Hi, please help me with the following. Thanks in advance! Some times goods are r

Question

Hi, please help me with the following.
Thanks in advance!

Some times goods are rationed, so people cannot buy as mu ch as they want at the announced price. A good example is Superbowl tickets, which are sold at a below-market price. Suppose that a consumer has Cobb-Douglas preferences: W (x1, x2) = x alpha 1 x beta 2 where alpha = 0. 2 and beta = 0. 8 Find the demand functions for the 2 goods. Assuming p1 = 2, p2 = 2, and I = 200. show that your answer in part a gives a demand for x1, equal to 20 units. Now assume the consumer can only buy 10 units what is the demand for good 2 if supply of good 1 is rationed at 10 units per customer? what is the consumer's MRS at the ralioned equilibrium? Suppose now that a black market opens up in good 1? How mu ch would the consumer pay for 1 more unit of good 1? (hint: think about MRS vs. relative price).

Explanation / Answer

For simple notation, let's set x1 = M and x2 = L. Also, let's say that the price of x1 is i while the price of x2 is j. U = M^a * L^B A. Max M^a * L^B subject to Mi + Lj = I The Lagrange function is: M^a * L^B - v(Mi + Lj - I) Take the first order conditions: M: a * M^(a-1) * L^B = vi (1) L: B * M^a * L^(B-1) = vj (2) Divide (1) by (2) (a/B) * (L/M) = i/j Solve for L L = M * (i/j) * (B/a) (3) Solver for M M = L * (j/i) * (a/B) (4) Substitute (3) into the budget constraint. Mi + Lj = I Mi + (M * (i/j) * (B/a))j = I Mi + (M * (i) * B/a) = I M*(1 + B/a) = I/i M = I/(i*(1 + B/a)) M = (I*a)/(i*(a + B)) Substitute (4) into the budget constraint. Mi + Lj = I (L * (j/i) * (a/B))*i + Lj = I (L * j * (a/B)) + Lj = I L * ((a/B) + 1) = I/j L = I/j*((a/B) + 1) L = (I*B)/(j*(a + B)) B. i = 2, j = 2, I = 200, a = 0.2, b = 0.8 M = (I*a)/(i*(a + B)) M = (200*0.2)/(2*(0.2+0.8)) M = 20 But if M is rationed at 10, then we will spend all of our remaining money on L. Mi + Lj = I L = (I - Mi)/j L = (200 - 10*2)/2 L = 90 At the rationed equilibrium. U = M^a * L*B MRS = Um/Ul Um = a * M^(a-1) * L^B Ul = B * M^a * L^(B-1) MRS = [a * M^(a-1) * L^B] / [B * M^a * L^(B-1)] MRS = (a/B) * L/B MRS = (0.2/0.8) * (90/20) MRS = 1.125 This value is not equal to 1 because of the constraint that M = 10. C. The price of M that is associated with M = 10 M = (I*a)/(i*(a + B)) i = (I*a)/(M*(a + B)) i = (200*0.2)/(10*(0.2 + 0.8)) i = 4

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