Following is the questions pasted into the web which include

some errors because the website doesn’t support the exponent.

1. Assume that utility function is given by

U(X,Y) = X1/3 Y1/2

(a) Obtain

the Marshallian demand functions for goods X and Y.

(b) Compute the indirect utility function and the expenditure function for this

case.

(c) Use the expenditure function calculated in part (b) together with Shephard’s lemma to compute the compensated demand function

for good X.

(d) Describe how the compensated demand curve for X is shifted by changes in income or by changes in the price ofthe other

good.

(e) Use the results from part (c) together with the uncompensated demand function for good Xto show that the Slutsky equation holds

for this case.

(9 marks)

2. The output of a farm is related to the inputs, labour and capital, by the function

0 = 100 L10 K1/4

where L

and K are labour and capital. Find

(a) The cost-minimising input levels for an output level 0 ifthe rental on capital is r and the wage

IS w.

(b) The long-run cost function.

(c) The effect of an increase in O on long run marginal cost.

(d) The demand for labour as a

function ofw, the level of capital inputs and the price of output, p (i.e., the short-run demand curve for labour).

(e) The demand for

labour and capital as functions ofthe factor prices, w and r, and the price of output, p (i.e., the long run factor demand equations).

(8

marks)

3. (a) Consider a Cournot model with two firms. The market inverse demand function is given by

P = 120 “ O

where O = q1 + q2

and qi

is the quantity produced by firm i. Each firm’s total cost is quadratic, given by

C1 = q12 and

C2 = q22, respectively.

Determine the

Cournot-Nash equilibrium, and compute the associated profits.

(b) Now suppose that the two firms in the duopoly are permitted to explicitly

collude by forming binding agreements with one another. The market inverse demand function is given by

P = 120 “ O

where O = q1

q2

and qi is the quantity produced by firm i. Each firm’s total cost is quadratic, given by

C1 = q12 and

C2 = q22, respectively.

Finally,

assume that the firms have equal bargaining power.

Determine the quantities which the firms would agree upon, and compute the associated

profits.

(8 marks)