + w. 3.9

30 So that there is no confusion later on, let u ( ) = v( u, w + w, w + w ) the Pareto frontier. From the Envelope Theorem. This is u u v = = µ <. u Hence, the Pareto frontier is downward sloping. In the graph below, I have drawn the Pareto frontier as concave, although we cannot guarantee this. Instead of thinking about the Pareto frontier as a function, it will be convenient sometimes to describe utility possibilities as a set: [ ( ) [ ( ) ( ) ] such that ( ) is an efficient allocation] U = u X = u, u X =,. In the graph below, each point in this space is a utility vector (u, u ). Points on the frontier are utility vectors generated by efficient allocations. Points under the frontier are feasible utility vectors, but they are generated by inefficient allocations. contract curve Β T Α R S u u u u(t) Pareto frontier u(s) u(r) u 3.3

31 The compensation criterion In order to define the compensation criterion, we first define the Pareto criterion. Definition: feasible allocation X is said to Pareto dominate another feasible allocation X if u i (X ) u i (X ) i N and u i (X ) > u i (X ) for some i N. In the graph below X Pareto dominates only those allocations that induce utility vectors in the shaded area. For eample, X dominates X. However, X and X are not comparable by the Pareto criterion. That is, X does not dominate X and X does not dominate X. This is what we mean by incompleteness. The Pareto criterion does not give us a basis to udge the desirability of all allocations relative to all others. u u ( X ) u ( X ) u ( X) u Definition: feasible allocation X is said to be potentially Pareto preferred to another feasible allocation X if there is some reallocation of X, say X, such that = M (X is a reallocation of X ) u X u X u X > u X for some. (X Pareto dominates X) i( ) i( ) and i( ) i( ) 3.3

32 This is the essence of the compensation criterion. social outcome X is preferred to X if there is a third outcome X attainable from X that Pareto dominates X. Note that the definition does not require that we actually move to X, it only requires its eistence. Put another way: lmost any public policy change is likely to produce winners and losers (i.e., some will benefit by the change and others will lose). The change is potentially Pareto preferred if the winners could compensate the losers (a reallocation) so that if the compensation takes place, no one is harmed by the change and some are strictly better off. The catch is that the compensation need not take place. To illustrate, consider two mutually eclusive outcomes: R (reduce emissions of some industrial pollutant) and NR (no environmental regulation). Relative to NR, option R will hurt the owners of some firms but will provide a cleaner environment for the enoyment of others. Consider our two-person society, and suppose that the Pareto frontiers in the graph below correspond to the two policy options. Suppose further that a policy option to achieve R results in allocation X and utility vector u(x ), while policy option NR will result in allocation X and utility vector u(x). Note that X does not Pareto dominate X. However, there is a third allocation X that is attainable with policy option R that does Pareto dominate X. Therefore, X is potentially Pareto preferred to X. So, according to the compensation criterion emissions of the industrial pollutant should be reduced even though person is harmed. u u ( X ) u ( X ) u ( X ) NR R u 3.3

33 Incompleteness The graph above illustrates the incompleteness of the compensation criterion. Here X is not potentially preferred to X, and X is not potentially preferred to X. However, relative to the Pareto criterion we can compare more outcomes using the compensation criterion. Inconsistency maor shortcoming of the compensation criterion is that it will generate inconsistent results in some situations. Suppose that the Pareto frontiers for the policy options we considered above are as in the graph below. Here, allocation X is potentially Pareto preferred to X through the reallocation X. So, according to the compensation criterion, emissions should be regulated. However, allocation X is potentially Pareto preferred to X through the reallocation X. So, according to the compensation criterion, emissions should not be regulated. In such a case, the compensation criterion gives us an inconsistent ranking. It tells us to regulate emissions, but also do not regulate emissions. u u ( X ) u ( X ) u ( X ) u ( X ) NR R u The compensation criterion and transferable utility It is often assumed in economics and game theory that utility can be transferred from one person to others. Utility is transferable only if there eists some commodity that enters each individual's utility function linearly and separately from all other commodities. For eample, utility is transferable in our two-person, two-good society only if the utility functions have the form u i ( i, i ) = v i ( i ) + i, i (, ). 39) 3.33

34 Utility functions of this form are called quasi-linear utility functions. Given quasi-linear utility functions, utility is transferable if individuals can make payments to each other using good. In the language of game theory, we say that side-payments are allowed. (Sometimes we assume that good is money). Note that the cost to an individual of increasing the utility of the other by one unit is one unit of utility. Now suppose that we want to choose an allocation that maimizes the sum of individual utilities subect to the resource constraints: ma V = u (, ) + u (, ) = v ( ) + + v ( ) +, s.t. + = w + w + = w + w, 4) [Note that V is a social welfare function]. If v ( ) and v ( ) are both monotonically increasing and strictly concave then a solution to 4) eists and it is unique. Given a solution to 4), we can define the indirect obective function V* = V( w + w w + w ),. This function gives us the maimal attainable utility for the two-person society. Now, if side-payments are allowed (utility is transferable), V* can be allocated in any way so that u + u = V*. 4) Recall that an allocation that maimizes a social welfare function is Pareto efficient. Define the set of utility vectors [ uv = u u u u V ] + U S = ( ) ( ), such that =. 4) This set describes the Pareto frontier when utility is transferable. The function describing the frontier in the (u, u ) space is u = V* - u 4') Note that this is a linearly decreasing function with slope equal to - and horizontal and vertical intercepts equal to V*. Recall that before we described the Pareto frontier as the set [ ux = [ u u ] X ( ) ], =, U = ( ) ( ) ( ) such that is an efficient allocation, 43) 3.34

35 and the function (,, ) u = v u w + w w + w. 43') Now, we did not allow side-payments when we generated 43) and 43'). In the case of the quasi-linear utility functions 39) with v ( ) and v ( ) monotonically increasing and strictly concave, the function 43') is decreasing and strictly concave in u. The graph below is of the two frontiers. Note that both frontiers are derived from the same individual utility functions and society's endowments of the two goods, w + w, (, ). The only difference between them is that side-payments are allowed along U S and are not allowed along U. V* u u ( X* ) u u ( X ) U V* U S u Suppose that X* is the allocation that maimizes social welfare as defined by 4). Without a transfer of utility between the individuals, the utility vector at this allocation u(x*) is the point of tangency between U S and U. Note that even though X efficient in the absence of side-payments, X* does not Pareto dominate X and vice-versa. However, suppose that the two players agree to the allocation X* and a transfer of utility from to so that the utility vector u is achieved. [ctually, the players can agree to this scheme or it can be imposed by a social planner]. Since both players are better off with X* and the transfer than they would be at X, we can say that X* is potentially Pareto preferred to X. Note the difference between what we have ust done and our definition of "potentially Pareto preferred". There an allocation X* was preferred to another X if there was a reallocation of consumption bundles that Pareto dominated X. With transferable utility, we do not reallocate the consumption allocation to reach a dominant outcome -- we choose the social welfare maimizing consumption allocation and then reallocate utility. 3.35

36 Ecercises [] We know that a Pareto efficient allocation is one from which there is no move that can make at least one person better off without harming another. Define a Pareto improving move as one that makes one person better off without harming another. Note that a Pareto improving move does not necessarily result in a Pareto efficient allocation. Define a Pareto superior allocation as one that results from a Pareto improving move. Note that this definition does not require a superior allocation to be an efficient one. In the following graph, the point W is the endowment point, while R, S and T are alternative allocations. Consider moves to and from these points to illustrate the concepts of Pareto efficiency, Pareto improving moves and Pareto superior allocations. Β R contract curve T S Α [] Consider a two-person pure echange economy with two goods. Suppose that does not value good at all. That is, only increases in the consumption of good will increase 's utility. Person 's indifference curves have the usual shape. In an Edgeworth o locate the contract curve and the core. [3] "Jack Sprat can eat no fat, his wife can eat no lean." In an Edgeworth o find the contract 'curve'. What about the core? [4] Discuss the relationship between: [a] efficient and core allocations [b] efficient allocations and endowments [c] core allocations and endowment [5] In a pure echange economy with two people ( and ) and two goods ( and ), suppose that u (, ) = + u (, ) = w = (, ) w = (, ) [a] Draw an Edgeworth bo to illustrate the economy. Draw a few indifference curves and the point W = (w, w ). [b] Solve for the Pareto efficient allocations. Illustrate them graphically. Illustrate the core allocations graphically. W u u 3.36

37 [6] Eplain how the Pareto efficient allocations in a two-person, two-good echange economy can be characterized with the appropriate constrained optimization problem. You don't have to go as far as deriving first-order conditions. Just set the problem up to fit the definition of Pareto efficient allocations. [7] Consider a two-person ( and ), two-good ( and ), pure echange economy. The consumers have identical utility functions, u i ( i, i ) = ln( i ) + i, i =,. Their endowments are (w, w ) = (, ) and (w, w ) = (, ). [a] Find the contract curve and draw it in an Edgeworth bo. [b] Can the allocation [(, ), (, )] = [(, ), (, )] be a competitive equilibrium allocation? [8] In a two-person, two-good, echange economy, the consumers have utility functions u i ( i, i ) = i i, i =,, and endowments (w, w ) = (, ) and (w, w ) = (, ). Find the competitive equilibrium allocation for prices (p, p ) = (, ). Is this allocation Pareto efficient? [9] In a pure echange economy with two people ( and ) and two goods ( and ), suppose that u (, ) = + u (, ) = w = (, ) w = (, ). Denote the prices of the two goods as p and p. [a] Find the competitive equilibrium (or equilibria). Verify the first welfare theorem. [b] ssume that (p, p ) = (4, ). Show that these prices cannot support a competitive equilibrium. Do the same for (p, p ) = (4, 3). [c] Verify that the allocation X = [(, ) = (5/3, /3), (, ) = (5/3, /3)] is an efficient allocation. Find the set of prices and endowments such that X is a competitive equilibrium allocation. Use this eercise to illustrate the second welfare theorem. [] Waldo and Penelope have preferences over liver and onions. In fact, both view liver and onions as perfect complements to be consumed in one-to-one proportions -- one onion for every pound of liver. ssume that Waldo is endowed with 5 onions and no liver, and Penelope is endowed with 5 onions and pounds of liver. [a] Find the contract curve (or space) and the core. [b] How does the contract curve and the core change if 5 pounds of liver is transferred from Penelope to Waldo. 3.37

38 [c] Find the set of competitive equilibria for the original endowment. How does this set change if the endowments change as in b)? [d] Comment on the necessity of equating marginal rates of substitution to find efficient allocations and competitive equilibria in this case and in general. [] Use the welfare theorems as applied to echange economies to discuss the relationships among core allocations, efficient allocations, and competitive equilibria. Use words, not math. [] Ken and arbie have preferences over quiche and Perrier. Ken only consumes quiche and Perrier in one-to-one proportions -- one bottle of Perrier for every slice of quiche. arbie has indifference curves that have the normal conve shape. Ken is endowed with two slices of quiche and two bottles of Perrier, while arbie is endowed with four slices of quiche and 4 bottles of Perrier. [a] Find the contract curve and the core. How does the contract curve and the core change if we transfer one slice of quiche from Ken to arbie. [b] For the original endowments, find the competitive equilibrium. [c] Verify that the allocation in which each person has three units of each good is an efficient allocation. Use this allocation to illustrate the second welfare theorem. [3] ny complaints about the operation of a competitive market system can be reduced to complaints about equity, and such complaints can be addressed by lump-sum transfers. State the welfare theorems in the contet of an echange economy and use them to evaluate this statement. [4] ennie and Joon have preference over Macadamia nuts and Listerine. Each considers Macadamia nuts and Listerine as complements to be consumed in one-to-one proportions -- one pound of Macadamia nuts for every gallon of Listerine. ennie is endowed with two pounds of Macadamia nuts and one gallon of Listerine, while Joon is endowed with one pound of Macadamia nuts and two gallons of Listerine. [a] Find the contract curve and the core. [b] Use the allocation in which both individuals consume.5 pounds of Macadamia nuts and.5 gallons of Listerine to illustrate the second welfare theorem. [5] If two allocations are not comparable by the Pareto criterion, they are not comparable by the compensation criterion. State the Pareto criterion and the compensation criterion, then tell me whether the statement is true or false and prove your conclusion. [6] Show that every Pareto efficient allocation is potentially Pareto preferred to every Pareto inefficient allocation. [7] The compensation criterion and the Pareto criterion are normative concepts that are used to eamine the 'desirability' of social states. Compare and contrast these two criteria. 3.38