Econ 30 Intermediate Microeconomics Prof. Marek Weretka Problem (Insurance) a) Solutions to problem set 6 b) Given the insurance level x; the consumption in the two states of the world is Solving for x from the second equation gives and plugging it into the rst one C NF = $0:000 0 x C F = $:000 + x 0 x = $:000 + 9 0 x x = 0 9 (C F $:000) C NF = $0:000 0:x = $0:000 + 9 $:000 9 C F = 5:560 The two extreme points are (0; 5:560) and (45:000) 9 C F c) Consumer with such Bernoulli utility function is risk averse. To see consider two lotteries (4:0) and (; ) and assume equally likely states. Observe that the two lotteries have the same average payo, but the rst one involves risk. We show on the graph that the risk makes it less attractive and hence agent is risk averse. d) MRS is given by MRS = 0: p C F = p CNF p = r CNF 0:9 p 9 CF 9 C C F F and hence at the endowment point! = (:000; 0:000) MRS = r CNF = p 0 = 0:3536 9 C F 9
Note that the MRS is di erent from the slope of the budget set equal to =9 e) First we nd optimal consumption levels: First secret of happiness implies that MRS is equal to the slope of the budget set. Therefore r CNF = 9 C F 9 ) r CNF C F = ) C NF C F = ) C NF = C F We can see that agent insures fully. Plugging it into the budget constraint and slowing for C F gives But then using the formula for x C F = 5:560 9 C F C F = C NF = 9 5:560 = 455:000 0 x = 0 9 (C F $:000) = = 0 (455:000 $:000) = 4000 9 therefore he covers 4000 f) With such premium the slope of the budget set is slope = 0: 0: = 4 hence budget line becomes steeper. Consequently optimal consumption is not on a 45 o degree line. Problem (Risk aversion and certainty equivalence) a) Yes, he is risk averse b) The expected value from the lottery is (see mark in graph above) c) The expected utility from this lottery is E (L) = 00+ 0 = U (L) = p p 00 + 0 = 5
d) Certainty equivalent CE is the amount of "sure" cash that makes Frank indi erent to the lottery. Lottery (CE; CE) is associated with utility U (CE) = p p p CE + CE = CE By de nition it must be equal to the utility of the original lottery U (L) = 5 hence p CE = 5 ) CE = 5 Therefore Frank is indi erent between $5 for sure, and risky lottery ($00; $0) e) He should choose $40 or more generally anything exceeding $5 f) Expected value of the lottery is unchanged and expected utility from this lottery is E (L) = 00+ 0 = U (L) = 00 + 0 = Utility from sure cash CE is given by U (CE) = CE which is in turn equal to : Consequently CE = itself. Frank should take the lottery. g) Expected value of the lottery is and expected utility is U(L) = (00) + 0 = 00 Utility from sure cash CE is hence U(CE) = CE (CE) = 00 ) CE = p 00 = 70: 7 Frank should take the lottery. Problem 3 (Standard Edgeworth Box) a) The total resources are! = (00; 0) b) The allocation! E = (0; 0) and! M = (90; 0) is not Pareto e cient. One way to see it is that the endowment is o the contract curve (see point c) 3
or c) In Pareto e cient allocation the slopes of indi erence curves must coincide. Therefore MRS E = MRS M x M 5x M = xe 5x E Elvis consumes x E = 00 x M and x E = 0 x M therefore we can write x M x M = 0 xm 00 x M Multiplying both sides by x M 00 x M gives x M 00 x M which can be reduced to = x M 0 x M or nally 00x M = 0x M x M = 0 xm Which de nes a straight line in the Edgeworth box d) Ce nition of Equilibrium: It is an allocation x E ; x E ;and x M ; x M and prices (p ; p ) such that ) for each consumer x i ; x i is optimal given prices (p ; p ) ) (p ; p ) are such that markets clear Remark: Equilibrium is our prediction of what we will observe in markets, when they are not a ected by any distortion. We nd it in a di erent way than the Contract curve. Later we show, however that the equilibrium allocation, among many others, is Pareto e cient and hence it is located on a contrct curve. Equilibrium determines only a relative price, therefore we can normalize p =. We also focus on market for good. (You can verify that market for good will clear automatically). With Cobb-Douglass utility functions, we nd optimal choices using magic formula (shortcut) rather than by deriving it from secrets of happiness. The optimal demand for MP 3 is given by x i = a m i a + b p where a = and b = 5 (they are the same for both traders Elvis and Miriam). Income m i is m E = p w E + p w E = p 0 + 0 = 0p + 0 m M = p w M + p w M = p 90 + 0 = 90p Consequently the demands for x are x E = 0p + 0 6 p x M = 90p 6 p By the second equilibrium condition prices assure markets clearing, that is the total demand for MP3, x E +x M is equal to the total supply w E + w M = 00. More formally 00 = x E + x E = 0p + 0 + 90p 6 p 6 p 4
Multiplying both sides by 6p or At such prices the optimal consumption is 600p = 0p + 0 + 90p = 0 + 00p p = x E = 0p + 0 6 p = 85 x M = 90p = 5 6 p The consumption of DVD is (again we use magic formulas and equilibrium prices p = ; p = ) x E = 5 0p + 0 6 p = 5 6 (0p + 0) = 8 x M = 5 90p = 5 90p = 6 p 6 p Hence allocation x E = (85; 8 ) and xm = (5; ) and prices (p ; p ) = ( ; ) is an equilibrium e) Any price system that is associated with the relative price equal to supports the same allocation as an equilibrium. For example (p ; p ) = (; ) or (p ; p ) = (; 00) f) Yes, they are e cient. To see it observe that MRS for Elvis and Miriam at equilibrium is equal to MRS E = xe 5x E MRS M = xe 5x E = = and hence their indi erence curves are tangent, so the allocation is Pareto e cient. g) Remark: In this case all allocations in Edgeworth box are Pareto e cient. Equilibrium: For any relative price p p < MRS i = 5 both trader spend their total income on x and no income on x - This results in excess demand for x. For p p > MRS i = 5 both traders spend their income on x and hence there is excess demand for x : Therefore markets can clear only for relative price p p = 5 But then all allocations on the budget set are optimal for each trader therefore any of them is an equilibrium allocation. 5
Problem 4 (Uncertainty and Asset Pricing) Endowment allocation is not Pareto e cient. It is also risky as it is associated with di erent payments in two states of the world. b) Equilibrium: Given the Cobb -Douglass preferences the optimal demand of each of the traders is x J = x B = a m J = p 00 + p 0 = a + b p p a m B = p 0 + p 00 = p a + b p p p We normalize p to one so that p =. The market clearing on the market for x requires that which gives a price x J + x B = 00 + p p = + p = 00 ) p = Hence the equilibrium prices are p = p = : Given the prices the equilibrium allocations are x J = ; x J = b m = a + b p and hence consumption of Benjamin is what is left on the market c) MRS for both traders is p 00 p = x B = 00 = x J = 00 = MRS J = MRS B = = = hence indi erence curves are tangent - the allocation is Pareto e cient. The equilibrium allocation is also not risky as the wealth is the same in the two states of the world (this is true for J and for B). Problem 5 (Irving Fisher Determination of Interest Rate) a) No, the initial allocation is not Pareto e cient. b) Given Cobb -Douglass preferences, the optimal demand for consumption "today" is 6
C J = C W = a m J = p 0 + p 000 a + b p + = 000p p 3 p a m B = p 000 + p 0 a + b p + = p 3 000 We normalize p to one so that p =. The market clearing for C requires that which gives a price C J + C W = 000 000 + 3 p 3 000 = 000 ) p = Hence the equilibrium prices are p = and p = : Given the prices the equilibrium allocations are and hence consumption of William is The interest rate is hence c) Yes equilibrium allocation is Pareto e cient as C J = 000p = 000 = 333:333 3 p 3 C J b m = = 000 = 333:333 a + b p 3 C B = 000 333:333 = 666:666 C B = 000 333:333 = 666:666 + r = p p = r = 00% MRS J = CJ C J MRS B = CB C B = 333:333 333:333 = 666:666 666:666 = = and hence the indi erence curves of two traders are tangent. d) Now the optimal consumptions are C J = C W = a m J = p 0 + p 000 = 000p a + b p + p p a m B = p 000 + p 0 = a + b p + p 000 Again normalize p to one so that p =. The market clearing on the market for C requires that which gives a price C J + C W = 000 000 + p 000 = 000 ) p = Hence the equilibrium prices are p = p = ; therefore the interest rate is + r = p p = 7
hence r = 0% With greater patients the willingness to save increases and the willingness to borrow goes down. In order to equilibrate the market the interest must go down. e) The demands now are C J = C W = a m J = p 0 + p 000 a + b p + = 000p p 3 p a m B = p 000 + p 0 a + b p + = p 3 000 We normalize p to one so that p =. The market clearing on the market for C requires that which gives a price C J + C W = 000 000 + 3 p 3 000 = 000 ) p = 4 Hence the equilibrium prices are p = 4 and p = : The interest rate is + r = p p = 4 hence r = 300% Intuition: Jane s have large endowment tomorrow and therefore because she wants to use it today she needs to borrow more. In order to equilibrate the savings market interest must go up. This partially reduces her willingness to borrow and also encourages William to lend. 8