# Microeconomic Theory � Econ 101 � 5 - DOC

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```					              ECON 6120: Microeconomics I – Midterm 1

Charles L. Baum II                                                          Fall 2010

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

1.     An individual consumes two goods and has indirect utility function

V(p1,p2,I) = (I+p2)2 / p1p2.
a) Derive the expenditure function (ignoring any corner solutions).
b) Derive the Marshallian demand for good 1 (ignoring any corner solutions).
2.     Nina is optimizing consumption over two periods. Thus, she is implicitly
deciding how much to save in period 1. Her utility function is separable:

U(C0,C1) = U(C0) + βU(C1)

where β is a preference parameter that tells us how much she values consumption today
relative to consumption tomorrow. Generally, we assume that 0 ≤ β ≤ 1, so that
consuming sooner is preferred to consuming later. Further, Nina faces the usual budget
constraint,

C1 = (1+r)(I0-C0) + I1,

where r is the interest rate.

a)      Set up the Legrangian and solve for Nina’s first order conditions.
b)      Using comparative statics, find the effect of β on C0.
c)      Does Nina consume more in the present or in the future? To answer this
question, consider the following three scenarios:
i)             Does Nina consume more in the present or in the future if
β(1+r) = 1?
ii)            Does Nina consume more in the present or in the future if
β(1+r) < 1?
iii)           Does Nina consume more in the present or in the future if
β(1+r) > 1?
3.      A 2005 U.S. Supreme Court ruling (referred to as Kelo) allows the government to
take private property for the purpose of economic development. Does the Kelo ruling
discourage private investment?

Suppose an entrepreneur has the following utility function,

U(x) = πv(x) + (1 – π)[c(x) + B] – x,

where π is the probability of the government not taking the business for economic
development, v(x) is the value of the business if not taken, c(x) is the compensation for
the business if taken by the government, B is the public use benefit to the entrepreneur
from the taking, and x is the investment by the entrepreneur.1 Let c be a function of the
actual value of the business such that c(x) = γv(x), where 1 > γ > 0. Assume the value
function, v(x), is concave in x.

a) What is (are) the entrepreneur’s choice variable(s)? Calculate the corresponding
first order condition(s).

b) Let the Kelo ruling be the equivalent of increasing the probability of the
government taking private property (or a business) for economic development.
(Perhaps the probability of this happening prior to the ruling was zero.) What is
the effect of a decrease in π on private business investment? Explain your
comparative static result.

c) Let the value of the business be v(x) = xα, where 1 > α > 0. Solve for the optimal

1
DeGennaro, Raymond P., and Tianning Li. (2010). “How Did Kelo Affect Business Formation?”
Working Paper. Knoxville, TN: University of Tennessee.
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. While the decade of the 1980s was a time of great continued growth for the United
States, some have been critical of this period because they feel that the 80s resulted in the
rich getting richer and the poor getting poorer. Some suggested that a redistribution of
income via the federal tax system should occur whereby the taxes paid by the rich should
be increased with the additional money raised being distributed to the poor in an attempt
to equalize incomes. I want to see what would happen if the government were to
introduce a tax system that completely equalizes incomes.

Below are shown two consumers -- a richer consumer who, when faced with a wage rate
of \$5.00 an hour, works 40 hours and a poorer consumer who, when faced with a wage
rate of \$5.00 an hour, works 20 hours. The time frame for this question is 112 hours and
the price of a unit of composite consumption good "C" is PC= \$1.00.

Implement a federal redistribution of income program where all people receive an income
of \$150.00 regardless of hours worked. Under this program, if anyone works enough
hours to earn more than \$150.00 the government will tax away income in excess of
\$150.00. If anyone doesn’t work enough hours to earn \$150, the government will
provide the needed cash to raise income to \$150.00.

a) Draw in the new budget constraint with new indifference curves (one for each type
of consumer) showing the new utility maximizing consumption bundle each individual
will consume under the redistribution of income program.
b) How will this redistribution of income affect the number of hours people work? In
your answer, state the sign of the substitution, income, and total effects (for each type of
consumer) that result from this government program. (Don’t attempt to graphically show
the substitution and income effect).
c) Describe the microeconomic assumption of nonsatiation. How does this
assumption come into play in this question?
d) According to microeconomic theory, would this putative "income equalizing"
program (if implemented in the 1980s) have allowed the continued growth experienced
while preventing the rich from getting richer and the poor from getting poorer?
Consumption                                                            Time
Good “C”

\$200                       UR

\$100                                                         UP

Hours of work
112            40                              20                              0

a) U* = (I+p2)2/ p1p2
So (I+p2)2 = U*p1p2 and I = (U*p1p2)1/2 – p2
E(p,U*) = (U*p1p2)1/2 – p2.

b) X1(p,I) = [(I+p2)2 / p12p2] / [2(I+p2) / p1p2] = (I+p2) / 2p1.

The LeGrangian for the life cycle model is

L(C0,C1,U(C0) + βU(C1) + ( I1 + (I0-C0)(1+r) – C1).

The first order conditions are:

LC0 = UC0 - (1+r) = 0
LC1 = βUC1 + (-1) = 0
L= I1 + (I0-C0)(1+r) – C1 = 0.

The second order conditions are

UC0C0      0         -(1+r)     C0

0      βUC1C1        -1       C1
Hb=
-(1+r)     -1          0         

For a maximum, UC0C0 and YC1C1 are < 0,

UC0C0                  0
0                βUC1C1

is > 0, and

UC0C0                   -(1+r)

βUC1C1      -1
b
H=
-(1+r)         -1         0

is > 0.
Now, let’s use comparative statics analysis to determine the effect of a change in β, the
time preference parameter, on current consumption:

UC0C0         0       -(1+r)            C0/β                       0

0       βUC1C1        -1             C1/β       =     -        Uc1

-(1+r)       -1          0               /β                       0

For simplicity, assume UC0C1 and UC1C0 = 0.

C0/β =

0          0       -(1+r)
-Uc1      βUC1C1        -1        /      Hb

0         -1          0

= Uc1[-(-1)(-(1+r)) / {+} = {-}/{+} < 0. That is, as β increases, preferences for future
consumption increase and C0 decreases.

From the first two first order conditions (dividing the first first order condition by the
second first order condition and cancelling common terms), we see that

Uc0/βUc1 = (1+r)

or Uc0/Uc1 = β(1+r).

If β(1+r) = 1, then Uc0/Uc1 = 1 or Uc0=Uc1. If the marginal utilities are the same, then
consumption in each period is the same: C0=C1.

If β(1+r) < 1, then Uc0 = β(1+r)Uc1. Since β(1+r) < 1, Uc1 must be larger than Uc0. If
Uc1>Uc0, then C0>C1. Here, the effect of time preferences dominates the effect of the rate
of return on capital. Since β < 1, it increases preferences for present consumption, and
the rate of return is too small to offset consumer impatience. Consequently, present
consumption is greater.

If β(1+r) > 1, then Uc0 = β(1+r)Uc1. Since β(1+r) > 1, Uc1 must be smaller than Uc0. If
Uc1<Uc0, then C0<C1. In this case, the effect of the rate of return dominates the effect of
time preferences. Recall again that β<1 prompts us to prefer present consumption, but
(1+r) is large enough in this instance to make waiting until the future to consume
attractive enough for future consumption to be larger.

The choice variable is x – the entrepreneur gets to decide how much to invest in his

Given the entrepreneur’s utility function,

U(x) = πv(x) + (1 – π)[c(x) + B] – x,

The first order condition is

U
 v' ( x)  (1   )c' ( x)  1  0 .
x

The second order condition (SOC) is

 2U
 v' ' ( x)  (1   )c' ' ( x)  0 .
xdx

The second order condition is negative because v' ' ( x) is negative (since it is concave in
x).

The comparative static result is

  2U 
        
x
    x 
      2U 
      
 xx 

x  v' ( x)  c' ( x)

        SOC

x  v' ( x)  v' ( x)
                      .
         SOC

Since 1 > γ > 0, the numerator is negative (with v' ( x)  0 ) and the denominator, which is
the second order condition, is negative. Thus,

x   
       0.
   {}
As the probability of the government taking a business increases, private business
investment decreases.

If v(x) = xα, then

U
  x1  (1   )x1  1  0
x

 x1  (1   ) x1  1

1
x1 
  (1   )

1
1        1
x                     .
  (1   )

If γ = 1, then from part b

x  v' ( x)  v' ( x)

         SOC

x  v' ( x)  v' ( x)

        SOC

x  0

 SOC

x
 0.


That is, if the government pays the full value of the property when confiscating, then the
entrepreneur’s investment is not affected by the probability of the government taking his
private property for economic development.

The new budget constraint is a horizontal line at I=\$150. Now that the
opportunity cost of leisure is zero (taking an hour of leisure comes at no loss in terms of
consumption or income), both consumers will stop working entirely. Their new
indifference curves will show utility being maximized when leisure equals 112 hours and
consumption good C = 150.
Neither consumer will work. The substitution effect for both consumers in terms
of hours of work is negative {-} because leisure is relatively cheaper (it now has zero
cost) so the consumers will substitute toward leisure. The income effect for the poorer
consumer is also negative {-} because the poorer consumer experiences an increase in
real income. When real income increases, we consume more of all normal goods,
including leisure, which means work fewer hours. However, the richer consumer has a
positive income effect {+}: the richer consumer has seen her real income decline.
Therefore, she consumes less of all normal goods including leisure, which means work
more. The assumption of nonsatiation says that more is always preferred to less. In this
problem, that means more leisure is preferred to less. Since leisure is purchased at zero
opportunity cost, there is no reason not to consume the maximum amount of leisure (112
hours) according to the assumption of nonsatiation. So, the richer consumers income
effect is completely dominated by her substitution effect.
Though it may be socially undesirable to see the income disparity in the United
States grow, without an incentive structure that rewards people for work, consumers will
not work under microeconomic assumptions. In order to successfully equalize incomes, a
program must be developed which leaves incentives for work in tact.

Consumption                                               Time = one week or 112 hrs.
Good “C”

UR U P

\$200                       UR

150

\$100                                                         UP

Hours of work
112            40                             20                                 0

Richer Consumer: SE is {-}           Poorer Consumer: SE is {-}
IE is {+}                             IE is {+}

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