Slutsky Decomposition

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1 Slutsky Decomposition Econ 2100 Fall 2015 Lecture 7, September 23 Outline 1 Connections between Walrasian and Hicksian demand functions. 2 Slutsky Decomposition: Income and Substitution Effects

2 Definition From Last Class Given a utility function u : R n + R, the Hicksian demand correspondence h : R n ++ u(r n +) R n + is defined by h (p, v) = arg min p x subject to u(x) v. x R n + Definition Given a continuous utility function u : R n + R, the expenditure function e : R n ++ u(r n +) R + is defined by for some x h (p, v). e(p, v) = p x Proposition Suppose u is continuous and locally nonsatiated. If v > u(0 n ), then: x (p, w) = h (p, v(p, w)); h (p, v) = x (p, e(p, v)). Next a Proof of this proposition.

3 Walrasian and Hicksian Demand Are Equal We want to show that x x (p, w) solves min p x subject to u(x) u(x ) We want to show that x h (p, v) solves max u(x) subject to p x p x Proof. Pick x x (p, w), and suppose x / h (p, u(x )): x is a utility maximizer but not expenditure minimizer. Then x s.t. p x < p x w and u(x ) u(x ) By local nonsatiation, x (close to x ) s.t. p x < w and u(x ) > u(x ) contradicting the fact that x is a utility maximizer. Therefore x must also be a expenditure minimizer. Since p x = w by full expenditure, we also have e (p, v(p, w)) = w. Pick x h (p, v), and suppose x / x (p, p x ): x an expenditure minimizer but not a utility maximizer. Then x s.t. u(x ) > u (x ) and p x p x Consider the bundle αx with α < 1. By continuity of u, u(αx ) > u (x ) v for α < 1, α 1. Therefore, p αx < p x p x contradicting the fact that x is an expenditure minimizer. Therefore x must also be a utility maximizer. Because u (x ) = v we have v (p, e(p, v)) = v.

4 Definition Support Function Given a closed set K R n, the support function µ K : R n R { } is µ K (p) = inf x K p x. It can equal since there might exists x K such that p x becomes unboundedly negative for a closed set. for example: K = {x R 2 : x 1 R, x 2 [0, )}, and p = ( 1, 0). if K is convex (closed and bounded in R n ), the support function is finite. When a set K is convex, one can recover it using the support function: given a p R n, {x R n : p x µ K (p)} is an half space that contains K ; furthermore, K is the intersection of all such half spaces (for all p). If K is not convex, the intersection of all sets {x R n : p x µ K (p)} is the smallest closed convex set containing K (this is called the convex hull). Theorem (Duality Theorem) Let K be a nonempty closed set. There exists a unique x K such that p x = µ K (p) if and only if µ K is differentiable at p. If so, µ K (p) = x. The support function is linear in p. e(p, v) is the support function of the set K = {x R n + : u(x) v}.

5 Properties of the Expenditure Function Proposition If u(x) is continuous, locally nonsatiated, and strictly quasiconcave, then e(p, v) is differentiable in p. Proof. Immediate from the previous theorem (verify the assumptions hold). Question 4 Problem Set 4 Using the result above and the fact that h (p, v) = x (p, e(p, v)), provide suffi cient conditions for differentiability with respect to p of the indirect utility function and extend the suffi cient conditions for continuity and differentiability in p of Walrasian demand to Hicksian demand.

6 Shephard s Lemma Proposition (Shephard s Lemma) Suppose u : R n + R is a continuous, locally nonsatiated, and strictly quasiconcave utility function. Then, for all p R n ++ and v R, h (p, v) = p e(p, v). Hicksian demand is the derivative of the expenditure function. There are diffrent ways to prove Shephard s Lemma: Use the duality theorem. Use the envelope theorem: let φ(x, q) = p x, φ(x (q); q) = e(p, v), and F (x, q) = u(x) v, then: D qφ(x ( q); q) = D qφ(x, q) x=x (q),q= q [λ ( q)] D qf (x, q) x=x ( q),q= q becomes pe(p, v) = h (p, v) 0 since F does not depend on p. Use a brute force and ignorance (bfi) argument (next slide).

7 Shephard s Lemma: Proof Proof. We want to show that h (p, v) = p e(p, v). Using the definition, and then the chain rule p e(p, v) = p [p h (p, u)] = h (p, u) + [p D p h (p, u)] The first order conditions of the minimization problem say p = λ p u (h (p, u)) Therefore p e(p, v) = h (p, u) + λ [ p u (h (p, u)) D p h (p, u)] At an optimum, the constraint must bind and so u (h (p, u)) = v Thus p u (h (p, u)) D p h (p, u) = 0 and therefore h (p, v) = p e(p, v). as desired.

8 Roy s Identity Proposition (Roy s identity) Suppose u is continuous, locally nonsatiated, and strictly quasiconcave and v is differentiable at (p, w) 0. Then x (p, w) = 1 p v(p, w), This can be also written as x k (p, w) = v (p,w ) w v (p,w ) v (p,w ) w, for all k Walrasian demand equals the derivative of the indirect utility function multiplied by a correction term. This correction normalizes by the marginal utility of wealth. There are diffrent ways to prove Roy s Identity Use the envelope theorem (last class). Use the chain rule and the first order conditions. Use a bfi argument (next slide).

9 Proof. Roy s Identity We want to show that v (p,w ) xk p (p, w) = k v (p,w ) w Fix some p, w and let ū = v ( p, w). The following identity holds for all p v (p, e (p, ū)) = ū differentiating w.r.t p k we get v + v e = 0 w and therefore v + v w h k = 0 Let x k = x k ( p, w) and evaluate the previous equality at p, w: v + v w x k = 0 Solve for x i to get the result.

10 Definition Slutsky Matrix The Slutsky matrix, denoted D p h (p, v), is the n n matrix of derivative of the Hicksian demand function with respect to price (its first n dimensions). Notice this says function, so the Slutsky matrix is defined only when Hicksian demand is unique. Proposition Suppose u : R n + R is continuous, locally nonsatiated, strictly quasiconcave and h ( ) is continuously differentiable at (p, v). Then: 1 The Slutsky matrix is the Hessian of the expenditure function: D p h (p, v) = D 2 ppe(p, v); 2 The Slutsky matrix is symmetric and negative semidefinite; 3 D p h (p, v)p = 0 n. Proof. Question 5, Problem Set 4.

11 Proposition Slutsky Decomposition If u is continuous, locally nonsatiated, and strictly quasiconcave, and h is differentiable, then: for all (p, w) D p h (p, v(p, w)) = D p x (p, w) + D w x (p, w)x (p, w) or h j (p, v(p, w)) = x j (p, w) + x j (p, w) xk (p, w), w for all j, k Proof. Remember that h (p, v) = x (p, e(p, v)). Take the equality for good j and differentiate with respect to p k hj = x j + x j e = x j + x j w w h k Evaluate this for some p, w and u = v (p, w) so that h ( ) = x ( ) h j = x j + x j w x k

12 Slutsky Equation Hicksian decomposition of demand Rearranging from the previous proposition: xj (p, w) = h j (p, v(p, w)) x j (p, w) xk (p, w) p }{{ k w }}{{} substitution effect income effect This is also known as the Slutsky equation: it connects the derivatives of compensated and uncompensated demands. If one takes k = j, the following is the own price Slutsky equation xj (p, w) = h j (p, v(p, w)) x j (p, w) xj (p, w) w }{{}}{{} substitution effect income effect

13 Normal, Inferior, and Giffen Goods The own price Slutsky equation x j (p, w) = h j (p, v(p, w)) x j (p, w) xj (p, w). w }{{}}{{} substitution effect income effect Since the Slutsky matrix is negative semidefinite, h k / 0; the first term is always negative while the second can have either sign; the substitution effect always pushes the consumer to purchase less of a commodity when its price increases. Normal, Inferior, and Giffen Goods A normal good has a positive income affect. An inferior good has a negative income effect. A Giffen good has a negative overall effect (i.e. x k / > 0); this can happen only if the income effect is negative and overwhelms the substitution effect x k w x k < h k 0.

14 Definition ε y, q = y q q y Slutsky Equation and Elasticity is called the elasticity of y with respect to q Elasticity is a unit free measure (percentage change in y for a given percentage change in q) that is often used to compare price effects across different goods. The own price Slutsky equation x j x j (p, w) = h j (p, v(p, w)) x j (p, w) xj (p, w) w Multiply both sides by p j /xj ( ) and rewrite as (p, w) p j xj (p, w) = h j (p, v(p, w)) p j xj (p, w) x j (p, w) w p j w xj (p, w) xj (p, w) w or p j xj (p, w) ε xj, pj = ε hj, pj ε xj, w w The elasticty of Walrasian (uncompensated) demand is equal to the elasticity of Hicksian (compensated) demand minus the income elasticity of demand multiplied by the share that good has in the budget.

15 Gross Substitutes The cross price Slutsky equation x j (p, w) = h j (p, v(p, w)) x j (p, w) xj (p, w) w Intuitively, we think of two goods as substitutes if the demand for one increases when the price of the other increases. Definition We say good j is a gross substitute for k if x j (p,w ) 0. Unfortunately the definition of substitutes based on Walrasian (uncompensated) demand is not very useful since it does not satisfy symmetry: we can have x j (p,w ) x > 0 and k (p,w ) < 0. There is a better definition that is based on Hicksian (compensated) demand.

16 Definition Net Substitutes We say goods j and k are net substitutes if h j (p,w ) 0 and net complements if h j (p,w ) 0. This definition is symmetric: h j (p,w ) 0 if and only if h k (p,w ) 0. Proof. h j (p, w) = e(p,v ) = 2 e(p, v) = 2 e(p, v) = h k (p, w) by Shephard s Lemma by Young s Theorem by Shephard s Lemma again

17 Econ 2100 Fall 2015 Problem Set 4 Due 28 September, Monday, at the beginning of class 1. Suppose u is twice continuously differentiable, locally nonsatiated, strictly quasiconcave, and that there exists ε > 0 such that x i ( p, w) > 0 if and only if x i (p, w) > 0, for all (p, w) such that ( p, w) (p, w) < ε. Also assume that H is nonsingular at ( p, w). Prove that under these conditions x is continuously differentiable at ( p, w). 2. Suppose a government has to decide on a budget. Its tax revenue in 2012 is Y and its tax revenue in 2013 will be ky, where k > 1 is the growth rate of the economy. The government has to decide how much to spend this year, x 1, and how much to spend next year, x 2. Because of the poor economy this year, the government will certainly run a deficit x 1 Y > 0. For each dollar of deficit, the government will have to pay bond holders 1 + r dollars next year, where r > 0 is the interest rate. Next year, the government can spend its 2013 tax revenue ky, minus what it will owe bond holders to finance the 2012 deficit. (The government cannot continue to run a deficit in 2013). The government s objective function is U(x 1, x 2 ) = v(x 1 ) + v(x 2 ), where v refers to the annual social benefit generated by government spending. Assume v is C 2, v (c) > 0 and v (c) < 0 for each c. Assume a strictly positive solution (i.e., x 0). (a) Write the government s budget constraint over 2012 and 2013 spending. (b) Using the Implicit Function Theorem, determine the comparative statics of 2012 and 2013 spending with respect to the interest rate and the growth rate. That is, find x 1, x 1, x 2, r k r x 2. as explicitly as possible, and determine the signs of these expressions as fully as you can. k Interpret briefly. 3. Prove that if u is continuous and locally nonsatiated, then the expenditure function is homogeneous of degree 1 and concave in p. 4. Using this fact and h (p, v) = x (p, e(p, v)), provide suffi cient conditions for differentiability with respect to p of the indirect utility function and extend the suffi cient conditions for continuity and differentiability in p of Walrasian demand to Hicksian demand. 5. Suppose u : R n + R is continuous, locally nonsatiated, strictly quasiconcave (equivalently, you need h to be single-valued) and h is continuously differentiable at (p, v). Let D p h (p, v) denote the n n derivative matrix of the Hicksian demand function h with respect to price (its first n dimensions), which we call the Slutsky matrix. Prove the following. (a) D p h (p, v) = D 2 ppe(p, v) (the Slutsky matrix is the Hessian of the expenditure function); (b) D p h (p, v) is symmetric and negative semidefinite (M is symmetric if M ij = M ji for all i, j = 1,..., n, and it is negative semidefinite if z Mz 0 for all z R n. ); (c) D p h (p, v)p = 0 n. 1

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