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lEonsider a household whose preferences. are described by the utility function UIXT X2} = X1X2 where X1 and X2 are household's consumption of goods...
1. Consider a household whose preferences are described by the utility function
U(X1, X2) = X1X2
where X1 and X2 are household's consumption of goods 1 and 2 respectively. Consider that household's budget constraint is:
P1X1 + P2X2 = I .
(a) Derive the household's demand functions for goods X1 and X2.
(b) Derive the household's compensated demand function for goods 1 and 2, i.e., obtain functions of the form
Xi = fi (P1, P2, U) , I = 1, 2
where U is the household's level of utility.
(c) Assume that in the initial situation the commodity prices, P1 and P2, and the household income level, I, are given by
P1 = $1, P2 = $1 and I = $2.
Sketch the compensated and uncompensated demand curves for good 2 with P1 held constant at the initial level. In the compensated case, U is held constant at the initial level while in the uncompensated case, I is held constant.
(d) By how much must I be increased if P2 increases to $2 (P1 remains at $1) and our household is to maintain its initial level of utility. Be sure to check your answer by examining the area under the compensated demand curve.
(9 marks)
2. A firm's production function is given as
Q = 10 L1/2 K1/2
where L and K are labour and capital. Firm's iso-cost function is
C = wL + rK.
(a) Using this production function, express the amount of labour employed by the firm as a function of the level of output it produces and the amount of capital it employs.
(b) Using your results from part (a), along with the iso-cost equation, determine the firm's short-run total cost function, assuming the price of capital, r, is $40, the price of labour, w, is $10, and the level of capital is held constant at 8 units.
(c) Using your results from part (b), along with the fact that the firm's long-run cost function is LRTC = 4Q, determine the level of output, Q, at which the firm's short-run total cost is equal to its long-run total cost. Also determine the optimal amount of labour that must be combined with the 8 units of capital to produce this level of output.
(d) Using your results from part (c), determine the values of short- and long-run costs.
(8 marks)
3. In a Stackelberg duopoly, one firm is 'leader' and one is 'follower'. Both firms know each other's costs and market demand. The follower takes the leader's output as given and picks his own output accordingly (i.e., the follower acts like a Cournot competititor). The leader takes the follower's reaction as given and picks his own output accordingly. Suppose that firms 1 and 2 face market demand, p = 100 - (q1 + q2). Firm costs are c1 = 10q1 and c2 = q22 .
(a) Calculate market price and each firm's profit assuming that firm 1 is the leader and firm 2 the follower.
(b) Do the same assuming the firm 2 is the leader and firm 1 is the follower.
(c) Given your answers in parts (a) and (b), who would firm 1 want to be the leader in the market? Who would firm 2 want to be the leader?
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