Introduction to General Equilibrium: Framework.

Transcription

1 Introduction to General Equilibrium: Framework. Economy: I consumers, i = 1,...I. J firms, j = 1,...J. L goods, l = 1,...L Initial Endowment of good l in the economy: ω l 0, l = 1,...L.

2 Consumer i : preferences over consumption set X i R L + represented by utility function u i. Typical consumption bundle of consumer i : x i = (x i1,..., x il ) X i.

3 Firm j: technology summarized by a production possibility set Y j R L. Typical production vector of firm j : y j = (y 1j,..., y Lj ) R L.

4 Total (net) amount of good l available for consumption in the economy, l = 1,...L: ω l + y lj.

5 What kind of production and consumption outscomes are feasible in this economy? Defnition: An economic allocation (x 1,...x I, y 1,..., y J ) is a specification of a consumption bundle x i X i for each consumer i = 1,...I, and a production plan y j Y j for each firm j = 1,...J. Defintion: The allocation (x 1,...x I, y 1,..., y J ) is feasible if x il ω l + y lj, l = 1,...L.

6 Effi ciency Definition: A feasible allocation (x 1,...x I, y 1,..., y J ) is Pareto optimal or Pareto effi cient if there is no other feasible allocation (x 1,...x I, y 1,..., y J ) such that u i (x i ) u i(x i ) for all i = 1,...I, AND u i (x i ) > u i(x i ) for some i. In other words, an allocation is Pareto effi cient if it is not possible to make any one better off without making someone worse off.

7 Utility possibility set: set of all utility profiles of consumers i = 1,...I that can be feasibly generated in this economy. U = {(u 1,.., u I ) : a feasible allocation (x 1,..x I, y 1,., y J ) such The utility profile generated in any Pareto effi cient allocation is on the "frontier" of the Utility Possibility Set.

8 Private Ownership (or Market) Economy. Initial endowments and technological possibilities (firms) are owned by consumers. Consumer i initially owns ω li 0 of good l, l = 1,...L, where ω li = ω l, l = 1,...L. Endowment vector of consumer i: ω i = (ω i1,..., ω il ) Share of firm j owned by consumer i : θ ij 0, j = 1,...J, where θ ij = 1, j = 1,...J. The share θ ij entitles consumer i to a fraction θ ij of firm j s profit.

9 (Pefectly) Competitive Economy A market exists for each of the L goods. All consumers and firms act as price takers i.e., assume that their individual actions do not affect market prices. Price vector: p = (p 1,..., p L ) R L. Note that given price vector p, the budget set of each consumer i is given by: {x i X i : px i pω i + θ ij (p y j )} Conumer s wealth depends on prices as prices determine the value of initial endowment as well as firms profits (value of consumer s shareholdings).

10 Competitive (Walrasian) Equilibrium. Definition: An allocation (x 1,.., x I, y 1,., y J ) and a price vector p = (p 1,..., p L ) RL constitute a competitive (or Walrasian) equilibrium if the following hold: (a) Profit maximization: Given price vector p,each firm j maximizes its profit by choosing the production plan y j i.e., y j solves max y j Y j p y j. (b) Utility Maximization: Given price vector p,each consumer i maximizes her utility over her budget set by choosing the consumption vector x i i.e.,x i solves max u i (x i ) x i X i s.t. p x i p ω i + θ ij (p y j ).

11 (c) Market Clearing: For each good l = 1,..., L x li = ω l + y lj.

12 Note that if the allocation (x 1,.., x I, y 1,., y J ) and price vector p 0 constitute a competitive equilibrium, then so do the allocation (x 1,.., x I, y 1,., y J ) and price αp for any scalar α > 0. So, we can normalize prices without any loss of generality - for instance, by setting one of the prices equal to 1 (the corresponding good is called the numeraire good).

13 Lemma 10.B.1: If the allocation (x 1,...x I, y 1,..., y J ) and a price vector p 0 satisfies the market clearing condition for all goods other than some good k, x li = ω l + y lj, l = 1,...L, l k, and if every consumer s budget constraint is satisfied with equality: p x i = p ω i + then the market for good k also clears. θ ij (p y j ), i = 1,...I.

14 Proof: The budget equality for each consumer ican be written as L l=1 p l x li = L l=1 p l ω li + so that p k x ki p k ω ki = l k = l k = l k p l x li l k p l ω li L θ ij l=1 θ ij p k y kj p l x li p l ω li p l l k l k p l x li ω li p l y lj θ ij l k θ ij y lj p l y lj θ ij y lj

15 i.e., p k (x ki ω ki = p l l k x li ω li θ ij y kj ) θ ij y lj

16 Adding this over i = 1,...I: = p k p l l k p l p l p l p l = l k = l k = l k = l k x ki ω ki x li ω li x li ω li x li x li ω l x li ω l θ ij y kj ω li θ ij y lj θ ij y lj θ ij y lj θ ij y lj y lj = 0, using the fact that all markets other than k clear. p k > 0, it follows x ki ω ki θ ij y kj = 0 As

17 i.e., which implies x ki ω k θ ij y kj = 0 x ki ω k y kj = 0 i..e., the market for good k also clears.

18 Partial Equilibrium Analysis. Analysis of market for one (or several) goods that form a small part of the economy.

19 Marshall (1920): consider one good that accounts for small fraction of consumer s total expenditure. The wealth (or income) effect on the demand for the good can be negligible. Substitution effect of change in the price of the good is dispersed among all goods and so prices of other goods are approximately unaffected. So, for the analysis of this market, we can take prices of all other goods as fixed. Expenditure on all other goods taken to be a composite commodity - the numeraire.

20 The Basic Quasi-linear Model: Consumers i = 1,...I. Two commodities: good l and the numeraire. x i : consumer i s consumption of good l. m i : consumer i s consumption of the numeraire (i.e., expenditure on all other goods). Consumption set of consumer i : R R +. (Allow negative consumption of the numeraire good - "borrowing" - assumption avoids dealing with corner solution).

21 Utility function: u i (m i, x i ) = m i + φ i (x i ), i = 1,..., I Assume: φ i (.) is bounded above, twice continuously differentiable, φ i (0) = 0, φ i (x i) > 0, φ i (x i) < 0, x i 0. Quasi-linear formulation: no wealth effect.

22 Normalize price of the numeraire good to equal 1. Let p be the price (relative price) of good l, Then, one can think of φ i (x i ) as measuring utility in terms of the numeraire good

23 . Firm j = 1,...J, produces q j units of good l using amount c j (q j ) of the numeraire good. c j (q j ) : cost function of firm j. Technology of firm j: Y j = {( z j, q j ) : q j 0, z j c j (q j )}. Assume: c j : R + R + is twice differentiable.

24 c j (q j) > 0 and c j (q j) 0 at all q j 0. [Think of c j (q j ) as derived from a cost minimization problem with fixed input prices.] Non-decreasing marginal cost curve (allows for constant and decreasing returns to scale). Also continuity of c j at 0 rules out any fixed cost that is not sunk (can be avoided by producing zero).

25 Initial endowment: No initial endowment of good l. Consumer i s initial endowment of the numeraire good : ω mi > 0. Let ω m = ω mi be the total endowment of the numeraire good in the economy.

26 Competitive Equilibrium: Profit max. Given equilibrium price p for good l, firm j s equilibrium output q j solves max q j 0 [p q j c j (q j )] Necessary and suffi cient first order condition: p c j (q j ), if q j = 0 (1) = c j (q j ), if q j > 0. (2)

27 Utility max. Given p and the solution to the firms profit maximization problems, consumer i s equilibrium consumption (m i, x i ) solves: m i + p x i ω mi + max [m i + φ i (x i )] m i R,x i R + s.t. θ ij (p q j c j(q j )) Budget constraint holds with equality in any solution to the above problem.

28 Rewrite the problem without of loss of generality as one of choosing only the consumption of good l: max [ω mi + x i R + or equivalently, x i θ ij (p q j c j(q j )) p x i + φ i (x i )] must solve and m i is determined by max [φ i(x i ) p x i ] x i 0 m i = ω mi + θ ij (p q j c j(q j )) p x i. A necessary and suffi cient first order condition: p φ i (x i ), if x i = φ i (x i ), if x i = 0, (3) > 0. (4) Thus, an equilibrium allocation is characterized fully by a price p of good l and the vector (x 1,..., x I, q 1,..., q J ) of consumption and production of good l.

29 Finally, market clearing for good l requires: x i = qj. (5)

30 Proposition: The allocation (x 1,..., x I, q 1,..., q J ) and price p constitutes a competitive equilibrium if and only if. p c j (q j ), if q j = 0 = c j (q j ), if q j p φ i (x i ), if x i = 0, = φ i (x i ), if x i > 0, j = 1,...J > 0,.i = 1,...I x i = q j. The above (I+J +1) conditions determine the (I+J +1) equilibrium values (x 1,..., x I, q 1,..., q J, p ). The equilibrium allocation and price of good l are entirely independent of the distribution of initial endowments and ownership shares of firms.

31 Observe: since φ i (x i) > 0, x i 0, it follows that the equilibrium price p > 0. Assume: max i φ i (0) > min c j (0). j Then, in equilibrium, total consumption and production of good l : x i = q j > 0. [If all consumers consume 0 and all firms produce 0, then so that c j (0) p φ i (0), i = 1,...I, j = 1,...J, a contradiction.] max i φ i (0) min c j (0), j

32 One can derive the equilibrium through traditional Marshallian demand-supply analysis.

33 Demand : For any p, each consumer s first order condition: p φ i (x i), if x i = 0, = φ i (x i), if x i > 0. Note φ i is a continuous and strictly decreasing function on R + with range [0, φ i (0)]. Individual Walrasian demand function of consumer i : x i (p) = 0, p φ i (0), = φ i 1 (p), p (0, φ i (0). Note individual demand x i (p) is independent of wealth, continuous & non-increasing in p and strictly decreasing in p on (0, φ i (0)).

34 Aggregate (market) demand for good l: x(p) = x i (p) - independent of endowment & the distribution of endowments, - continuous & non-increasing in p and - strictly decreasing in p on (0, max i φ i (0)). Note that individual and aggregate demand is infinite at zero price. Also, aggregate demand is zero for p max i φ i (0).

35 Supply: For any p, firm j s profit max yields the following first order condition: p c j (q j), if q j = 0 = c j (q j), if q j > 0. If p < c j (0), firm s supply is zero. Suppose c j is strictly convex (upward sloping marginal cost) and c j (q) as q. Then, for each p > c j (0), there is a unique q j such that p = c j (q j).

36 The firm s supply curve in that case: q j (p) is q j (p) = 0, p c j (0), = c 1 j (p), p > c j (0). - continuous and non-decreasing in p & - strictly increasing for p > c j (0).

37 The aggregate (market) supply curve is given by: q(p) = q j (p). Note that q(p) = 0 for p min j c j (0). For p > min j c j (0), q(p) is strictly positive, strictly increasing and continuous.

38 The market equilibrium price p is given by the point where aggregate demand and supply intersect i.e., Let z(p) = x(p) q(p). Then, x(p ) q(p ) = 0 (6) z(p) = x(p) > 0, p min j c j (0) = q(p) < 0, p max i φ i (0). z(p) is continuous and strictly decreasing in p on (min j c j (0), max i φ i (0)). Unique p (min j c j (0), max i φ i (0)), such that z(p ) = 0 i.e.,(6) holds.

39 The equilibrium allocation is given by setting x i = x i(p ), i = 1,...I, q j = q j(p ), j = 1,...J. If c j is convex but not strictly convex (for example, linear), there may not be unique solution to the profit max problem and so q j (p) is a correspondence (upper hemicontinuous, using Maximum Theorem). Similar analysis goes through with more technical arguments.

40 Important case: Constant returns to scale. c j (q j ) = c j q j where c j > 0 is the constant average as well as marginal cost ("unit cost"). Firm j s supply function q j (p) = 0, p < c j [0, ), p = c j =, p > c j. The aggregate supply is infinite (not well defined as a real number) for p > c j. If all firms have constant returns to scale technology with cost functions c j (q j ) = c j q j,j = 1,...J, then the unique equilibrium price p = min j c j. Only firms with the minimum unit cost can produce in equilibrium (such firms are indifferent between all levels of output at that price).

41 The total quantity of good l produced and consumed is given by the aggregate demand function and equals x(p ). If there are muliple firms with the minimum unit cost, the way the total quantity demanded x(p ) is produced across these firms is not uniquely determined.

42 For any y > 0, the inverse of the aggregate supply function given by q 1 (y) indicates the equalized marginal cost of all firms that produce this output: q 1 (y) is the industry s marginal cost curve.

43 Define the industry s aggregate cost of producing any level of total output y by: s.t. C(y) = min q j,,...j c j (q j ) q j = y, q j 0, j = 1,..J. Lagrangean: L(q 1,...q J, λ) = c j (q j ) + λ(y q j )

44 First order necessary and suffi cient conditions: c j ( q j ) = λ, q j > 0 λ, q j = 0 - all firms that produce strictly positive output, marginal cost is equalized to λ -all firms that produce zero output, marginal cost at zero is no larger than λ For any p, letting λ = p, we can see that q j = q j (p), j = 1,...J, must minimize industry s aggregate cost of producing y = q j (p).

45 Further, using envelope theorem: C (y) = λ = c j ( q j ), j such that q j > 0. Thus, industry s marginal cost of producing y = q j (p) is given by c j (q j(p)), for all j such that q j (p) > 0 i.e., q 1 (y), the inverse aggregate supply curve. Therefore, C(q) = C(0) + = C(0) + q 0 q 0 C (y)dy q 1 (y)dy

46 q 0 q 1 (y)dy = C(q) C(0) The area under the aggregate supply curve is equal to the (minimized) total variable cost of the industry (i.e., the total cost excluding the sunk cost). The area under individual supply curve is the total variable cost of the firm.

47 Let P (y) = x 1 (y) be the inverse aggregate demand function. Then, for any y, P (y) = φ i (x i(p (y))), i such that x i (p) > 0. In other words, P (y) represents the marginal benefit from consumption of good l to a consumer that consumes strictly positive quantity when the price is P (y) and total quantity y is consumed. So, P (y) represents the marginal benefit to soiety from total consumption of amount y.

48 For any y > 0, define the (maximum) social benefit from total consumption of amount y : B(y) = max x i,,..,i s.t. Define Lagrangean φ i (x i ) x i = y, x i 0, i = 1,..I. L(x 1,...x I, µ) = (φ i (x i )) + µ(y x i )

49 First order necessary and suffi cient conditions: φ i ( x i ) = µ, x i > 0 µ, x i = 0 so that for all consumers that comsume strictly positive amount of good l, marginal utility is equalized to µ and for all consumers that consume zero amount, marginal utility at zero does not exceed µ. For any p, letting µ = p, we can see that x i = x i (p), i = 1,...I, must maximize society s benefit from consuming y = x i (p).

50 Further, using envelope theorem: B (y) = µ = φ i ( x i ), i such that x i > 0. Thus, society s marginal benefit from consuming y = x i (p) is given by φ i (x i(p)), i such that x i (p) > 0 (the height of the individual demand curves at x i (p)) i.e., P (y), the inverse aggregate demand curve. Therefore, (using B(0) = 0) B(x) = = x 0 x 0 B (y)dy P (y)dy The area under the aggregate demand curve is equal to the (maximum) social benefit from consumption of good l. The area under individual demand curve is the total benefit to the individual consumer.

51 To sum up: 1. Profit maximization by price taking firms ensures that the total output produced by the industry at any price minimizes the industry s cost of producing this amount (i.e., market distributes total output across firms optimally). 2. Utility maximization by price taking consumers ensures that the total consumption in society is distributed across consumers so as to maximize the total benefit to society (optimal distribution of consumption).

52 3. The height of the aggregate supply curve indicates the industry s marginal cost of production. The area under the aggregate (inverse) supply curve indicates the industry s total variable cost. 4. The height of the aggregate demand curve indicates the society s marginal benefit from consumption. The area under the aggregate (inverse) demand curve indicates society s total benefit from consumption of good l.

53 Pareto Optimality & Competitive Equilibrium. Fix the consumption and the production levels of good l at (x 1,..., x I, q 1,..., q J ) where x i = q j. The total amount of the numeraire available for distribution amount consumers is ω m c j (q j ).

54 Quasilinear utility: transferable utility. By transferring numeraire good across consumers in various ways one can generate a utility possibility set: {(u 1, u 2,..., u I ) : u i φ i (x i )+ω m c j (q j ) The right hand side of the inequality defining the set is a constant (given (x 1,..., x I, q 1,..., q J )). So, the frontier of this utility possibility set is is a hyperplane with normal vector (1, 1,..., 1). Changing the consumption and production levels of good l i.e., the vector (x 1,..., x I, q 1,..., q J ) shifts the frontier of the utility possibility set in a parallel fashion.

55 The frontier moves outward or inward according to whether φ i (x i ) + ω m c j (q j ) increases or decreases when we change the vector (x 1,..., x I, q 1,.. As long as the frontier can be shifted outwards by change in the vector (x 1,..., x I, q 1,..., q J ) the original situation is not Pareto optimal.

56 Thus, every Pareto optimal allocation must involve consumption and production profile ( x 1,... x I, ỹ 1,..., ỹ J ) for good l so as to shift the frontier as far out as possible i.e., ( x 1,... x I, ỹ 1,..., ỹ J ) solves x i = max (x 1,...,x I ) 0 (q 1,...q J ) 0 s.t. q j [ φ i (x i ) c j (q j )] The maximand above is often called the Marshallian aggregate surplus (or total surplus). It measures the net benefit to society from producing and consuming good l. There exists a solution to the maximization problem. The solution ( x 1,... x I, ỹ 1,..., ỹ J ) is called the optimal consumption and production levels for good l.

57 While there are a conmtinuum of Pareto optimal allocations corresponding to points on the highest utility possibility frontier, they all involve this very same consumption and production vector for good l - the only difference in various Pareto optimal allocations arise from differences in distribution of the numeraire good (that can transfer utility from one agent to another unit for unit).

58 Lagrangean: L = φ i (x i ) c j (q j ) + µ[ q j First order necessary and suffi cient condition (maxiand is concave, feasible set is convex): x i ] µ c j ( q j ), if q j = 0 = c j ( q j ), if q j > 0, j = 1,...J. φ i ( x i ) µ, if x i = 0, = µ, if x i = 0, i = 1,...I q j = Setting µ = p, we see that these conditions are satisfied by the production and consumption profile for good l in any competitive equilibrium allocation. x i

59 The First Fundamental Theorem of welfare Economics. Proposition. If the price p and the allocation (x 1,..., x I, q 1,..., constitutes a competitive equilibrium, then this allocation is Pareto optimal.

60 Conversely, consider the set of Pareto optimal allocations. The production and consumption levels of good l in any such allocation must necessarily be the same as the level sustained in competitive equilibrium i.e., (x 1,..., x I, q 1,..., q J ). Only consumption of the numeraire good and the utilities of consumers can differ across various Pareto optimal allocations. Now in the competitive equilibrium - the equilibrium price, the equilibrium consumption and production of good l and the profits of firms are unaffected by any transfer of the numeraire good from one agent to another. Transferring the numeraire good from one agent to another changes the utility of the agents by exactly the amount of the transfer. Can attain any point on the Pareto optimal boundary of the utility possibility set by transferring numeraire good across consumers.

61 The Second Fundamental Theorem of Welfare Economics. Proposition. For any Pareto optimal levels of utility (u 1,..., u I ), there are transfers of the numeraire commodity (T 1,...T I ) satisfying I T i = 0 such that a competitive equilibrium reached from the endowments (ω m1 + T 1,..., ω mi + T I ) yields precisely the utilities (u 1,..., u I ).

62 Welfare Analysis in Partial Equilibrium. Social welfare function: assigns social welfare value (real number) to each profile of utility levels (u 1, u 2,...u I ) : W (u 1, u 2,...u I ) (Utilitarian welfare). Assume: W is strongly monotonic in its arguments. For any given consumption and production levels of good l, (x 1,...x I, q 1,..., q J ), where x i = vectors that are attainable are given by: {(u 1, u 2,..., u I ) : u i φ i (x i )+ω m q j,the utility c j (q j )}. As the boundary of this set expands, the maximum social welfare W attainable on this set (through redistribution of the numeraire good) increases (strictly).

63 Thus, *For any strongly monotonic social welfare function W, a change in the consumption and production of good l leads to an increase in (the maximum attainable) social welfare if and only if it increases the Marshallian surplus: S(x 1,...x I, q 1,...q J ) = [ φ i (x i ) c j (q j )]. Thus, social welfare analysis of changes in the consumption and production of good l can be carried out exclusively in terms of the Marshallian surplus.

64 Indeed, as we have seen, Pareto effi ciency also requires that the consumption and production of good l must satisfy max S(x 1,...x I, q 1,...q J ) (x 1,...,x I ) 0 (q 1,...q J ) 0 s.t. x i = q j.

65 Consider a consumption and production vector of good l, ( x 1,... x I, q 1,... q J ) such that for ŷ = (i) ( x 1,... x I ) solves: max [ x i,,..i s.t. φ i (x i )] x i x i = ŷ, x i 0, i = 1,..I. (ii) ( q 1,... q J ) solves min q j,,...j s.t. c j (q j ) q j = ŷ, q j 0, j = 1,..J.

66 We have seen that: φ i ( x i ) = P (ŷ) = B (ŷ), i such that x i > 0 c j ( q j ) = C (ŷ), j such that q j > 0, where P is the inverse aggregate demand function, B (.) is the industry marginal benefit and C (.) is the industry marginal cost (or the aggregate inverse supply function).

67 Now, Note: = [ S( x 1,... x I, q 1,... q J ) φ i ( x i ) = B(ŷ) C(ŷ) = = ŷ 0 ŷ 0 c j ( q j )] B (y)dy C(0) P (y)dy ŷ 0 ŷ 0 C (y)dy C (y)dy C(0) ŷ = [ [P (y) C (y)]dy] S(0) 0 ŷ [ 0 [P (y) C (y)]dy]

68 is the area between the aggregate demand and supply surves and can be written as : ŷ [ 0 [P (y) C (y)]dy] ŷ = [ [P (y) ŷp (ŷ)] + [ŷp (ŷ) (C(ŷ) C(0))] 0 = CS(P (ŷ)) + P S(P (ŷ)) where CS(p) and P S(p) denote the aggregate consumer and producer surplus generated in a (hypothetical) market with price taking consumers and producers at price market price p. Therefore, in partial equilibrium analysis, social welfare maximization, Marshallian surplus maximization and Pareto effi ciency are equivalent and eventually reduce to maximization of CS + PS.

69 ŷ It is easy to see that [ [P (y) C (y)]dy] is maximized at the output where: 0 P (y ) = C (y ) i.e., social marginal benefit equates industry s marginal cost.

70 As C (y) is inverse aggregate supply curve, this is also the aggregate output consumed and produced in a competitive equilibrium (supply=demand). Thus, competitive equilibrium outcome is equivalent to Marshallian surplus maximization.

71 All of this assumes no externalities or other distortions (taxes, subsidies etc). Welfare loss due to distortions is measured by the change in CS +PS i.e., the area between aggregate demand and the supply (or industry MC curve). Sometmes, called deadweight loss.

72 Example. Welfare loss due to a distortionary tax (in a competitive market). Sales tax on good l: t per unit paid by consumers. Tax revenue returned to consumers through lump sum transfer (non distortionary spending).

73 Let (x 1 (t),..., x I (t), q 1 (t),..., q J (t)) and p (t) be the competitive equilibrium allocation and price given tax rate t. FOC: φ i (x i (t)) = p (t) + t, for all i such that x i (t) > 0. c j (q j (t)) = p (t), for all j such that x j (t) > 0. Let x (t) = x(p (t) + t) = x i (t). Market clearing: x(p (t) + t) = q(p (t))

74 * Easy to check that (over the range where a strictly positive quantity is traded) p (t) is strictly decreasing in t and that (p (t) + t) is strictly increasing in t. x (t) is strictly decreasing in t (as long as it is strictly positice) and x (t) < x (0), t > 0. Let S (t) = S(x 1 (t),..., x I (t), q 1 (t),..., q J (t)). We have that S (t) = [ x (t) 0 [P (y) C (y)]dy] S(0) W elfare change = S (t) S (0) = x (t) x (0) [P (y) C (y)]dy which is negative since x (t) < x (0) and P (y) > C (y) for all y [0, x (0)).