Question

Consider a simple underwater exchange economy where two consumers, Bob (b) and Sandy (s), consume Jellyfish (j) and Krabby Patty burgers (k). Assume that the economy starts with the following initial allocation:

Bob has 6 and Sandy has 2 Jellyfish (??=6,??=2)

Bob has 4 and Sandy has 12 Krabby Patty burgers (??=4, ??=12)

1. a. Use an Edgeworth box to represent this simple allocation economy. Make sure to locate the initial endowment and label it E. Measure goods for Bob from the lower left corner. Measure Krabby Patty burgers on the vertical sides and Jellyfish on the horizontal sides of the box. (Draw this carefully enough and large enough so that you can clearly add your answers for part (c) of this problem).

1. b) Assume that Bob’s preferences are captured by the following utility function, ?=?∗? and that Sandy’s preferences can be represented by the utility function ?=8?∗?+12. Solve for a mathematical expression for the contract curve.
2. c) Now, on your graph from part (a), graph the contract curve you found. Now note that at their initial endowments Bob has a utility level of 24 and Sandy has a utility level of 204. Roughly draw the indifference curves associated with the initial endowments and show on the graph the points on the contract curve that both consumers prefer to their initial endowment.

1. d) Suppose that instead of bartering, after the initial allocation, Bob and Sandy use a price system to trade Krabby Patty burgers and Jellyfish. Consider potential prices ??=2 and ??=4. What (individual) quantities would Bob and Sandy choose under this price trading system?

1. e) Now consider different potential prices, ??=3 and ??=9. What (individual) quantities would Bob and Sandy choose under this price trading system?

1. f) Do the prices in parts (d) and (e) support competitive equilibria in this simple exchange economy? Why?

Answer 1

A.

Total jellyfish in the economy = Jellyfish that Bob has + Jellyfish that Sandy has = 6+2 = 8

Total Krabby Patty burgers in the economy = Krabby Patty burgers that Bob has + Krabby Patty burgers that Sandy has = 4+12 = 16

Figure 1 plots the Edgeworth box. E denotes the initial endowment.

B. Bob’s preferences are captured by the following utility function, ?b=?_b∗?_b, where Ub is the utility that Bob gets by consuming jb amount of jellyfish and kb amount of Krabby Patty burgers

Sandy’s preferences can be represented by the utility function ?s =8?_s∗?_s+12, where Us is the utility that Sandy gets by consuming js amount of jellyfish and ks amount of Krabby Patty burgers.

MRS of Bob = MUj_b / MUk_b = k_b/j_b

MRS of Sandy = MUj_s / MUk_s = 8k_s/8j_s = k_s/j_s

At the contract curve, IC of Bob and Sandy are tangent to each other. So, MRS of Bob should be equal to MRS of Sandy.

k_b/j_b = k_s/j_s  

k_{b *} j_s = k_s *   j_b ....... (1)

Total jellyfish in the economy = Jellyfish that Bob has (jb) + Jellyfish that Sandy has (js) = 6+2 = 8

j_s = 8- j_b

Similarly, k_s=16-k_b (total endowment of Krabby Patty burgers is 16)

Putting value of js and ks in equation 1

k_{b *}   (8- j_b) = (16-k_b) * j_b

8k_b - k_bj_b = 16j_b - k_bj_b

k_b = 2 j_b

therefore, the contract curve is given by k_b = 2 j_b

C) Contract curve has been indicated in figure 2 as the red line.

Bob's IC at a utility level of 24, is given by 24=?∗? and is plotted in the blue line in figure 2

Sandy's IC at a utility level of 204, is given by 204=8?∗?+12 and is plotted in the green line in figure 2.

The points on the contract curve that both consumers prefer to their initial endowment are indicated by the portion of contract curve that is highlighted in purple color.

D) If ??=2 and ??=4,

then income of Bob = value of his endowment = Pk * kb + Pj * jb = 2* 4 + 4 * 6 = 32

then income of Sandy = value of his endowment = Pk * ks + Pj * js = 2* 12 + 4 * 2 = 32

At optimal, Bob would equate slope of his IC and slope of his budget line

MRS = Price ratio

k_b/j_b = 4/2

k_b = 2 j_b

Put k_b = 2 j_b in the budget line of Bob

Pk * kb + Pj * jb = 32

2 * 2 j_b+ 4 * j_b = 32

j_b= 4

k_b = 2 j_b = 8

Bob would like to choose 4 jellyfish and 8 Krabby Patty burgers

At optimal,

demand for jellyfish by Bob + demand for jellyfish by Sandy = total endowment of jellyfish

4 + js = 8

js = 4

demand for Krabby Patty burgers by Bob + demand for Krabby Patty burgers by Sandy = total endowment of Krabby Patty burgers

8 + ks = 16

ks =8

Sandy would like to choose 4 jellyfish and 8 Krabby Patty burgers.

As per guidelines, first four questions have to be answered. Thanks!