# Microeconomic Theory � Econ 101 � 5 - Download as DOC

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```					               ECON 612: Microeconomics I – Midterm 1

Charles L. Baum II                                                            Fall 2005

Instructions: Answer all five questions, which are equally weighted. You may write on
the backs of these pages when necessary.

1.      Janet’s utility function is U(X,Y) = (X+1)Y where X and Y denote her
consumption of the goods X and Y. Janet purchases the goods at market prices pX and pY
and she has income I.

a) Derive Janet’s Hicksian demand functions and her expenditure function
(assuming an interior optimum).
b) Use the expenditure function to derive her indirect utility function.
c) Use her indirect utility function to derive the Marshallian demand function for
good X.
2.     Consider the simple labor supply model in which the individual consumes two
goods: leisure (L) and consumption (X). The individual has a total allocation of T hours,
which can be allocated for leisure or work. The individual receives a wage w for each
hour worked, pays the market price p for each unit of consumption X, has unearned
income I, and pays a tax of t on all earned income.

a) State the individual’s constrained optimization problem. Derive the first order
conditions for an internal optimum.
b) Use comparative statics to determine the effect of a change in unearned
income (I) on the optimal amount of consumption (X). Assume consumption
is a normal good.
c) Use comparative statics to determine the effect of a change in the earned
income tax (t) on the optimal amount of consumption (X).
d) Assume the individual is at an interior optimum. Using your comparative
static result in part c, identify the substitution effect and its sign and the
income effect and its sign.
e) Explain the intuition behind the substitution and income effects prompted by
an increase in t.
3.     Suppose Walt consumes two items: good X and good Y. The price of good X is
\$25, the price of good Y is \$6.25, and Walt has income I=100. Walt’s utility function is
U = (XY)1/2.
a) Find the utility maximizing amount of good X and good Y for Walt to consume.
b) What is Walt’s utility at this optimum?
c) Now suppose Walt seeks to identify the lowest-cost way of attaining utility of U =
2. How much of good X and good Y should Walt consume?
d) Now suppose Walt seeks the lowest-cost way of attaining utility of U = 4. How
much of good X and good Y should Walt consume?
the cost minimization setup.
f) Explain under what circumstances the utility maximization setup is equivalent to
the cost minimization setup. Also, explain when, if ever, they’re different.
4. Alan’s preferences satisfy the von Neumann-Morgenstern axioms, and he is
strictly risk averse. His von Neumann-Morgenstern utility is a function of income
spent on consumption, U(Y), where Y denotes income (let the price of consumption
be normalized to 1). With probability p, Alan develops no health problems and gets
U = U(Y); however, with probability (1-p), he gets develops health problems and dies
(and gets utility of U(0) = 0). However, Alan is considering whether to start smoking
cigarettes. Let the probability of dying be an increasing function of smoking (use S to
represent the number of cigarettes smoked), p’(S)<0.

a) Set up Alan’s expected utility function. Given this setup, how much will Alan
smoke? Explain why intuitively.
b) Now, suppose that smoking affects utility such that U = U(Y, S), where S is the
number of cigarettes smoked with US > 0 and where the first argument in the
utility function is redefined to be income spent on non-cigarette consumption.
Calculate Alan’s first order condition. What is the marginal benefit of smoking?
What is the marginal cost of smoking? Identify these from your first order
condition.
c) Now, revise your model to incorporate cigarette prices, pS. Show your new
expected utility function.
d) Let T be the tax on cigarettes. If T is increased, will Alan be more or less likely
to smoke? Show mathematically. Explain your results intuitively. What does
each term in your comparative static result mean?
5.      Now suppose Liz has the option of taking a job as an executive assistant that pays
\$62,000 with certainty or choosing a career in law. If Liz chooses the career in law, there
is a 50% chance she will not make partner and will only earn \$45,000. On the other
hand, if Liz chooses a career in law, there is a 50% chance she will make partner and will
earn \$80,000 annually. Liz's utility function is U(I) =     I, where I represents annual
income in dollars.
a) Given that Liz is risk-averse, should she become an executive assistant or choose
a career in law?
b) Now assume the job as an executive assistant is no longer an option. How much
is Liz willing to pay for insurance to protect herself from the variable income
associated with a career in law?
c) Graph Liz's utility function with respect to income.
a) Min pXX + pYY + {U* - (X+1)Y}.

First order conditions:
LX (X,Y, = pX - Y = 0
LY (X,Y, = py - (X+1) = 0
L (X,Y, = U* - (X+1)Y = 0

Thus, (X+1)/Y = pY/pX or Y = (X+1)pX/pY. By the constraint, U* = (X+1)2 pX/pY
or X+1 = [U*(pY/pX)]1/2 or X = [U*(pY/pX)]1/2 -1. Thus, Y = [U*(pX/pY)]1/2. E(pX,pY,U*)
= 2 [pXpYU*]1/2 – pX.

b) Since the expenditure and indirect utility functions are inverses, V = [(I + pX)/2]2
(1/pXpY) = (I+pX)2/4pXpY

c) You can use Roy’s identity: X(p,I) = -V(p,I)/ pX / V(p,I)/ I. X(p,I) = { (I+pX)/2
(1/pXpY) – [(I+pX)/2]2 (1/pX2pY) } / (I+pX)/2 (1/pXpY) = (I+pX)/2pX – 1 = (I+pX-2pX)/2pX
= (I-pX)/2pX

a)     From L(p,I) = U(X,Y) +  (I – pxX – pyY) or L(p,I) = (X,Y)1/2 +  (I – 25X –
6.25Y), we find that X* = 2 and Y* = 8.
b)     U* = (2*8)1/2 = 4.
c)     X* = 1 and Y* = 4.
d)     X* = 2 and Y* = 8.

e)     From L(p,I) = pxX + pyY +  ( U - U(X,Y)) or L(p,I) = 25X + 6.25Y +  (4-
(X,Y)1/2), we get the following first order conditions:

LX : 25 - (0.5)(XY)-1/2Y = 0

LY: 6.26 - (0.5)(XY)-1/2X = 0


Lλ: U - U(X,Y) = 0.

Dividing the first first order condition by the second yields Y = (px/py)X.

Plugging this into the third and solving for X yields

X = (py/px)1/2- U .

So, X* = 2 and Y* = 8.

f)     Assuming the same prices (px and py), the same income (I), and the same
utility functions (U(X,Y)) in both setups, utility maximization [L(p,I) =
U(X,Y) +  (I – pxX – pyY)] yields the same answers (X*,Y*) as cost
                    
minimization [L(p,I) = pxX + pyY +  ( U - U(X,Y))] when U (from the cost
minimization setup) is the same as U* (maximized utility from the utility

maximization problem). They yield different answers (X*,Y*) when U is
not equal to U*. And, of course, they would yield different answers if the
prices, incomes, and/or utility functions were different between the two
setups.

a)     She should become an executive assistant because (62,000)1/2 = 248.99 >
247.48 = 0.5(80,000)1/2 + 0.5(45,000)1/2.
b)            (62,500 – X)1/2 = 247.48. So, X = \$1253.64.
c)

Liz's Preferences

12
10
8
Utility

6                                                        Risk-Averse
4
2
0
1   11    21   31    41   51   61    71   81   91
Income

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