EC487 Advanced Microeconomics, Part I: Lecture 5

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1 EC487 Advanced Microeconomics, Part I: Lecture 5 Leonardo Felli 32L.LG October, 207

2 Pareto Efficient Allocation Recall the following result: Result An allocation x is Pareto-efficient if and only if there exists a vector of weights λ = (λ,..., λ I ), λ i 0, for all i I and λ h > 0 for at least one h I, such that x solves the following problem: max x,...,x I s.t I λ i u i (x i ) i= I x i ω i= () Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

3 Pareto Efficient Allocation (cont d) Recall we proved the only if statement. Proof: If: If x is Pareto-efficient then there exist λ such that x solves (2). To prove this implication we need the Second Welfare Theorem and the following Result. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

4 Pareto Efficient Allocation (cont d) Result Let U : R L + R be continuously differentiable, concave and monotonic. Consider the following problem: max U(x) s.t. p x p ω x R N + Then there exists a µ > 0 such that U(x) x l = µ p l l =,..., L Proof: By Kuhn-Tucker Theorem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

5 Pareto Efficient Allocation (cont d) We can now re-state the if statement above as follows. Result (If:) Assume that x is a Pareto-efficient allocation with x,i > 0 for all i I, and that u i ( ) are monotonic, concave and continuously differentiable. Then there exists an I -tuple λ,..., λ I > 0 such that x solves the planner s problem (2). Moreover, λ i is the inverse of the marginal utility of income. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

6 Pareto Efficient Allocation (cont d) Proof: By Second Welfare Theorem, since x is Pareto-efficient it is a Walrasian Equilibrium for endowments x,i = ω i and a price vector p > 0. Therefore for a given p consumers maximize their utility subject to budget constraint by choosing x,i. In other words, by the result above, there exists a I -tuple: γ,..., γ I > 0 such that: u i (x,i ) x i l = γ i p l i, l Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

7 Pareto Efficient Allocation (cont d) Consider now the central planner s Problem: max x,...,x I s.t I λ i u i (x i ) i= I x i ω i= (2) It is a concave problem: concave objective function and linear constraint, therefore x solves it if we can find an L-tuple α,..., α L > 0 such that: [ I ] λ i u i (x,i ) i= x i l = α l i, l Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

8 Pareto Efficient Allocation (cont d) or λ i u i(x,i ) x i l = α l i, l Choosing now λ i = γ i α l = p l and noticing that γ i is the marginal utility of income concludes the proof. Notice that α l are the shadow prices of the feasibility conditions, and according to the result above correspond to the Walrasian equilibrium prices. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

9 Production Economy Consider now an economy with I consumers and J producers characterized by their production possibility set Y j. Define a production economy with private ownership as follows: } E = {(ω,..., ω I ); u i ( ); Y j ; θ j i, i I, j J where θ j i is the share owned by consumer i of firm j, of course: I θ j i = i= j J Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

10 Walrasian Equilibrium with Production: Definition Define a Walrasian equilibrium for a production economy E as: { } p ; (x,,..., x,i ); (y,,..., y,j ) such that: x,i solves for every i I : max u i (x i ) s.t. p x i p ω i + x i J j= θ j i ( p y,j) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

11 Walrasian Equilibrium with Production: Definition (cont d) y,j solves for every j J: max p y j s.t. y j Y j y j and market clearing conditions are satisfied: I J I x,i y,j ω j 0 i= j= i= Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, 207 / 53

12 Aggregate Excess Demand Define aggregate excess demand for this economy as: Z(p) = I J x i (p) y j (p) I i= j= i= ω i where x i (p) is consumer i s Marshallian demand and y j (p) is firm j s net supply: y j l (p) > 0 is the supply of commodity l while yκ(p) j < 0 is minus unconditional factor demand of commodity κ. Notice that Z(p) is homogeneous of degree zero in p. Both x i (p) and y j (p) are homogeneous of degree zero in p. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

13 Walras Law Walras Law: p Z(p) = 0. This is obtained once again by summing each consumer s budget constraint: p x i (p) p ω i J j= θ j i ( p y j (p) ) = 0 That is: I I I p x i (p) p ω i J i= i= i= j= θ j i ( p y j (p) ) = 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

14 Walras Law (cont d) In other words: ( I I J I ) (p p x i (p) p ω i θ j i y j (p) ) = 0 i= i= j= i= or I I J p x i (p) ω i y j (p) = 0 i= i= j= We can now state the three main Theorems we have proved in a pure exchange economy for the production economy E. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

15 Existence Theorem Theorem (Existence Theorem) Consider a Z(p) that satisfies the following conditions:. Z(p) is non-empty; 2. Z(p) is convex valued; 3. Z(p) is upper-hemi-continuous; 4. Z(p) is homogeneous of degree 0; 5. Z(p) satisfies Walras Law; 6. Z(p) is bounded; then there exists p such that Z(p ) 0. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

16 Properties of Walrasian Equilibrium We now move to the properties of Walrasian equilibrium in the production economy E. Definition An allocation is feasible for a production economy E if and only if there exists a production plan y j for every firm j J such that y j Y j j J and I x i i= I ω i + J i= j= y j Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

17 Properties of Walrasian Equilibrium (cont d) Definition An allocation is Pareto-efficient for a production economy E if and only if it is feasible and there does not exists an alternative feasible allocation ˆx that Pareto-dominates it. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

18 First Welfare Theorem Theorem (First Fundamental Theorem of Welfare Economics) Let E be a production economy with consumer preferences satisfying weak monotonicity. Let { p ; (x,,..., x,i ); (y,,..., y,j ) } be a Walrasian equilibrium for E. Then x is a Pareto-efficient allocation. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

19 Second Welfare Theorem Theorem (Second Welfare Theorem) Assume that x, such that x,i > 0, is Pareto-efficient and that preferences u i ( ) are strongly monotonic; preferences are convex: u i ( ) are quasi-concave; technology Y j is convex for every j J and 0 Y j. Then there exists a redistribution of endowments ω i for i I and a vector of shares θ j i, for i I and j J such that { } p ; (x,,..., x,i ); (y,,..., y,j ) is a Walrasian equilibrium for the economy Ē. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

20 Second Welfare Theorem: Lemma Lemma Assume that x, such that x,i > 0, is Pareto-efficient and that: preferences are strongly monotonic; preferences are convex; Y j are convex for every j J; 0 Y j for every j J. Then there exists a y,j for every firm j J and a vector p such that: I J I a) x,i = y,j + ω i ; i= j= i= b) u i (x i ) > u i (x,i ) implies p x i > p x,i ; c) p y,j p y j for every y j Y j and every j J. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

21 Second Welfare Theorem: Proof Proof: Given this Lemma we just need to find the re-distribution ω i, θ j i, i, which at p gives exactly ( p x,i) to every consumer. Let V = I p x,i and i= V i = p x,i. Let also s i = V i V be i s share of the total value. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

22 Second Welfare Theorem: Proof (cont d) We prove the Theorem by setting: and ω i = s i I i= ω i i I θ j i = s i i I j J. By construction this choice of ω i and θ j i implies that the budget constraint of each agent is satisfied at p : J p ω i ( + p y,j) I J = s i p ω i + y,j = j= = s i p ( I i= θ j i x,i ) = s i ( I i= i= p x,i ) j= = s i V = V i = p x,i Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

23 Externalities Definition An externality is any indirect effect that either a production or a consumption activity has on a utility function, a consumption set or a production set An indirect effect is an effect that is: created by an economic agent other than the one who is affected; not transmitted through prices. Example: two firms that pollute each other environment, each one imposes a negative external effect on the other. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

24 Externalities (caveat) Notice that: If the two firms merge the pollution effect on each other is not an external effect any more but part of the firm s technology. If a market in pollution rights is created then firm i must buy from firm j a pollution right such as it would buy any other intermediate input: the externalities are incorporated into the market transactions. Notice, however, that the general equilibrium model does not treat as endogenous the size of agents and the number of markets. It takes them for given. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

25 Externalities: Examples. A firm polluting a river and thus decreasing the possibilities of consuming the water is an externality: the external effect of a production activity on a consumption set. In this case the consumption feasible set is a correspondence that depends on the production levels of the polluting firm X i (y j ) In general a production externality is such that: X i (y,..., y J ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

26 Externalities: Examples (cont d) 2. The noise emanating from the stereo system of one s neighbor is a typical consumption externality. The utility function of the concerned consumer i depends on consumer s j s music consumption x j m: u i (x i, x j m) In general, a consumption externality is characterized by: u i (x,..., x I ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

27 Externalities: Examples (cont d) 3. Meade (952) s famous example of the beekeeper and the orchard is a typical example of a mutual production externality. The production function of each firm depends on the input of the other firm: f (x, x 2 ), f 2 (x, x 2 ) In general, the PPS of both firms is a correspondence that depends on the production plan of all firms: Y j (y,..., y J ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

28 Externalities: Robinson Crusoe s Economy Consider a Robinson Crusoe s economy. There exists a single consumer, two goods and two firms. Clearly in this economy there is no issue of who owns the firms. Assume that there are two externalities imposed on firm 2: an externality generated by the consumer s consumption of good : x ; an externality generated by firm s production of good : y. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

29 Externalities: Robinson Crusoe s Economy (cont d) The consumer s preferences are: u(x, x 2 ). Firm s technology: y = f (y2 ) (differentiable and concave). Recall that by sign convention y2 is negative. Firm 2 s technology: y2 2 = f 2 (y 2, y, x ) (differentiable and concave). Recall that by sign convention y 2 is negative. Let ω = (ω, ω 2 ) be the consumer s endowment vector. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

30 Externalities: Robinson Crusoe s Economy (cont d) We consider first the Pareto efficient allocation. This is the solution to the following central planner s problem: max U(x, x 2 ) {x,x 2,y,y 2,y 2,y 2 2} s.t. x ω + y + y 2 x 2 ω 2 + y 2 + y 2 2 y = f (y 2 ) y 2 2 = f 2 (y 2, y, x ) Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

31 Externalities: Robinson Crusoe s Economy (cont d) Consider now the necessary and sufficient first order conditions: U f 2 λ + µ 2 = 0 x x U x 2 λ 2 = 0 λ µ + µ 2 f 2 y λ 2 + µ f y 2 = 0 = 0 λ 2 µ 2 = 0 λ + µ 2 f 2 y 2 = 0 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

32 Externalities: Robinson Crusoe s Economy (cont d) They can be re-written as: U + U f 2 x x 2 x U x 2 f 2 = y 2 + f 2 d f y d y2 = d f d y 2 This corresponds to the equality of: the social marginal rate of substitution, that takes the consumption externality into account, the social marginal rate of transformation of firm 2, that coincides with the private one, the social marginal rate of transformation of firm, that takes the production externality into account. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

33 Externalities: Robinson Crusoe s Economy (cont d) Consider now the Walrasian equilibrium of this economy. Notice that the key assumption is that each individual agent considers as parameters not only the prices but also the other variables that characterize his decision set. In particular, these variables y and x for firm 2 must be equal to the choices of the other agents. Let p = (p, p 2 ) be the vector of prices in this perfectly competitive economy. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

34 Externalities: Robinson Crusoe s Economy (cont d) Firm s maximization problem is clearly not affected by externalities: max {y,y 2 } p y + p 2y 2 s.t. y = f (y 2 ) The private marginal rate of transformation equals the price ratio: d f = p p 2 d y 2 Let y, = (y,, y, 2 ) be the solution to this problem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

35 Externalities: Robinson Crusoe s Economy (cont d) The consumer s utility maximization problem is also not affected by externalities: max {x,x 2 } U(x, x 2 ) s.t. p x + p 2 x 2 p ω + p 2 ω 2 + Π(p, p 2 ) where Π(p, p 2 ) = (p y, + p 2 y, 2 ) + (p y,2 + p 2 y,2 2 ) The private marginal rate of substitution equals the price ratio: U/ x U/ x 2 = p p 2 Let x = (x, x 2 ) be the solution to this problem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

36 Externalities: Robinson Crusoe s Economy (cont d) Firm 2 is affected by the externalities, from firm 2 s view point y, and x are given, it cannot control them: max {y 2,y 2 2 } p y 2 + p 2y 2 2 s.t. y 2 2 = f 2 (y 2, y,, x ) The private marginal rate of transformation equals the price ratio: f 2 (y 2, y,, x ) y 2 = p p 2 Let y,2 = (y,2, y,2 2 ) be the solution to this problem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

37 Externalities: Robinson Crusoe s Economy (cont d) Therefore the Walrasian equilibrium of this economy is a vector of prices p = (p, p 2) and an allocation such that: {x, y,, y,2 } The allocation {x, y,, y,2 } solves the three problems above given the vector of prices p ; Markets clear: x = ω + y, + y,2 x2 = ω 2 + y, 2 + y,2 2 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

38 Externalities: Robinson Crusoe s Economy (cont d) The key comparison is the one between the two marginal conditions that define the Pareto efficient allocation: U + U f 2 x x 2 x U x 2 f 2 = y 2 + f 2 d f y d y2 = d f d y 2 and the two marginal conditions that define the Walrasian equilibrium allocation: U x U x 2 f 2 = y 2 = d f d y 2 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

39 Failure of First Welfare Theorem Result The Walrasian equilibrium of an economy with externalities is not Pareto efficient. In other words in the presence of externalities the First Welfare Theorem does not hold. In general, economic decisions appear to be too decentralized at a Walrasian equilibrium allocation: they do not take into account the external effect that individual decisions have on other agents. In general: a firm exercising a negative externality will produce too much, a firm exercising a positive externality will produce too little. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

40 Incomplete Markets One way to interpret the inefficiency we identified is in terms of incomplete markets In other words, in the economy we considered two markets do not exist: the market through which firm acquires the right to exert an externality on firm 2; the market through which the consumer acquires the right to exert an externality on firm 2. One way to amend the inefficiency is to establish firm 2 s ownership rights on its production activity and hence create the missing markets. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

41 Completing Markets Assume that both these markets are perfectly competitive (very strong assumption). Let q,2 be the price at which firm must buy from firm 2 the right to exercise its externality, p,2 be the price at which the consumer must buy from firm 2 the right to exercise her externality. Let now (p, p 2, q,2, p,2 ) be the vector of prices of this redefined perfectly competitive economy. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

42 Completing Markets (cont d) Firm s maximization problem is now: max {y,y 2,y,2 } p y + p 2y2 q,2 y,2 s.t. y = f (y 2 ) y,2 = y Enforcement of ownership rights implies that y,2 = y. The marginal condition is now: d f = (p q,2 ) p 2 d y 2 Let (ŷ, ŷ 2, ŷ,2 ) be the solution to this problem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

43 Completing Markets (cont d) The consumer s utility maximization problem is now: where max {x,x 2,x,2 } U(x, x 2 ) s.t. p x + p 2 x 2 + p,2 x,2 p ω + p 2 ω 2 + ˆΠ x,2 = x ˆΠ = (p ŷ +p 2 ŷ 2 q,2 ŷ,2 )+(p ŷ 2 +p 2 ŷ 2 2 +q,2 ŷ,2 +p,2 ˆx,2 ) The marginal condition is then: U/ x = p + p,2 U/ x 2 Let (ˆx, ˆx 2, ˆx,2 ) be the solution to this problem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53 p 2

44 Completing Markets (cont d) Firm 2 controls the supply of the externality rights (ȳ,2,2, x therefore its maximization problem is now: {y 2,y 2 2 max,ȳ,2, x,2 } p y 2 + p 2y2 2 + q,2 ȳ,2 + p,2 x,2 s.t. y 2 2 = f 2 (y 2, ȳ, x ) ) The marginal conditions are then: p 2 f 2 y 2 + p = p 2 f 2 ȳ,2 + q,2 f 2 = p 2 x,2 + p,2 = 0 Let (ŷ 2, ŷ 2 2,2,ˆȳ, ˆ x,2 ) be the solution to this problem. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

45 Completing Markets (cont d) Therefore the Walrasian equilibrium of this economy is a vector of prices (p, p 2, q,2, p,2 ) and an allocation such that: {(ˆx, ˆx 2, ˆx,2 ), (ŷ, ŷ 2, ŷ,2 ), (ŷ 2, ŷ 2 2,ˆȳ,2, ˆ x,2 )} The allocation solves the three problems above given the vector of prices; Markets clear: x = ω + y. + y,2 x 2 = ω 2 + y. 2 + y,2 2 ˆȳ,2 = ŷ,2 ˆ x,2 = ˆx,2 Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

46 Completing Markets (cont d) Putting together the marginal conditions that define the Walrasian equilibrium we now conclude that: U x + U x 2 f 2 x U x 2 f 2 = y 2 + f 2 d f y d y2 = d f d y 2 In other words, when markets are complete the Walrasian equilibrium allocation is Pareto efficient: the First Welfare Theorem holds. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

47 Observation Observation (Coase 960) Provided markets are complete it does not matter for Pareto efficiency how property rights are allocated. Assume that firm is allocated ownership rights on a quantity Q of externality and any reduction in this quantity has to be purchased from firm. Similarly assume that the consumer is allocated ownership rights on an amount x of externality and any reduction has to be purchased from the consumer. Let once again (p, p 2, q,2, p,2 ) be the vector of prices of this redefined perfectly competitive economy. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

48 Observation (cont d) The consumer s utility maximization problem is then: max {x,x 2,x,2 } U(x, x 2 ) s.t. p x + p 2 x 2 p ω + p 2 ω 2 + Π + p,2 ( x x,2 ) x,2 = x where Π is the total profit of firm and 2, and ( x x,2 ) is the amount of externality that the consumer supplies. The budget constraint can then be re-written as: p x + p 2 x 2 + p,2 x,2 p ω + p 2 ω 2 + Π + p,2 x Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

49 Observation (cont d) Therefore the marginal condition is the same as the one above: U/ x = p + p,2 U/ x 2 Similarly for firm. Putting together the marginal conditions that define the Walrasian equilibrium we obtain once again: U + U f 2 x x 2 x U x 2 f 2 = y 2 p 2 + f 2 d f y d y2 = d f d y 2 In other words, the allocation of property right affects the distribution of surplus but does not affect the Pareto efficiency of the allocation (we will come back to the Coase Theorem). Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

50 Public Goods Definition A good is public if its use by one agent does not prevent other agents from using it. Individual consumption does not exhaust the good. Examples: national defense, pollution abatement programs etc... In general this definition is not as clear cut as it seems: voluntary subscription, size of the group affected, congestion. However, we will focus on pure public good: non-rivalry and non-exclusive. Subscription is not required, everyone is affected collectively, and there is no possibility of congestion. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

51 Public Goods (cont d) Consider an economy with I consumers and two good: one private z and one public y. Good y is produced, using the private good in the quantity z, by means of the following DRS technology: y = g(z), g > 0, g < 0 Preferences: U i (x i, y) Let ω be the aggregate endowment of the private good. We characterize first the Pareto-efficient allocation. Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

52 Public Goods (cont d) The central planner s problem: max {x i, i,y,z} s.t. I λ i U i (x i, y) i= I x i + z ω i= y = g(z) The first order conditions are: λ i U i x i = µ, i, I i= λ i U i y = ν, ν g (z) = µ Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

53 Public Goods (cont d) We then obtain the following Bowen-Lindahl-Samuelson marginal condition: I U i / y U i / x i = g (z) i= In other words, the sum of all consumers marginal rate of substitution between the public and the private good must equal the marginal rate of transformation (in the technology) between these two goods. Key question: are there decentralized resource-allocation mechanisms that yield a Pareto-optimal allocation? We will come back to the answer to this question Leonardo Felli (LSE) EC487 Advanced Microeconomics, Part I 27 October, / 53

Competitive Equilibrium

Competitive Equilibrium Econ 2100 Fall 2017 Lecture 16, October 26 Outline 1 Pareto Effi ciency 2 The Core 3 Planner s Problem(s) 4 Competitive (Walrasian) Equilibrium Decentralized vs. Centralized Economic