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Homework answers / question archive / (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)

(20%) For each of the following utility functions, compute the Walrasian demand function *ξ*(*p,w*), Hicksian demand function *h*(*p,v*) = (*h*_{1}(*p,v*)*,h*_{2}(*p,v*)), the expenditure function *e*(*p,v*) and the substitution or Slutsky matrix *S*(*p,w*). a)

*u*(*x*_{1}*,x*_{2}) = 2ln(*x*_{1}) + ln*x*_{2}

√ √

*u*(*x*_{1}*,x*_{2}) = 4*x*_{1 }+*x*_{2 }(Skip the Slusky matrix in this part)*u*(*x*_{1}*,x*_{2}) = min(*x*_{1}*,*2*x*_{2})*u*(*x*_{1}*,x*_{2}) = max(*x*_{1}*,x*_{2})

(20%) Each of the following four functions is a possible Marshallian demand function for two commodities at prices *p*_{1 }and *p*_{2}, 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)

(20%) Let *u *: R* ^{N }*→ R be continuous and twice continuously differentiable on

. Assume that for every

0 and *D*^{2}*u*(*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.

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

Let

.

- Show that
*V*is non-empty. - If p belongs to
*V*, find an expression for*Dx*(*p*) in terms of the derivatives of*u*.^{[1]} - 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 *S _{c}*(

- Provid an intuitive explanation of
*S*(_{c}*p*). (Hint: Using that*λ*is th shadow value of money.) f Show that*DS*(_{c}*p*) = −*x*(*p*).

g Show that the function *S _{c}*(

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 *S _{p}*(

Let *p _{E }*be the equilibrium price vector defined by the equation

h Show that *p _{E }*is the unique price vector that minimizes the function h(p).

(15%) A utility function is *homothetic *if

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.

(15%)

- Write down the definition for utility function to be quasilinear. (Please clarify the domain and range of the utility function.)
- Prove that there is no income effect when the utility function is quasilinear. ( Assuming the solution is interior in this part.)
- 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)?

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

- In depression in which all prices are reduced by 30%, as a consumer, do you like your wage to be reduced by 30 %?
- 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.

[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.