I Micro Final, April 26, 2012
1. Suppose that a gambler discounts future utility at rate > 0 per period, yielding discount factor = 1/(1 + ) < 1
and has felicity function u(c) = c1 /(1 ) where > 0 and = 1. In that case, the sum of disconted felicity can
be regarded as a von Neumann-Morgenstern expected utility function.
Consider a gamble that pays off as follows. The payoff starts at $1 and a fair coin is ipped. If the coin turns up
heads, the game ends and the gambler gets the current payoff. If the coin turns up tails, the gambler gets nothing
this period and the payoff is doubled. Keep repeating until the game ends.
a) How much will the gambler be willing to pay to play this game?
b) How does the gamblers willingness to pay change as and vary?
Answer:
a) The probability of a payoff in period 0 is 1 , in period 2 is 1 1 = 1 , etc. Thus the probability of a payoff
2
2
2
4
at time t, t = 0, 1, 2, . . . , is t = 1/2t+1 . If the payoff occurs at time t, the amount of the payoff is ct = 2t ,
yielding felicity u(ut ) = 2t(1) /(1 ). If the payoff doesnt occur, felicity is zero. The felicity must be
discounted to the present, yielding t1 2t(1) /(1 ). This is the discounted utility obtained if the payoff
occurs at time t. To nd expected utility, we multiply by the probability of the payoff at time t and sum. In
other words, we sum t1 2t /2(1 ). This gives
EU =
1
2(1 ) t=0
t1 2t =
1
.
2(1 )(2 )
We can use the certainty equivalent e to measure willingness to pay for the gamble. We nd e by setting
u(e) = EU. Then e(, ) = (2+1 2)1/(1) . This is the maximum that the gambler would be willing to pay.
b) As increases, decreases. Since 2(2 ) is raised to a positive power when > 1 and negative power
when < 1, the willingness to pay, e, will increase when > 1 and decrease when < 1.
The case of varying is more complex than anticipated. This can be answered by considering ln e =
( 1)1 ln(2+1 2). and taking its derivate with respect to .
2. Consider a two-person, two-good exchange economy. Consumer 1 has utility u1 (xi ) = min{x1 , x1 } and consumer 2 has
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utility u2 (x2 ) = x1 + 2x2 . The endowments are 1 = (6, 1) and 2 = (0, 2).
a) Find all the equilibria of this economy.
b) Find all Pareto optima.
Answer:
a) Here we exploit the facts that the equilibrium must be Pareto optimal and lie on the budget line. If consumer
two is not at a corner point, the price vector must be p = (1, 2) (or a multiple). Then incomes are m1 = 8 and
m2 = 4. Consumer one demands x1 = (8/3, 8/3) and x1 = (10/3, 1/3) is worth $4, so x2 is a demand
point for consumer two at these prices. It follows that we have an equilibrium at p = (1, 2).
Note that if either price is zero, consumer two may obtain innite utility, so both prices must be positive.
Now if 2p1 > p2 , x2 = 0. Then x1 = (6p1 + p2 )/(p1 + p2 ) < 6. This excess supply means p1 = 0, which is
1
1
impossible.
Similarly, if 2p1 < p2 , x2 = 0. Then x1 = (6p1 + p2 )/(p1 + p2 ), which must be 3 since we cannot have
2
2
excess supply. This implies p1 = (2/3)p2 , which is impossible since 2p1 < p2 .
b) If x1 = x1 , consumer one has an excess of one of the goods. Giving the excess to consumer two is a Pareto
1
2
MICRO I FINAL, APRIL 26, 2012
Page 2
{(x1 , x2 )
improvement. This leaves P =
:=
=
=
3} as the only possible
Pareto optima. Since there is a utility tradeoff between the points of P, all points in P are Pareto optima.
x1
1
x1 , x2
21
6 x1 , x2
12
3 x1 , 0
1
x1
1
3. Consider the Ramsey (one-sector) model of capital accumulation with production function f(a) = 2a and felicity
function u(c) = ln c. The discount factor obeys 0 < < 1. Let x > 0 be the initial stock (there is no other
endowment).
a) Find the optimal paths of consumption and capital accumulation.
b) Find the (Malinvaud) corresponding price path pt .
c) Does the transversality condition hold: limt pt at = 0.
Answer:
a) Let pt be the sequence of prices, so the consumers budget constraint is p0 x = pt ct . The rms
t=0
maximization problem shows that pt = 2t p0 (the rm maximizes pt+1 f(at ) pt at ). The consumers rstorder conditions are t+1 u (ct+1 )/ = t u (ct )/pt . This can be rewritten (2)/ct+1 = 1/ct or ct+1 = (2)ct .
Thus ct = (2)t c0 . It follows that the budget constraint is p0 x = t p0 c0 = p0 c0 /(1 ), or c0 = (1 )x.
t=0
We may normalize prices so p0 = 1.
b) The Malinvaud prices were found in part (a). They are pt = 2t .
c) Since c0 = (1 )x, a0 = x. Then f(a0 ) = 2x. Since c1 = 2c0 , a1 = 22 x. By induction, at = 2t t+1 x
and pt at = t+1 0, establishing the transversality condition.
4. Consider a production economy with 2 goods and 2 consumers. There is one rm with technology set Y = {(y1 , y2 ) :
2y2 y1 , and 2y1 y2 }. Both consumers have equal-weighted Cobb-Douglas utility: ui = (xi xi )1/2 . Endowments
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are 1 = (1, 2) and 2 = (0, 3).
a) Find all equilibrium allocations.
b) How the equilibrium change if the endowments were 1 = (2, 1) and 2 = (3, 0)?
Answer:
a) Since utility is Cobb-Douglas, we know that no price can be zero. We normalize prices so p = (1, p). The
constant returns to scale production means that prot is always zero, so the only income is from the endowments.
The incomes are m1 = 1 + 2p and m2 = 3p. Then consumer demands are x1 (p) = (1 + 2p)(1/2, 1/2p) and
x2 (p) = 3p(1/2, 1/2p) so consumer demand is (1 + 5p)(1/2, 1/2p). The aggregate endowment is = (1, 5).
We rst check if there is an equilibrium without production. It would require (1 + 5p)/2 = 1, or p = 1/5.
But at that price, it is optimal to use good 2 to produce good one. If we produce good one, the price must be
p = 1/2. Then consumer demand is (7/4, 7/2). With an endowment of (1, 5), this requires production vector
y = (3/4, 3/2), which is feasible and yields zero prot when p = (1, 1/2). (Note that producing good two
yields negative prot.) The resulting allocation of consumption is x1 = (1, 2) and x2 = (3/4, 3/2).
b) Here the incomes are 2 + p and 3, so consumer demand is (5 + p)(1/2, 1/2p). The aggregate endowment
is = (5, 1). Without production, the equilibrium price would be p = 5. At that price, it is protable to
production good 2 from good 1. But if we produce good 2, the price must be p = 2. In that case market demand
is (7/2, 7/4), which requires production vector y = (3/2, 3/4). This is feasible and prot-maximizing and
gives us the equilirium. Thus prices are p = (1, 2), production is y = (3/2, 3/4) and consumption is
x1 = (2, 1) and x2 = (3/2, 3/4).
5. Consider an exchange economy with 2 consumers, 1 good, and 3 states of the world. Let xi denote consumer is
s
MICRO I FINAL, APRIL 26, 2012
Page 3
consumption of the one good in state s. Each consumer has utility function
3
u(xi ) =
ln xi .
s
s=1
The endowments are 1 = (1, 2, 3) and 2 = (3, 2, 1). Find the Arrovian securities equilibrium. Be sure to indicate
the spot market prices, prices of the securities, and the equilibrium consumption by each consumer.
Hint: Since we can normalize each spot market separately, we can set ps = 1 for each s.
Answer: With ps = 1 for every s, both consumers consume their income in state s. I.e., xi s = i s + zi . Thus indirect
s
1
1
utility is vi (z) = 3=1 ln(i s + zi ). The rst-order conditions are then qs = 1/(i s + zi ) or i s + zi = 1/(qs ).
s
s
s
s
1
1
1
Summing over i and using asset market clearing, we nd 4 = 2/(qs ). The qs must be equal, so we may set qs = 1
for every security s. Now = 1/2. Substituting back in the rst-order conditions, we nd 2 = i s = zi . Thus
s
1
z1 = (1, 0, 1) and z2 = (1, 0, 1). The corresponding goods allocations are x1 = (2, 2, 2) and x2 = (2, 2, 2).