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)
