A consumer has the utility function U(x, y)= x + 2y^1/2.The price of good x is 2 and the price of good y is 1.The consumer's income is 20.If the price of good y rises to 2, then entire change in demand for y is due to the substitution effect.

Question

Quizplus · Accepted Answer

To determine whether the entire change in demand for y is due to the substitution effect, we can use the Slutsky equation:
∆xy = ∆x + (∂x/∂p)y * (p1/p2)
Where ∆xy is the total change in demand for y, ∆x is the substitution effect, (∂x/∂p)y is the income effect, and (p1/p2) is the ratio of the new and old prices.
Since the problem tells us that the price of y has risen to 2, and all other variables are constant, we can use this equation to solve for the substitution effect:
∆xy = ∆x + (∂x/∂p)y * (1/2)
We know that the income of the consumer is 20, and the prices of x and y are 2 and 1 respectively. Plugging these values into the utility function, we can solve for the demand functions for x and y:
x = 5 - 2y
y = 10 - 5√(2-x/5)
Using these demand functions, we can solve for the substitution effect:
∆x = (∂x/∂p)y * (p2 / p1) * ∆p
Since the price of y has risen from 1 to 2, and the price of x is constant, we have:
∆x = (∂x/∂p)y * (1/2) * (-1)
Taking the derivative of the demand function for x with respect to the price of y, we get:
(∂x/∂p)y = -2/5
Substituting in the values, we get:
∆x = (2/5) * (1/2) * (-1) = -1/5
Thus, the substitution effect is ∆x = -1/5.
Since the total change in demand for y is equal to the substitution effect (as given in the problem), and the income effect is zero, we can conclude that the entire change in demand for y is due to the substitution effect. Therefore, the answer is A.

A Consumer Has the Utility Function U(x, Y)= X