Download Fall 2009 Final KEY

ECON 311
Prof. Hamilton
Fall Quarter, 2009
NAME:_______________________
FINAL EXAM
200 points
1. (30 points). Consider two ways for the government to collect tax revenue: a sales tax of on food, or
an income tax. If you currently buy both food and clothes, which tax would you prefer if your total
tax payment would be the same under either tax? Support your answer with a graph.
If you currently buy both food and clothes, you will always be better off with the income tax. Under
the sales tax, there is both an income effect (you purchase less clothes and food, because you have less
after-tax income) and a substitution effect (you buy less clothes to avoid paying the tax, as clothes are
now relatively more expensive). Under the income tax, there is only an income effect. To see why
you are better off, consider the following figure:
AOG
A0
tax revenue
AS
U0
AI
UI
US
BCS
FS
FI
F0
BCI
BC0
Food
First consider a sales tax on food. The total movement (income plus substitution effect) is from A =
(F0, A0) to B = (FS, AS). How much money has the government taken? At point B, the representative
consumer now consumes CS units of clothes. The revenue taken by the government is summarized by
the difference between the amount of AOG you used to be able to buy on your old budget line at CS,
less what you can buy now on your new after-tax budget. Your utility under a sales tax on clothes is
US.
An income tax lowers your income the same amount regardless of the market basket chosen. If
government collects the same amount of tax revenue from you as under the sales tax, the IRS could
increase the income tax, which would create a parallel shift in your budget constraint through point B.
You will now change your consumption to C = (FI, AI). Compared to the outcome under a sales tax,
you will now consume more clothes and less other goods, but be on the higher indifference curve, UI.
The sales tax has both an income effect and a substitution effect, while an income tax has only an
income effect and is preferable because it causes less of a distortion. The income tax is better by
the amount: UI - US > 0.
2. (20 points). Write out (or describe in words) the condition which must hold at a cost-minimizing
input mix for a firm that employs capital and labor in production. Show the cost-minimizing
outcome on a diagram. Briefly provide intuition on why the condition you state must hold.
K
K*
A
B
Q0
KB
IC
0
L*
LB
L
At a cost-minimizing input mix, the firm’s isoquant must be tangent to the isocost curve. The cheapest
way to produce Q0 units of output is shown as point A in the above diagram, with K* units of capital
and L* units of labor.
The tangency between isoquant and isocost implies that the marginal rate of technical substitution
(MRTS) must be equal to the ratio of factor prices, or MPK/r = MKL/w. This condition states that the
marginal product per dollar must be the same for a dollar invested in capital or labor. If this condition
did not hold, the firm could lower cost by shifting dollars away from the input with the lower marginal
product per dollar and towards the input with the higher marginal product per dollar.
3. (30 points) (a) Depict an industry in long-run competitive equilibrium on a 2-panel diagram. In one
panel, show the profit-maximizing output level and profit for an individual firm. In the other
panel, show market supply and market demand.
MARKET
Individual firm
$/Q
MC
$/q
S1
S0
AC
SR profit
P1
P1
P0
P0
d = P1
d = P0
D
QE
q0
Quantity (millions)
q1
quantity (units)
(b) Suppose your diagram depicts the situation of an individual firm in the U.S. selling in a global
market. Now suppose an earthquake hits firms in China that compete with the U.S. firm in the global
market and temporarily removes their output from global supply. Show on your diagram: (i) the effect
of the supply disruption on the global market price, (ii) the short-run profit-maximizing output level of
the U.S. firm, and (iii) the short-run profit level of the U.S. firm.
Initial profit in the long run equilibrium is zero. After the supply disruption, market supply shifts back
from S0 to S1, raising the market price along the global demand curve from P0 to P1. The individual
firm responds to the higher global price by increasing short-run output from q0 to q1 until the new
global price equals marginal cost. Marginal cost exceeds average cost at this price, so profit, which is
shown as the shaded region in the right panel of the figure, is positive.
4. (40 points). An individual derives utility from consumption goods, X, and leisure time, N, to
maximize the daily utility level, u = XN. Outside income per day is $m and the price of
consumption goods is $p. The individual can earn money each day by working at a wage rate of
$w per hour, but working of course takes away from leisure time (which implies it causes
disutility). The individual’s utility maximization problem is constrained by two things: (i) the
budget constraint, pX = m + wL, and (ii) the “time constraint”, L + N = 24 .
(a) Write a single equation for the budget constraint in terms of N, X, and m and interpret.
The utility is constrained by two things: the budget constraint and the “time constraint”; pX = m +
wL, and L + N = 24. This can be written in terms of N, X and prices by substituting in for L = 24 – N.
Doing so gives: pX = m + w(24 – N) or
24w + m = pX + wN.
The left-hand side is “potential income”, the amount of income the individual could have if she
devoted all her time to work. The right-hand side is her total expenditure on goods and “leisure”,
where the price of leisure time is the wage that is foregone when lounging (opportunity cost).
(b) Derive the demands for goods and leisure (X* and N*)
CP:
Max. XN
MUX = N;
MUN = X
s.t
pX + wN = 24w + m
MRS = price ratio => MUX / MUN = p/w or N/X = p/w
⇒ X = N(w/p)
Plug this into the budget constraint: 24w +m - wN - wN = 0
=> N* = (24w+m) / 2w
X* = (24w+m) / 2p
(c) Solve for the individual’s labor supply function, L* = 24 – N*. Graph the labor supply function and
verify that labor supply slopes upward in the wage rate.
Labor supply: L* = 24 - N* = 24 - (24w+m) / 2w = 24 - 12 - m/ 2w = 12 - m/2w
Notice: (1) If w ≤ m/24 (the wage is sufficiently small relative to the individual’s outside income), then
L* = 0 (since negative labor is not possible)
(2) dL* / dw = m / 2w2 > 0 (L-supply slopes upwards)
w
L0
L1
m0/24
Labor
(d) What is the effect of an increase in outside income (m) on labor supply? Show the effect of a
decrease in m (for instance due to graduation) on the individual’s labor supply.
dL* / dm = -1 / 2w < 0, so a decrease in outside income (i.e., from m0 to m1 < m0) increases labor
supply. An increase in labor supply is represented by a rightward shift in the labor supply curve, as
more labor is now supplied at every wage. This is shown as a movement from L0 to L1.
5. (40 points). Consumer utility for calculators (x) and all other goods (y) is given by the utility
function
u ( x, y ) = 100 x − 1.5 x 2 + y ,
where p is the price of calculators and the price of all other goods is $1.
(a) Compute consumer demand for calculators.
CP:
Max. u ( x, y ) = 100 x − 1.5 x 2 + y
MUx = 100 - 3x;
s.t
px + $1y = I
MUy = 1
MRS = price ratio => MUx / MUy = p/1, or
(100 - 3x)/1 = p/1 ⇒ 100 - 3x = p (inverse demand)
Demand for calculators is: x = (100 - p) / 3
The production function for a firm in the business of calculator assembly is given by
x = 5K0.5L0.5
(b) If both inputs are variable and w = $20 and r = $5, derive the cost function.
Firm’s Problem (FP):
MPL = 2.5K0.5L-0.5;
min. C = wL + rK
s.t.
x = 5K0.5L0.5
MPK = 2.5K-0.5L0.5;
MRTS = factor price ratio => MPL / MPK = w/r or
K/L = w/r
⇒ K = L(w/r) = 4L
Plug this into the production function: x = 5(4L)0.5L0.5 = 10L => L* = x / 10
And, since K* = 4L*
K* = 4L* = 4x / 10
The long-run cost function is: c(x) = 20L* + 5K* = 2x + 2x = 4x
(c) Using the demand and cost information above, derive the competitive equilibrium quantity and
price in the calculator market. Show the competitive equilibrium on a diagram.
Q/$
DWL under Monopoly
Pm
Pc
P*= 4
Qm
Q*
Q
In competitive equilibrium, the firm maximizes: π = px - 4x => FOC: p* = 4 (supply is a horizontal
line at p* = 4). The equilibrium quantity is where demand = supply:
p = 100 - 3x = 4
=> 3x = 96 => ( x* = 32, p* = 4)
(d) If the calculator business is controlled by a monopoly firm, calculate the monopoly price and
quantity of calculators in the market. Show the monopoly equilibrium on your diagram above.
Shade the region that represents deadweight loss under monopoly.
In a monopoly market, monopoly profit is: π = p(x)x - 4x = (100 - 3x)x - 4x
FOC: 100 - 6x - 4 = 0, or 6x = 96
=> xm = 16
The monopoly price is a point on the demand curve: p = 100 - 3(16) => pm = $52
The deadweight loss is the shaded triangle, which has the area: DWL = ½(32 - 16)(52 - 4) = $384.
6. (20 points). Suppose Nokona and Rawlings compete in a Cournot duopoly market for baseball
gloves with the cost functions c(qN) = 5qN + qN2 for Nokona and c(qR) = 5qR + qR2 for Rawlings.
Player’s view Nokona and Rawling gloves to be perfect substitutes and inverse demand for
baseball gloves is P = 85 − Q, where Q = qN + qR is the total quantity of gloves produced by both
firms.
(a) Calculate the reaction functions, qNR and qRR. Show the equilibrium quantity of gloves produced by
each firm as the intersection of two reaction functions on a graph.
Nokona’s Profit:
πN = [85– qN – qR]qN – 5qN - qN2
FOC: πN' = 85 – 2qN – qR – 5 - 2qN = 0
Reaction function: qNR = 20 – ¼qR
Similarly:
qRR = 20 – ¼qN
qN
80
qRR
20
Cournot equilibrium
16
qNR
16
20
80
qR
(b) Find the equilibrium quantity of gloves produced in a Cournot duopoly. What is the market price
of a baseball glove in a Cournot duopoly?
Cournot equilibrium is the intersection of the reaction functions.
qN = 20 – ¼[20 - ¼qN] = 15 + (15/16)qN
=> qN* = 16 = qB*
Market price: P* = 85 – Q* = 85 – (qN* + qR*) = 85 - 32
=> P* = $53
7. (20 points). Billy John Pigskin of Mule Shoe, Texas has a von Neumann-Morgenstern utility
function of u (c) = c . Billy John weighs 400 pounds and can outrun jackrabbits and pizza
delivery trucks. Billy John is beginning his senior year of college football. If he is not seriously
injured, he will receive a $1,000,000 contract for playing professional football. If an injury ends
his football career, he will receive a $10,000 contract as a refuse removal facilitator. There is a
10% chance that Billy John will be injured badly enough to end his career. If Billy John can buy
an insurance policy that pays him $1,000,000 if he suffers a career ending injury while in college at
a price $p, what is the largest price that Billy John would be willing to pay for such an insurance
policy? Derive Billy John’s maximum willingness to pay and show this value on a diagram.
U(w)
U(w)
910
WTP = $171,900
ρ (risk premium)
$10K
CE = $828.1K
E(w)
$1m
Wealth
Suppose BJ elects not to buy insurance. In this case his expected utility is:
No insurance: E(u) = 0.90( 1,000,000 ) + 0.10( 10,000 ) = 900 + 10 = 910 utils.
If BJ elects to buy insurance, he will have $1,000,000 - p for sure, so that his expected utility is:
With insurance: E(u) = 1,000,000 − p utils
BJ should buy insurance if
1,000,000 − p > 910
The maximum WTP for insurance is the value of p that satisfies
1,000,000 − p = 910
Squaring both sides:
1,000,000 – p = 828,100 Î p* = 1,000,000 - 828,100 = $171,900
BJ would pay a maximum insurance payment of $171,900
THE END. Have a great winter break!