solution to advanced micro midterm undergrad

Solution to the Sample Questions for the Mid-term Exam

Draft: March 22, 2011.

1. (Marshalliand & Hicksian Demand, Indirect Utility Function, Cobb-Douglaus)
(1) A consumer’s Marshallian demand function specifies what the consumer would buy in each price and wealth (or income), assuming it perfectly solves the utility maximization problem
.

(1)

The problem has a unique optimal solution when
(2)
since the utility function in (1) represents well-behaved preference. By putting
1 into (2) and then rearranging the equation, we have the MRS of
(3)
At a consumption bundle on the budget line that satisfies the condition in (3), the utility-maximization problem (1) has the solution and we find Jenny’s Marshallian demand function for each good is
,

(4)

.
(2) The indirect utility function prices , and income I, i.e.

where

and

is the maximum utility attained with

solves the utility maximization problem in (1). Therefore,

1

1

(5)
(3) Given her utility level , Jenny’s Hicksian demand function (or compensated demand function) for each good solves the expenditure minimization problem
(6)
As in Question 1.(1), we find a unique solution when the rate of exchange between two goods of equals the MRS of
, i.e., where the utility function is tangent to the expenditure function. By rearranging the condition in (3) with respect to and putting it into the budget constraint in (6), we have

Safely assuming that Jenny only consumes a positive amount of each good, we derive a Hicksian demand function for good 1 and good 2 as follows.
(7)
.
(4) By putting the given values ( = $1,
= $2,
= $400) into the equations in (4) and (5), we find that her maximum utility is 20,000 at
= (200, 100). That is,
Jenny’s utility is maximized when she consumes 200 units of beef and 100 units of rice with her (ordinal) utility of 20,000.
(5) Excise tax of $1 imposed on beef consumption increases from $1 to $2. By putting the changed