Question

Ken and Daniel always eat miso soup and sushi in exact proportions: Ken eats one miso per two sushi; Daniel eats one miso per one sushi. They throw away any extra soup or sushi as it would not give them any pleasure. Once, they were stuck in the office together for hours, marking students’ assignments. Ken had one miso soups and 9 sushi with him; Daniel had 9 miso and one sushi.

(For the purpose of this question, assume that miso and sushi can be divided into little pieces so one-half miso or one-hundredth of sushi makes sense)

a) Is ken's and Daniel's indifference cure "L" shaped? why or why not ?

b) Draw the Edgeworth box for Ken and Daniel. Please put Ken at the origin and Daniel “upside down”; put miso on horizontal axis and sushi on vertical. Show their endowment point.

c) Show all Pareto optimal allocations of miso and sushi between Ken and Daniel in the Edgeworth Box. Please note that this question is a bit tricky, though not difficult. Draw a bunch of indifference curves and look for tangencies.

Answer 1

A.) Yes, Ken and Daniel both will have L shaped indifference curves (IC). The L-shaped ICs are associated with consumption of complementary goods. These goods are always consumed in a fixed ratio. In our case this fixed ratio is 1:2 for Ken (Ken eats one meso per two sushi) and Daniel is 1:1 (Daniel eats one miso per sushi). Any change in this ratio and the consumer does not get any utility. So, any increase in consumption of one good without increasin the quantity of other in fixed proportion is a waste for the consumer. These kind of preferences are represented through L shaped ICs only.

The indifference map of both cosumers are shown below.

b.) The edgeworth box and initial endowment point is shown below in the diagram as point W.

Please note that ox is Ken's origin point and oy is Daniel's origin point which is upside down. The horizontal axis measures the total endowment of Miso with both Ken and Daniel so length of edgeworth box is the maximum miso that one can have. Similalry, the vertical axis measures the total endowment of sushi.

The initial point of endowments is represented by the point W where Ken has 1 miso and 9 sushi and simultaneously Daniel has 9 miso and 1 sushi.

C.) Pareto optimal allocations are also shown in the above diagram as the line joining the points e1, e2 and e3. It is because along this line, indifference curves are tangent to each other.

Pareto optimality is defined as the points where ICs are tangent to each other. If ICs are not tangent to each other then there are inefficiencies in allocation and Pareto optimality ensures no inefficiency in the system. So all points where ICs are tangent to each other are pareto optimal points.

To explain the graph, IC(K1), IC(K2) and IC(K3) are Ken's indifference curves which have regular shape (downward sloping and convex to origin).

IC(D1), IC(D2) and IC(D3) are Daniel's indifference curve which are also regular shape but turned upside down (because Daniel's axis is turned upside down). So, if you ll look it from the axis oy, these ICs will also look regular.

e1, e2 and e3 point show the tangency of ICs of Ken and Daniel. Hence the line joining these points shows Pareto optimum points.