A small economy has only two consumers, Ivan and Marilyn. Ivan's utility function is U(x, y) . Marilyn's utility function is U(x, y) = x + 3y. Ivan is endowed with 135 units of x and 60 units of y. They make trades to reach a Pareto optimal allocation of resources in which both persons consume positive amounts. How much y does Ivan consume?

Question

A small economy has only two consumers, Ivan and Marilyn. Ivan's utility function is U(x, y)  . Marilyn's utility function is U(x, y) = x + 3y. Ivan is endowed with 135 units of x and 60 units of y. They make trades to reach a Pareto optimal allocation of resources in which both persons consume positive amounts. How much y does Ivan consume?

Quizplus · Accepted Answer

For a Pareto optimal allocation, the marginal utilities of both individuals must be equal. Therefore, we need to set the partial derivative of Ivan's utility function with respect to y equal to Marilyn's marginal utility, which is 3. 
So, 
 ∂U(x,y)/∂y = e^x/2y = 3
Solving for y, we get y=2e^(1/2x).
Now, we can use Ivan's endowment to calculate x. 
x + y = 195 (total endowment of x and y)
x + 2e^(1/2x) = 195
We can solve for x numerically, using trial and error or graphing. The solution is x ≈ 56.61. 
Therefore, Ivan consumes y = 2e^(1/2x) ≈ 8.98 or rounded to the nearest whole number, 9 units of y.

A Small Economy Has Only Two Consumers, Ivan and Marilyn