Question

Example Consider a society with 2 individuals A and B and 2 goods 1 and 2. The total available amount of good 1 and 2 is 9 units and 12 units. The utility function for the consumers is given by U(x)=(x,1)(x) for i=A,B. a) Show this economy in an Edgeworth box, including indifference curves. b) Define the meaning of the following notions; Pareto- efficient allocation, Pareto set, and contract curve. c) Find and draw the contract curve for this economy. d) Suppose that initial endowments are as follows: A has 4 units of good 1 and 4 units of good 2, and B has 5 units of good 1 and 8 units of good 2. Find the competitive equilibrium price e) Calculate the optimal allocation of consumption for both consumers A and B, and their utilities at this allocation. f) Does the equilibrium distribution of the goods lie on the contract curve? Why would you expect this result?

Answer 1

a. The following diagram represents the given economy inside an Edgeworth Box:

12 units IC 9 units

b. A Pareto Efficient allocation is such an allocation where resources are allocated in an efficient manner. Neither consumer can be made better off without making the other worse off.

A Pareto Set is the set of all Pareto Efficient allocations in an exchange economy.

Contract curve is a diagrammatical representation of the set of Pareto Efficient allocations in an Edgeworth Box.

c. The condition for Pareto Efficiency in this economy is that marginal rate of substitutions for both consumers should be equal. Therefore:

$\dpi{80} \fn_phv \small MRS_1=\frac{y_1}{x_1}, MRS_2=\frac{y_2}{x_2}, MRS_1=MRS_2\rightarrow \frac{y_1}{x_1}=\frac{y_2}{x_2}$

Since total endowment of good x is 9 units and of good y is 12 units,

$\dpi{80} \fn_phv \small y_1(9-x_1)=(12-y_1)x_1 \rightarrow \frac{y_1}{x_1}=\frac{12}{9}$

Plotting this line inside the Edgeworth Box gives the contract curve:

12 units Contract Curve IC1 0, 9 units

d. Assuming that the price of good y is 1 per unit, consumer 1's problem is:

$\dpi{80} \fn_phv \small max \ x_1y_1 \ s.t. \ px_1+y_1=4p+4$

At equilibrium, marginal rate of substitution is equal to the price ratio:

$\dpi{80} \fn_phv \small \frac{y_1}{x_1}=p \rightarrow y_1=x_1p$

Substituting this into the consumer's budget equation:

$\dpi{80} \fn_phv \small p_x_1+px_1=4p+4 \rightarrow x_1=\frac{2p+2}{p}, y_1=2p+2$

Consumer 2's problem is:

$\dpi{80} \fn_phv \small max \ x_2y_2 \ s.t. \ px_2+y_1=5p+8$

At equilibrium, marginal rate of substitution is equal to the price ratio:

$\dpi{80} \fn_phv \small \frac{y_2}{x_2}=p \rightarrow y_2=x_2p$

Substituting this into the consumer's budget equation:

$\dpi{80} \fn_phv \small px_2+px_2=5p+8 \rightarrow x_2=\frac{5p+8}{2p}, y_2=\frac{5p+8}{2}$

Total demand for x is equal to the total endowment of x:

$\dpi{80} \fn_phv \small \frac{2p+2}{p}+\frac{5p+8}{2p}=12 \rightarrow p=\frac45$

Hence, the competitive equilibrium prices are: $\dpi{80} \fn_phv \small (p_x,p_y)=(\frac45,1)$