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#### (20%) For each of the following utility functions, compute the Walrasian demand function ξ(p,w), Hicksian demand function h(p,v) = (h1(p,v),h2(p,v)), the expenditure function e(p,v) and the substitution or Slutsky matrix S(p,w)

###### Economics

(20%) For each of the following utility functions, compute the Walrasian demand function ξ(p,w), Hicksian demand function h(p,v) = (h1(p,v),h2(p,v)), the expenditure function e(p,v) and the substitution or Slutsky matrix S(p,w). a)

1. u(x1,x2) = 2ln(x1) + lnx2

√       √

1. u(x1,x2) = 4              x1 +       x2 (Skip the Slusky matrix in this part)
2. u(x1,x2) = min(x1,2x2)
3. u(x1,x2) = max(x1,x2)

# 2

(20%) Each of the following four functions is a possible Marshallian demand function for two commodities at prices p1 and p2, respectively and when wealth is w. In each case, determine whether it is the demand function of a consumer with a locally non-satiated, continuous, and strictly quasi-concave utility function. If it is, say what the utility function is.

Otherwise, give a reason.

a)

!

b)

c)

d)

# 3

(20%) Let u : RN R be continuous and twice continuously differentiable on

. Assume that for every

0 and D2u(x) is

negative definite. For every

, Frisch demand, x(p), is defined to be the set of vectors that solve the problem

where λ is a fixed positive number.

1. Show tht if , then x(p) contins a single point.

Let

.

1. Show that V is non-empty.
2. If p belongs to V , find an expression for Dx(p) in terms of the derivatives of u. [1]
3. Show that the N × N matrix Dx(p) is symmetric an negative definite, so that Frisch demand curves slope downward.

Define consumer surplus to be Sc(p) = λ−1u(x(p)) − p · x(p).

1. Provid an intuitive explanation of Sc(p). (Hint: Using that λ is th shadow value of money.) f Show that DSc(p) = −x(p).

g Show that the function Sc(p) is strictly convex.

Suppose there is a fixed supply of the N commodities, represented by the N-vector y, where

). Define producer surplus to be the market value of y, that is, the producer

surplus is Sp(p) = p · y. Then the total surplus is

# h(p) = Sc(p) + Sp(p)

Let pE be the equilibrium price vector defined by the equation x(pE) = y.

h Show that pE is the unique price vector that minimizes the function h(p).

# 4

(15%) A utility function is homothetic if

# u(ax) = au(x) for all a > 0

a Prove, when the utility function is homothetic and the Walrasian demand is singlevalued, the demand is in the form ξ(p,w) = g(p)w for some function g. [2]b Prove that if the utility function is homothetic, then there is no Giffen good . c Explain briefly why you would or would not expect utility functions to be homothetic.

# 5

(15%)

1. Write down the definition for utility function to be quasilinear. (Please clarify the domain and range of the utility function.)
2. Prove that there is no income effect when the utility function is quasilinear. ( Assuming the solution is interior in this part.)
3. Now, we think about a utility funtion that is both homothetic and quasilinear. Due to (4a), the demand ξ(p,w) = g(p)w has an inome term w in it. So it seems that the inome effect presents in this case. Is there an inconsistency between (4a) and (5 b)?

# 6

(10%) Answer the following questions very briefly by no more than a short paragraph.

1. In depression in which all prices are reduced by 30%, as a consumer, do you like your wage to be reduced by 30 %?
2. In a depression in which all prices are reduced y 30%, as an employer, will you reduce your employees’ wages by 30%? If yes, please give a reason. If no, please mention what you will do as an employer.

# 7

[1] Feel free to assume x(p) is differentiable. In addition, note that a negative definite matrix is always invertible, as the eigenvalues are away from zero.

[2] Due to this property, homothetic utility functions are very convenient in some applications.

[3] Quasi-linear utility functions are prevalent in many cases. For instance, in cooperative games and in mechanism design, the linear term can be represented as the transfers.