Q: Consider a two-good exchange economy with two individuals, A and B. A\'s pref
ID: 1252926 • Letter: Q
Question
Q: Consider a two-good exchange economy with two individuals, A and B. A's preference is represented by uA(xA1, xA2) = 0.3ln(xA1) + 0.7ln(xA2), and B's is represented by uB(xB1, xB2) = 0.8ln(xB1) + 0.2ln(xB2). A's initial holding is (wA1, wA2) = (10,4) and B's is (wB1, wB2) = (8,12).a) Calculate the demand function for each of A and B.
b) Find the competitive equilibrium relative price p1/p2, by equating the demand for good 1 (or good 2) to the supply of it.
c) Derive the equation which describes the set of efficient allocations (ie, contract curve).
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Explanation / Answer
A) Let's say that good 2 is the numerair. That is P2 = 1 Let's do A first. For notation, x1 = N and x2 = M. Max 0.3*ln(N) + 0.7ln(M) st. P*N + M = P*10 + 4 The Lagrange function is: 0.3*ln(N) + 0.7ln(M) - v*(P*N + M - P*10 - 4) where v is a Lagrange multiplier Take the first order conditions. N: 0.3/N = vP (1) M: 0.7/M = v (2) Divide (1) by (2) (3/7)*(M/N) = P (3) N = (3/7)*(M/P) Substitute into the budget constraint. P*N + M = P*10 + 4 P*(3/7)*(M/P) + M = P*10 + 4 (3/7)*M + M = P*10 + 4 M*(10/7) = P*10 + 4 M = P*7 + 2.8 from (3): M = N*P*(7/3) Substitute into the budget constraint. P*N + M = P*10 + 4 P*N + N*P*(7/3) = P*10 + 4 N*P*(10/3) = P*10 + 4 N = 3 + 12/10P Now, let's do B. Max 0.8*ln(N) + 0.2*ln(M) s.t. P*N + M = P*8 + 12 The Lagrange function is: 0.8*ln(N) + 0.2ln(M) - v*(P*N + M - P*8 - 12) where v is a Lagrange multiplier Take the first order conditions. N: 0.8/N = vP (1) M: 0.2/M = v (2) Divide (1) by (2) 4*(M/N) = P (3) N = 4*(M/P) Substitute into the budget constraint. P*N + M = P*8 + 12 P*4*(M/P) + M = P*8 + 12 4*M + M = P*8 + 12 5*M = P*8 + 12 M = P*(8/5) + 12/5 from (3): M = N*P*(1/4) Substitute into the budget constraint. P*N + M = P*8 + 12 P*N + N*P*(1/4) = P*8 + 12 P*N*(5/4) = P*8 + 12 N = 8*(4/5) + 12*(4/5)/P N = 6.4 + 9.6/P B) Here we have: NA = 3 + 12/10P NB = 6.4 + 9.6/P MA = P*7 + 2.8 MB = P*(8/5) + 12/5 NA + NM = 10 + 8 3 + 12/10P + 6.4 + 9.6/P = 18 9.4 + 10.8/P = 18 P = 10.8/(18-9.4) P = 1.256 MA + MB = 4 + 12 P*7 + 2.8 + P*(8/5) + 12/5 = 16 P*8.6 + 5.2 = 16 P = (16 - 5.2)/8.6 P = 1.256 Remember that we made good 2 the numeraire. So, P1/P2 = 1.256 C) This is a Pareto problem. So, we need to calculate the initial utility of one of the firms. Let's pick firm B. UB = 0.8ln(8) + 0.2ln(12) UB = 2.16 for notation, =/> means "greater than or equal to." Max 0.3*ln(NA) + 0.7ln(MA) s.t. (1) 0.8ln(NB) + 0.2ln(MB) =/> 2.16 (2) NA + NB =/< 18 (3) MA + MB =/< 16 We can substitute (2) and (3) into (1) to make this easier. Max 0.3*ln(NA) + 0.7ln(MA) s.t. 0.8ln(18 - NA) + 0.2ln(16 - MA) =/> 2.16 Now, the Lagrange function is: L = 0.3*ln(N) + 0.7ln(M) + v(0.8ln(18 - N) + 0.2ln(16 - M) - 2.16) where v is a Lagrange multiplier and we are suppressing the "A" notation. N and M represent the amounts consumed by person A. Take the first order conditions. N: 0.3/N = v0.8/(18 - N) (1) M: 0.7/M = v0.2/(16 - M) (2) Divide (1 ) by (2) (3/7)*(M/N) = 4*(16 - M)/(18 - N) Solve for M M = 4*(16 - M)*N*(7/3)/(18 - N) M/(16 - M) = N*(28/3)/(18 - N) (16 - M)/M = (3/28)(18 - N)/N 16/M - 1 = (3/28)(18 - N)/N 16/M = (3/28)(18 - N)/N + 1 M = 16/((3/28)(18 - N)/N + 1) You can clean this up if you want, but this is the contract curve. That is, a contract is efficient if it satisfies M = 16/((3/28)(18 - N)/N + 1)Related Questions
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