Question

a. U(r, 2)xfr + a)°(x2 + b)1-a d. U(,)( h. U(, 2) 1. For each of the utility functions above, find the consumer's opti mal consumption bundle when prices of goods 1 and 2 are pi and P2, and the consumer has an income m 2. For each of the utility functions above, find the consumer's opti mal consumption bundle when prices of goods 1 and 2 are pi and P2, and the consumer has an endowment (e1, e2) of the two goods 3. For each of your answers in question 2. above, write down the consumers net demands for the two goods at prices pi and p2 Find the ratio of prices for which the net demand for good 1 is zero. Also find the price ratio at which the net demand for good 2 is zero 4. Can you explain your answer to question 3. above intuitively?

Answer 1

xix, U2 (X + a) (x2+ b), u xx

1.

Consumer 1's problem:

max x xs.t. piX1P2X2 m

At equilibrium, marginal rate of substitution is equal to the ratio of the prices of the goods:

2x1P x2 P x2 MRS P2 x 2x1 al& leea

Substituting this value in the consumer's budget line:

2m m P1X1+2x1PmX1 23p2

Consumer 2's problem:

max (x1 a (X2 + b) s.t. pix1 P2X2m

a(x1+ a)(x b) (1 a)(x1 +a) (X2 + b) a(x2+ b) (1 a)(X +a) P2

Pr(1-aX a)bp2a X2

Substituting this value in the consumer's budget line:

m+apia apix1P (1 a)(x1 + a) bp2a amx - api+p

m+apia a2p1a- a 1 bp2a X2 P2a

Consumer 3's problem:

max x X s.t. piX1 P2x2 m

X1 X2= 0 if p <p2 if p Pz P2 X1 0, X2 El6

2.

When the consumer has been endowed with some units of both goods, his income will be in the form of his endowment:

For consumer 1:

2(P1 e1+P2e2) X2 P1 e1P2e2 Зр X1 Зр.

For consumer 2:

Pie+Pzez+ apja] ap +p

PieP2e2 apia apia-a 1- bp2a X2= P2a

For consumer 3:

Pe+Pzez x2= 0 if pi< P X1 pieP2e if p, P2 X1 0, X2 P2

3.

For consumer 1:

P1 P1 X 0 if0, p P2 0, X2 - 0 if 0. p P 0 P2 P2

For consumer 2:

P1 P1 X 0 if0, p P2 0, X2 - 0 if 0. p P 0 P2 P2

For consumer 3:

$x_1=0 \ if \ \frac{p_1}{p_2}>1, x_2=0 \ if \ \frac{p_1}{p_2}<1$

4.

The first two cases are of Cobb Douglas utility functions. In these cases, demand for both goods will be zero only if the prices of both goods are zero. In the third case, the consumer treats the two goods as substitutes and buys the cheaper good only.