Microeconomics II Proofs PDF

Microeconomics II Proofs Chapter 2: Pure Exchange Economy Lemma 2.2: Relative Prices Theorem 1: Relative Prices For all consumers, price vectors and endowments, xi (p, pωi ) = xi (αp, αpωi )...

Microeconomics II Proofs Chapter 2: Pure Exchange Economy Lemma 2.2: Relative Prices Theorem 1: Relative Prices For all consumers, price vectors and endowments, xi (p, pωi ) = xi (αp, αpωi ) for all α > 0. Proof by contradiction. Suppose that the lemma is not true. Then, there exists some consumer i, price p, endowment ω and number α > 0, such that xi (p, pωi ) 6= xi (αp, αpωi ). Define a := xi (p, pωi ) and b := xi (αp, αpωi ). Step 1: Since a = xi (p, pωi ) it follows that kj=1 pj aj = kj=1 pj ωij. Multiplying both sides by α P P we get k k αpj ωij. X X j j αp a = j=1 j=1 Thus, a is in the budget set when prices are αp. Similar, since b = xi (αp, αpωi ) it follows that k k. 1 αpj wij X X j j αp b = · α j=1 j=1 k k pj wij. X X j j p b = j=1 j=1 Thus, b is in the budget set when prices are p. Step 2: If ui (a) > ui (b), then it cannot be b = xi (αp, αpωi ) as a is in the budget set and gives higher utility. A contradiction. If ui (a) < ui (b), then it cannot be a = xi (p, pωi ) as b is in the budget set and yields higher utility. A contradiction. If ui (a) = ui (b), but a 6= b, then the demand functions are not unique, as assumed. 1 Microeconomics II Proofs Proposition 2.3: Walras’ Law Theorem 2: Walras’ Law For any price vector p, we have pz(p) ≡ 0, i.e., the value of the excess demand is identically zero. By definition of z(p) we have n X z(p) = [xi (p, pωi ) − ωi ] i=1 Xn pz(p) = p [xi (p, pωi ) − ωi ] i=1 n X = [pxi (p, pωi ) − pωi ] i=1 By definition of xi (p, pωi ) it must be that pxi (p, pωi ) = pωi hence n X pz(p) = = 0. i=1 Proposition 2.4: Market Clearing Theorem 3: Market Clearing If demand equals supply in k − 1 markets, and pk > 0, then demand must equal supply in the k-th market. By Walras’ law we have pz(p) = 0 " n n # X X p xi (p, pωi ) − ωi = 0 i=1 i=1 k " n n # xji (p, pωi ) ωij X X X j p − =0 j=1 i=1 i=1 k−1 " n n # " n n # xji (p, pωi ) ωij X X X X X j k p − +p xki (p, pωi ) − ωik =0 j=1 i=1 i=1 i=1 i=1 Since demand equals supply in k − 1 markets they will clear, k−1 " n n # X X j X j j p xi (p, pωi ) − ωi = 0 j=1 i=1 i=1 2 Microeconomics II Proofs but then this implies: n k " # X X k p xki (p, pωi ) − ωik = 0. i=1 i=1 And since pk > 0, we have n n " # X X xki (p, pωi ) − ωik = 0 i=1 i=1 Proposition 2.5: Free Goods Theorem 4: Free Goods If (p∗ , x∗ ), is a Walrasian equilibrium and z j (p∗ ) < 0, then p∗,j = 0. That is, if some good is in excess supply at a Walrasian equilibrium it must be a free good. By the definition of the Walrasian equilibrium n X n X xi (p∗ , p∗ ωi ) ≤ ωi i=1 i=1 n X n X xi (p∗ , p∗ ωi ) − ωi ≤ 0. i=1 i=1 This condition holds for each market, that is: n X n X xli (p∗ , p∗ ωi ) − ωil ≤ 0. i=1 i=1 Since prices are non-negative, " n n # X X p∗l xli (p∗ , p∗ ωi ) − ωil ≤ 0. (∗) i=1 i=1 But then " n n " ## X X X p∗l xli (p∗ , p∗ ωi ) − ωil ≤ 0. (∗∗) l6=j i=1 i=1 By Walras’ law " " n n ## " n n # X X X X j X j ∗l l ∗ ∗ l ∗j ∗ ∗ p xi (p , p ωi ) − ωi + p xi (p , p ωi ) − ωi = 0 l6=j i=1 i=1 i=1 i=1 Applying (∗∗) from above, it follows that " n n # X j ωij ≥ 0. X p∗j xi (p∗ , p∗ ωi ) − i=1 i=1 3 Microeconomics II Proofs But since (∗) holds for any l it also holds for l = j, thus " n n # X j X j p∗j xi (p∗ , p∗ ωi ) − ωi = 0, i=1 i=1 | {z } z j (p∗ ) 0. Observe that k X px̂i = pl x̂li l=1 pl xli + pj x̂ji X = |{z} l6=j =0 = pxi. Thus, if bundle xi was in the budget of agent i, then so is x̂i. Since the utility function of agent i is strongly increasing, then ui (x̂i ) > ui (xi ). But then it cannot be that xi = xi (p, pωi ) contradicting that (x, p) is a Walrasian equilibrium. Proposition 2.9: Intuition: Brouwer fixed-point Theorem Theorem 6: Brouwer fixed-point Theorem If f : S k−1 → S k−1 is a continuous function from the unit simplex to itself, there is some x in S k−1 such that x = f (x). 4 Microeconomics II Proofs For S 1 we have (p1 , 1 − p1 ) ∈ S 1 if p1 ∈ [0, 1]. Hence, it is sufficient to look at the mapping function f (p1 ) : [0, 1] → [0, 1]. Since f maps from [0, 1] to [0, 1] both f (0) ≤ 1 and f (1) ≥ 0. Graphically: Proposition 2.10: Existence of Walrasian Equilibria Theorem 7: Existence of Walrasian Equilibria If z : S k−1 → Rk is a continuous function that satisfies Walras’ law, pz(p) ≡ 0, then there exists some p∗ such that z(p∗ ) ≤ 0. Proof: Define a map g : S k−1 → S k−1 by pi + max{0, zi (p)} gi (p) = 1 + kj=1 max{0, zj (p)} P Since pi ≥ 0 and max{0, zi (p)} ≥ 0 for all i, then gi (p) ≥ 0 for all i and all p. Thus, g(p) ∈ Rk+ for all all p. 5 Microeconomics II Proofs Observe that k k X X pi + max{0, zi (p)} gi (p) = Pk i=1 i=1 1 + j=1 max{0, zj (p)} Pk Pk i=1 pi + max{0, zi (p)} = Pk i=1 1 + j=1 max{0, zj (p)} 1 + ki=1 max{0, zi (p)} P = 1 + kj=1 max{0, zj (p)} P = 1. Hence, g indeed maps into the simplex S k−1. Since the function z is continuous by assumption and the max function is also continuous, it follows that g is also continuous. Thus, we can apply the Brouwer’s fixed-point theorem to say that there exists p∗ such that p∗ = g(p∗ ), that is p∗i + max{0, zi (p∗ )} p∗i = (∗) 1 + kj=1 max{0, zj (p∗ )} P We will show that p∗ is a Walrasian equilibrium. Start with (∗) and multiply by the denominator, we get k X p∗i + p∗i max{0, zj (p∗ )} = p∗i + max{0, zi (p∗ )} j=1 k X p∗i max{0, zj (p∗ )} = max{0, zi (p∗ )} ∀ i = 1,..., k. j=1 Multiply each equation with zi (p∗ ) k X zi (p∗ )p∗i max{0, zj (p∗ )} = zi (p∗ ) max{0, zi (p∗ )} ∀ i = 1,..., k. j=1 We have a system of k equations:  z1 (p∗ )p∗1 kj=1 max{0, zj (p∗ )} = z1 (p∗ ) max{0, z1 (p∗ )}  P    z2 (p∗ )p∗2 kj=1 max{0, zj (p∗ )} ∗ ∗ P  = z (p ) max{0, z (p )}  2 2 .. k equations.     Pk  zk (p∗ )p∗k ∗ )} ∗ ∗  j=1 max{0, zj (p = zk (p ) max{0, zk (p )} Sum op these k equations " k # k  k X X X ∗ ∗  ∗  zi (p )pi max{0, zj (p )} = zi (p∗ ) max{0, zi (p∗ )} i=1 j=1 i=1 | {z } =0 by Walras’ law 6 Microeconomics II Proofs Then we get: k X zi (p∗ ) max{0, zi (p∗ )} = 0 i=1
2 Each element of this sum is either 0 if zi (p∗ ) ≤ 0 or is zi (p∗ ) if zi (p∗ ) > 0, thus each element of the sum is weakly positive. Suppose that there existed some i such that zi (p∗ ) > 0. Then, we could write the sum as k ∗
2 X zi (p ) + zj (p∗ ) max{0, zj (p∗ )} = 0 | {z } j6=i >0 | {z } ≥0 But then the sum would have to be strictly positive! A contradiction. Hence, ∀ i = 1,..., k it must be that zi (p∗ ) ≤ 0 and by definition p∗ forms a Walrasian equilibrium. Proposition 2.13: Equivalence of WPE and SPE Theorem 8: Equivalence of WPE and SPE Suppose that the utility functions of all agents are continuous and strongly increas- ing. Then an allocation is weakly Pareto efficient if and only if it is strongly Pareto efficient. For equivalence, we need to prove two things: that WPE implies SPE and that SPE implies WPE. SPE ⇒ WPE: By contradiction. Suppose not, then ∃ an allocation x which is SPE but not WPE. If x is not WPE, then there exists a feasible allocation x0 which all agents strictly prefer to x. But then all agents also weakly prefer x0 to x, and any single agent also strictly prefers it to x. But then x is not SPE. A contradiction. WPE ⇒ SPE: By contradiction. Suppose not, then ∃ an allocation x which is WPE but not SPE. Since x is not SPE, ∃ a feasible allocation x0 such that ui (xi ) ≤ ui (x0i ) ∀ i, and for at least one j, uj (xj ) < uj (x0j ). But consider the following allocation: agent j receives θx0j for θ ∈ (0, 1) while all other agents i receive x0i + n−1 1−θ 0 xj. By continuity of u, there exists some θ close enough to 1 such that uj (xj ) < uj (θx0j ) Since utility functions are strongly increasing:   1−θ 0 ui (xi ) ≤ ui (x0i ) 0 < ui xi + x. n−1 j We have constructed a feasible allocation that all agents prefer to x. But then x is not WPE. A contradiction. QED. 7 Microeconomics II Proofs Proposition 2.14: First Theorem of Welfare Economics Theorem 9: First Theorem of Welfare Economics If (x, p) is a Walrasian equilibrium and the utility functions of all agents are strongly increasing, then x is Pareto efficient. Proof: By contradiction. Suppose not. Then there exist a Walrasian equilibrium (x, p) such that x is not Pareto efficient. Then, there exists a feasible allocation x0 which all agents prefer. Since (x, p) is an equilibrium, then it must be that x0i is not in the budget set of any consumer. That is k k j wij pj ∀ i X X x0 i p j > j=1 j=1 Summing up over all agents n X k n X k j wij pj X X x0 i pj > i=1 j=1 i=1 j=1 k n k " # " n # j wij X X X X x0 i pj > pj. (∗) j=1 i=1 j=1 i=1 Since x0 is feasible, then n n j wij X X = x0 i ∀j i=1 i=1 " n # " n # j wij pj = X X x0 i p j. i=1 i=1 Summing up over j, we get k " n # k " n # j wij pj = X X X X x0 i pj. (∗∗) j=1 i=1 j=1 i=1 If we then combine (∗∗) and (∗) we obtain k " n # k " n # X X j X X j wi p j > wi pj. j=1 i=1 j=1 i=1 A contradiction. 8 Microeconomics II Proofs Proposition 2.15: Second Theorem of Welfare Economics Theorem 10: Second Theorem of Welfare Economics Suppose x∗ is a Pareto efficient allocation and that preferences are represented by continuous and strongly increasing utility functions. Suppose further that a competitive equilibrium exists from the initial endowments ωi = x∗i and let it be given by (p0 , x0 ). Then, in fact, (p0 , x∗ ) is a competitive equilibrium. Proof: Since ωi = x∗i , then for any price vector p, it must be pωi = px∗i. In particular p0 ωi = p0 x∗i. Thus, for all i, x∗i is in consumer i’s budget set. Since x0 is an equilibrium allocation, then for all i it must be ui (x0i ) ≥ ui (x∗i ). (∗) Since utility functions are continuous and monotonic, then SPE and WPE are equivalent. Then, as x∗ is Pareto efficient, there does not exist an agent i such that ui (x0i ) > ui (x∗i ). Combining this with (∗), we obtain that, for all i, it must hold ui (x0i ) = ui (x∗i ). But then if x0i maximizes agent i’s utility subject to the budget constraint, so does x∗i , and for all i. Then, since x∗ is obviously feasible, (p0 , x∗ ) is a competitive equilibrium. QED Proposition 2.16: Welfare maximization implies Pareto efficiency Theorem 11: Welfare maximization implies Pareto efficiency If x∗ maximizes a social welfare function, then x∗ is Pareto efficient. Proof: By contradiction. Suppose not. Then, there exists a feasible allocation x0 such that ui (x0i ) ≥ ui (x∗i ) ∀i and ul (x0l ) > ul (x∗l ) for some l Since W is increasing in all its arguments by assumption, W u1 (x∗1 ),..., ul (x0l ),..., un (x∗n ) > W u1 (x∗1 ),..., un (x∗n ).

By the same argument, W u1 (x01 ),..., un (x0n ) ≥ W u1 (x∗1 ),..., ul (x0l ),..., un (x∗n ).

Combining the two inequalitites, we obtain W u1 (x01 ),..., un (x0n ) > W u1 (x∗1 ),..., un (x∗n ).

which is in contradiction with the assumption that x∗ maximizes W. QED 9 Microeconomics II Proofs Proposition 2.17: Pareto efficiency maximizes some welfare function Theorem 12: Pareto efficiency maximizes some welfare function Let x∗ be a Pareto efficient allocation with x∗,j i > 0 for all i and all j. Let the utility functions be concave, continuous and strongly P increasing. Then, there is ∗ ∗ some choice of weights ai such that x maximizes ∗ ai ui (xi ) subject to resource constraints. Proof: By the STWE, x∗ is also a Walrasian equilibrium for some price vector. Call this price vector p∗. Then, by the definition of the Walrasian equilibrium, for each agent i, x∗i maximizes utility given the budget constraint p∗ x∗i. Consider that maximization problem max ui (xi ) xi s.t. p∗ xi = p∗ x∗i The Lagrangian of this problem is Li = ui (xi ) − λi (p∗ xi − p∗ x∗i ) and the first order condition is ∂Li ∂ui (xi ) ! = − λi p∗,j = 0. ∂xji ∂xji Since x∗i maximizes the above problem, it has to satisfy all the FOCs. Hence ∂ui (x∗i ) = λi p∗,j , ∀ i. (∗) ∂x∗,j i Now consider the problem of maximizing the welfare function: n X max ai ui (xi ) x i=1 n n x∗,1 X X s.t. x1i = i i=1 i=1... n n x∗,k X X xki = i i=1 i=1 The Lagrangian is given by n n n n n ! ! x∗,1 x∗,k X X X X X 1 LW = ai ui (xi ) − µ x1i − i −... − µ k xki − i i=1 i=1 i=1 i=1 i=1 10 Microeconomics II Proofs We have a FOC for each i, j, ∂LW ∂ui (xi ) ! = ai · − µj = 0 ∂xji ∂xji The problem is that a single number, µj , has to satisfy n equations, one for each i. However, we can rewrite the FOCs as ∂ui (xi ) 1 j = µ ∂xji ai But, if ai = λ1i for (∗) and xi = x∗i for all i, then the FOCs are satisfied for µj = p∗,j ! Since ui are concave for all i, this is a sufficient (see “Kuhn-Tucker sufficiency” on page 503 of Varian) condition for x∗ to be the solution to the welfare maximization problem. QED 11 Microeconomics II Proofs Chapter 3: Production Economy Proposition 3.1: Aggregate profit maximization Theorem 13: Aggregate profit maximization An aggregate production plan y maximizes aggregate profit if and only if each firm’s production plan y` maximizes its individual profit. We need to prove two implications: 1. If y maximizes aggregate profits, then yl maximizes firm l0 s profits. Proof: Suppose not. Then there exists a firm l and a plan yl0 such that pyl0 > pyl. But then y does not maximize aggregate profits because a plan (y1 ,..., yl0 ,..., yf ) achieves higher ag- gregate profits. A contradiction. Pf 2. If yl maximizes profits for each firm l, then y = l=1 yl maximizes aggregate profits. Proof: Suppose not. Then, there exists a plan y0 such that py0 > py f X f X p yl0 > p yl l=1 l=1 f X f X pyl0 > pyl. l=1 l=1 But then, there exists at least one firm l such that pyl0 > pyl which implies that yl is not profit-maximizing for firm l. QED. 12 Microeconomics II Proofs Chapter 5: The core of an economy Proposition 5.3: Walrasian equilibirum is in the core Theorem 14: Walrasian equilibirum is in the core If (x∗ , p) is a Walrasian equilibrium with initial endowments ω, then x∗ ∈ C(ω). Proof: Suppose not. Then x∗ ∈ / C(ω). Thus, there exists a coalition S ≤ N and allocation 0 x such that ui (x0i ) > ui (x∗i ) ∀ i ∈ S X X and x0i = ωi. i∈S i∈S Multiply the last equality with p: X X p x0i = p ωi. (∗) i∈S i∈S Since x∗ is an equilibrium, then px0i > pωi ∀i ∈ S. Summing up, we obtain X X p x0i > p ωi , i∈S i∈S which contradicts (∗). Proposition 5.5: Equal treatment in the core Theorem 15: Equal treatment in the core Suppose that preferences can be represented with strictly quasiconave, strongly increasing and continuous utility functions. Then, if x is an allocation in the core of an r-fold replica of the economy, then any two agents of the same type must receive the same bundle. Let x be an allocation in the core and index the 2r agents with A1 ,..., Ar , B1 ,..., Br. Suppose the proposition is not true. Then, there exist two agents of the same type who do not receive the same bundle. (∗). Pick two agents i and j such that uA (xAj ) ≤ uA (xAl ) ∀ l ∈ {1,..., r} uB (xBi ) ≤ uB (xBl ) ∀ l ∈ {1,..., r} 13 Microeconomics II Proofs There may be multiple agents satisfying this condition, but at least one exists. Let r 1X xA = xAl r l=1 r 1X xB = xBl. r l=1 Since x is feasible, then r r r r 1X 1X 1X 1X xAl + xBl = ωAl + ωBl r r r r l=1 l=1 l=1 l=1 1 1 = rωA + rωB r r = ωA + ωB. Hence xA + xB = ω A + ω B (∗∗) By (∗) we know that there exist two agents of the same type who do not get the same bundle. Suppose, without loss of generality, that they are of type A. Then, by strict quasiconcavity of uA we have uA (xAj ) < uA (xA ) (The above statement is not obvious and should be proven). Further, by strict quasicon- cavity of uB we have uB (xBi ) ≤ uB (xB ). Since u is continuous and monotonic, there exists some ε > 0 such that uA (xAj ) < uA (xA − 1ε) and uB (xBi ) < uB (xB + 1ε). Since xA − 1ε + xB + 1ε = xA + xB is feasible by (∗∗), then agents Aj and Bi form a blocking coalition. 14 Microeconomics II Proofs Proposition 5.6: Shrinking core; or Edgeworth-Debreu-Scarf Theorem Theorem 16: Shrinking core; or Edgeworth-Debreu-Scarf Theorem Suppose that preferences can be represented with strictly quasiconave, strongly increasing and continuous utility functions. Further suppose that there is a unique Walrasian equilibrium x∗ from initial endowments ω. Then, if x 6= x∗ there exists a replication r such that x is not in the r-core. Sketch of the proof Let y be a point in the core, but not be an equilibrium. Then the line connecting ω and y has to cut the indifference line. This means that there exists an allocation gA which A-types prefer to yA. We can represent it by gA = θωA + (1 − θ)yA for θ ∈ (0, 1). By continuity of u, we can suppose that θ = VT for some integers T and V. Replicate the economy V times. Consider the coalition of all A-type agents and V − T B-type agents. Consider an allocation z, where all A-type agents receive gA and all B- type agents in the coalition receive yB. Since all A-type agents strictly prefer z to y and B-type are indifferent, this forms a blocking coalition if the allocation z is feasible. Note 15 Microeconomics II Proofs that:   T T V gA + (V − T )yB = V ωA + (1 − )yA + (V − T )yB V V = T ωA + (V − T )yA + (V − T )yB = T ωA + (V − T )(yA + yB ) = T ωA + (V − T )(ωA + ωB ) = T ωA + (V − T )ωA + (V − T )(ωB ) = V ωA + (V − T )ωB. Which is exactly the endowment of our coalition. 16 Microeconomics II Proofs Chapter 7: Public goods Proposition 7.1: Efficient provision of a public good Theorem 17: Efficient provision of a public good Consider the allocation xi = ωi and gi = 0 for i = 1, 2. It is a Pareto improvement to provide a public good if and only if r1 + r2 > c. We need to prove two statements: Step 1: If it is a Pareto improvement to provide the public good, then r1 + r2 > c. Proof: Direct proof. If it is a Pareto improvement to provide the public good, then there exist some contributions g1 and g2 , such that g1 + g2 ≥ c and u1 (1, ω1 − g1 ) > u1 (0, ω1 ) u2 (1, ω2 − g2 ) > u2 (0, ω2 ) By definition of ri , we then have u1 (1, ω1 − g1 ) > u1 (1, ω1 − r1 ) u2 (1, ω2 − g2 ) > u2 (1, ω2 − r2 ) Since u is strongly increasing then ω1 − g1 > ω1 − r ω2 − g2 > ω2 − r2 or r1 > g1 r2 > g2 adding them up, we get r1 + r2 > g1 + g2 and since g1 + g2 ≥ c, we get r1 + r2 > c. Step 2: If r1 + r2 > c, then it is Pareto improving to provide the public good. Proof: There exists some ε > 0, sucht that r1 + r2 − 2ε > c. By definition of ri , we have ui (1, ωi − ri ) = ui (0, ωi ) Since ui is increasing then ui (1, ωi − ri + ε) > ui (0, ωi ). 17 Microeconomics II Proofs But then, contributing gi = ri − ε provides the public good and leads to a strictly higher utility for both agents. 18