Question

Hi.I need your answer for all from A to G for this question

2*. Assume that Bob has a budget constraint p1x1 + p2x2 = m, and that his preferences are represented by the Cobb-Douglas utility function U(x1, x2) = x1 c x2 d , where c>0 and d>0. State Bob’s optimization (utility maximization) problem.

a) Set up the Lagrangian function.

b) Derive the necessary conditions (the first-order conditions) for an optimal interior solution.

c) Show that the MRS (the slope of the indifference curve) is equal to the slope of the budget line at the optimal solution.

d) Derive Bob’s demand functions for good 1 and for good 2.

e) Is good 1 a normal good for Bob? f) Now assume that c=1/4 and d=3/4, m=160, p1=4 and p2=2. Calculate the income and substitution effects from an increase in price of x1 from p1=4 to p1=5.

g) Illustrate these effects in a graph.

Answer 1

C)

A consumer will be in equilibrium when his utility is maximized within the given budget constraint.

At optimal choice, budget line is tangent to the Indifference curve. Therefore, at this point, the slope of the indifference curve (M.R.S.) will be equal to the slope of the budget line.

This equality is required because MRS measures rate at which consumer is willing to exchange one good for other, while price ratio is the rate at which consumer exchange one good for others.

If the ratios are not equal, the consumer will change his consumption bundle.

E) Since the price and quantity are inversely related, therefore, both goods are normal goods.