Question

14. The utility function is as follows: ? = (? + 2)(? + 1) ???????? ?? = ?? ,?? = ?? ??? ? = ? a. Write the Lagrangian function, find the x* and y* in terms of the parameters Px, Py and B. check whether the second order sufficient condition holds. b. Make a comparative-static analysis to find the effect of a respective change in B, Px and Py on x* and y*.

Answer 1

14. (a) The utility function is and the budget constraint would be as . The Lagrangian function would be as . The FOCs are as below.

or or or or .

or or or or .

or or or .

Comparing the first two FOCs, we have , which is the utility optimizing combination of goods, and we have this as or . Putting this in the third FOC, we have or or or or or , and since , we have or or or or or or or or . These are the required optimal values of x and y for given B and prices of x and y.

To check the second order sufficient condition, the bordered hessian matrix would be as below.

or . Since there are two variable and one constraint, ie n=2 and k=1, we have to check the sign of n-k=1 leading principal minor, which is |H| itself. We have or or . Hence, the critical points (x*,y*) are at maximum. The second order condition is hence verified.

(b) For optimal x, we have

• , meaning that for a marginal increase in B, x* increases by .
• , meaning that for a marginal increase in Px, x* decreases by .
• , meaning that for a marginal increase in Py, y* increases by .

For optimal y, we have

• , meaning that for a marginal increase in B, y* increases by .
• , meaning that for a marginal increase in Px, y* increases by .
• , meaning that for a marginal increase in Py, y* decreases by .