Partial Answers to Homework #1 3.D.5 Consider again the CES utility - - PDF document

SLIDE 1

Partial Answers to Homework #1

3.D.5 Consider again the CES utility function of Exercise 3.C.6, and assume that α1 = α2 = 1. Thus u(x) = [xρ

1 + xρ 2]1/ρ.

a) Compute the Walrasian demand and indirect utility functions for this utility function. We start by restricting our attention to the case (ρ < 1). We maximize the equivalent utility function [xρ

1+xρ 2]. The first-order conditions for an interior solution are ρxρ−1 1

= λp1 and ρxρ−1

2

= λp2 where λ is the multiplier corresponding to the budget constraint. Note that for ρ < 1, all solutions must be interior, otherwise xρ−1

i

would be undefined. (If ρ ≥ 1, the solutions will be at the corners.) Dividing the first-order conditions to eliminate λ yields xρ−1

1

xρ−1

2

= p1 p2

• . Thus

x1 = p1 p2

• 1

ρ−1

x2 . Substitute this expression in the budget constraint and solve for x2. Then substitute the result into the above equation to obtain x1. This gives the Walrasian demand functions: x1(p, w) = w p1 + p

1 1−ρ

1

p

ρ ρ−1

2

x2(p, w) = w p2 + p

1 1−ρ

2

p

ρ ρ−1

1

. These can also be written x1(p, w) = p

1 ρ−1

1

w p

ρ ρ−1

1

+ p

ρ ρ−1

2

x2(p, w) = p

1 ρ−1

2

w p

ρ ρ−1

1

+ p

ρ ρ−1

2

. Substituting into the actual utility function, we find the indirect utility v(p, w) = w[p

ρ ρ−1

1

+ p

ρ ρ−1

2

]

1−ρ ρ .

Using the shorthand δ = ρ/(ρ − 1) we can write these as xi(p, w) = wpδ−1

i

/[pδ

1 + pδ 2] and v(p, w) =

w[pδ

1 + pδ 2]−1/δ. Since ρ < 1, δ < 1.

b) Verify that these two functions satisfy all the properties of Propositions 3.D.2 and 3.D.3. For λ > 0, xi(λp, λw) = λwλδ−1pδ−1

i

/λδ[pδ

1 + pδ 2] = xi(p, w), so demand is homogeneous of degree zero. Also,

p1x1 + p2x2 = wpδ

1/[pδ 1 + pδ 2] + wpδ 2/[pδ 1 + pδ 2] = w, establishing Walras’s Law.

Concerning 3.D.3, v(λp, λw) = λwλ−1[pδ

1 + pδ 2]−1/δ = v(p, w), showing that indirect utility is

homogeneous of degree 0 in (p, w). As the quotient of non-zero continuous functions, v is continuous. Clearly v is increasing in w and decreasing in each pl. This leaves quasi-convexity, which can be shown by considering the bordered Hessian. c) Derive the Walrasian demand correspondence and indirect utility function for the case of linear utility and the case of Leontief utility (see Exercise 3.C.6). Show that the CES Walrasian demand and indirect utility functions approach these as ρ approaches 1 and −∞, respectively.

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PARTIAL ANSWERS TO HOMEWORK #1 Page 2

d) The elasticity of substitution between goods 1 and 2 is defined as ξ12(p, w) = −∂[x1(p, w)/x2(p, w)] ∂[p1/p2] p1/p2 x1(p, w)/x2(p, w). Show that for the CES utility function, ξ12(p, w) = 1/(1 − ρ), thus justifying its name. What is ξ12(p, w) for the linear, Leontief, and Cobb-Douglas utility functions? In part (a), we found x1/x2 = (p1/p2)1/(ρ−1). Thus ∂(x1/x2)/∂(p1/p2) = p1 p2

• 1

ρ−1 −1

. It follows that ξ12 = 1/(1 − ρ). 3.D.6 Consider the three-good setting in which the consumer has utility function u(x) = (x1 − b1)α(x2 − b2)β(x3 − b3)γ. a) Why can you assume that α+β +γ = 1 without loss of generality? Do so for the rest of the problem. For α + β + γ > 0, the transformation ψ(u) = u1/(α+β+γ) is increasing. Thus ψ(u) represent the same preferences and the exponents sum to 1. b) Write down the first-order conditions for the UMP, and derive the consumer’s Walrasian demand and indirect utility functions. This system of demands is known as the linear expenditure system and is due to Stone (1954). The utility function doesn’t make sense if xl < bl, so we will focus on the interior case and assume that p1b1 + p2b2 + p3b3 < w, which makes the interior case feasible. Ignoring corners (which can’t

• ccur here) the Lagrangian is L = (x1 − b1)α(x2 − b2)β(x3 − b3)γ − λ(p1x1 + p2x2 + p3x3 − w). The

first-order conditions are αu(x)/(x1 − b1) = λp1, βu(x)/(x2 − b2) = λp2, and γu(x)/(x3 − b3) = λp3. These reduce to p1/p2 = α(x2 −b2)/[β(x1 −b1)] and p1/p3 = α(x3 −b3)/[γ(x1 −b1)]. Substituting in the budget constraint and solving, we find x1 = b1 + α(w − p · b)/p1. Then x2 = b2 + β(w − p · b)/p2 and x3 = b3 + γ(w − p · b)/p2 where b = (b1, b2, b3). The indirect utility function is then v(p, w) = (α/p1)α(β/p2)β(γ/p3)γ(w − p · b). c) Verify that these demand functions satisfy the properties listed in Propositions 3.D.2 and 3.D.3. This is straightforward. 3.G.3 Consider the (linear expenditure system) utility function given in Exercise 3.D.6. a) Derive the Hicksian demand and expenditure functions. Check the properties listed in Propositions 3.E.2 and 3.E.3. Solving the expenditure minimization problem leads to the first-order conditions p1 = αλu(x)/(x1− b1), p2 = βλu(x)/(x2 − b2), and p3 = γλu(x)/(x3 − b3). These reduce to p1/p2 = α(x2 − b2)/[β(x1 − b1)] and p1/p3 = α(x3 − b3)/[γ(x1 − b1)]. Setting u(x) = u, and substituting for (x2−b2) and (x3−b3), we find h1−b1 = uαα−1ββγγp1−α

1

p−β

2 p−γ 3 .

Thus h1 = b1 + u(p1/α)α(p2/β)β(p3/γ)γ(α/p1), h2 = b2 + u(p1/α)α(p2/β)β(p3/γ)γ(β/p2), and h3 = b3 + u(p1/α)α(p2/β)β(p3/γ)γ(γ/p3). The expenditure function is then e(p, u) = p · b + u(p1/α)α(p2/β)β(p3/γ)γ. b) Show that the derivatives of the expenditure function are the Hicksian demand function you derived in (a). This is easily verified.

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PARTIAL ANSWERS TO HOMEWORK #1 Page 3

c) Verify that the Slutsky equation holds. d) Verify that the own-substitution terms are negative and that the compensated cross-price effects are symmetric. e) Show that S(p, w) is negative semidefinite and has rank 3.