Question

Ann and Bob own beer and peanuts in their initial endowment.

x represents the number of beer bottles and y represents the number of peanut boxes. The utility functions of Ann and Bob are as follows.
uₐ(xₐ,yₐ) = xₐyₐ⁴ and uᵦ (xᵦ,yᵦ) = 2xᵦ + yᵦ
Find the relative price and optimal consumption that satisfy the general equilibrium for the following three cases. And draw the balance in detail in the Edgeworth box.
Ann and Bob each own 10 bottles of beer.(Basic conditions)
a) Only Bob owns 20 boxes of peanuts.
b) Bob owns 15 boxes of peanuts and Ann owns 5 boxes of peanuts.
c) Only Ann owns 20 boxes of peanuts.

Answer 1

The relative price and optimal consumption for each case are as follows: a) Ann: 10 beer, 0 peanuts; Bob: 0 beer, 10 peanuts. b) Ann: 5 beer, 5 peanuts; Bob: 5 beer, 10 peanuts. c) Ann: 0 beer, 10 peanuts; Bob: 10 beer, 0 peanuts.

Step-by-step explanation:

The general equilibrium in this case can be found by comparing the marginal utilities of beer and peanuts for each individual. The relative price is determined by the ratio of the marginal utilities of the two goods for each person. The optimal consumption for each person is determined by their relative price and initial endowment.

1. In case a: Since only Bob owns peanuts, we can calculate the relative price by comparing the marginal utility of beer for Ann with the marginal utility of peanuts for Bob. The optimal consumption for Ann is 10 bottles of beer and 0 boxes of peanuts, while for Bob it is 0 bottles of beer and 10 boxes of peanuts.

2. In case b: In this case, both Ann and Bob own peanuts. We can calculate the relative price by comparing the marginal utility of beer for Ann with the marginal utility of peanuts for Bob. The optimal consumption for Ann is 5 bottles of beer and 5 boxes of peanuts, while for Bob it is 5 bottles of beer and 10 boxes of peanuts.

3. In case c: Since only Ann owns peanuts, we can calculate the relative price by comparing the marginal utility of beer for Ann with the marginal utility of peanuts for Bob. The optimal consumption for Ann is 0 bottles of beer and 10 boxes of peanuts, while for Bob it is 10 bottles of beer and 0 boxes of peanuts.

By plotting these optimal consumption points for Ann and Bob in an Edgeworth box, we can visualize the balance between the two individuals' consumption of beer and peanuts.