Question

3. (10 points) Income and substitution effects A consumer's utility is given by U(x, y) = xy. Income is m and prices are given by p and Py (a) Find the demand functions for x and y. (b) What is demand for each good if p 2 and py 1 and income is m = (c) If price of x fell to pa 1, what is the consumer's new bundle? (d) How much of the response in the consumption of x is due to the income effect and how much is due to the substitution effect?

Answer 1

a) Optimal demand is determined where MRS = Px/Py
MRS = MUx/MUy = $\frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}} = \frac{y}{x}$
So, y/x = Px/Py
So, y = x(Px/Py)

Budget constraint: xPx + yPy = m
So, xPx + x(Px/Py)Py = m
So, xPx + xPx = 2xPx = m
So, x = m/2Px

y = x(Px/Py) = (m/2Px)*(Px/Py) = m/2Py
So, y = m/2Py

b. px = 2; py = 1; m = 30
So, x = m/2Px = 30/(2*2) = 30/4 = 7.5
And, y = m/2Py = 30/(2*1) = 30/2 = 15

c. px' = 1; py = 1; m = 30
So, x" = m/2Px' = 30/(2*1) = 30/2 = 15
And, y = m/2Py = 30/(2*1) = 30/2 = 15

d. As Px has decreased so we find new income so that original bundle is just affordable.
Change in income = (New Px - Px)*x = (1 - 2)*7.5 = -7.5
So, new income, m' = m + Change in m = 30 + (-7.5) = $22.5
Thus, x' = m'/2Px' = 22.5/2*(1) = 22.5/2 = 11.25

So, Substitution effect on x = x' - x = 11.25 - 7.5 = 3.75 units

Income effect on x = x" - x' = 15 - 11.25 = 3.75 units