Question

Consider a student who purchases education (x) and other goods (y). The student has preferences over these goods given by u(x, y) = ln(x) + 3ln(y). The prices of education and other goods are, respectively, px = 10 and py = 5, and the student’s income is I = 20.

1. Using the first-order conditions you derived, find the student’s optimal consumption of education x ∗ and other goods y ∗ and derive an algebraic expression for the student’s income expansion path.

2. Graph the budget constraint, IEP, optimal bundle (x ∗ , y∗ ), and the indifference curve passing through the optimal consumption bundle. Label all curves, axes, slopes, and intercepts. Put education on the x-axis.

3. Suppose now that the student receives a voucher for 1 free unit of education from the government. In other words, the price of her first unit of education is zero, and the price of any additional units is the original price px = 10. The student cannot sell the voucher. Both I and py are unchanged. Draw the new budget constraint under the voucher program. Be sure to label any important point(s) on the constraint and label the slope(s) of the constraint. Put education on the x-axis. And find the optimal consumption of x and y for the student when she receives the voucher.

Answer 1

1) max U (x,y) given the income constarint and for this setting up Lagrange,

L = U(x,y) + ( I-.x-.y)

L = ln(x) + 3 ln(y) + (20-10x-5y)

1/x = 10 (equation 1)

similarly,

3/y = 5 (equation 2)

Dividing 1 by 2,

y = 6x

substituting the value of y from income constraint;

4 - 2x = 6x

4 = 8x

x* = 1/2

y* = 12

therefore, optimal consumption bundle = (1/2 , 12)

Now, finding an algebraic consumption path,

y = 6x( income expansion

(2)