Reference no: EM13843257
1. A shipping firm sells q cases of wine from Chile to consumers in Canada. They receive a price according to the demand function ps = a1 - 1 3 q, while paying pb = a2 + 1 6 q for the wine. Further, transportation costs them γ per case shipped. Assume that a1, a2 and γ are positive.
(a) Express the firm's profit as a function of cases sold
(b) Assuming that a1 - a2 - γ > 0, find the number of cases sold that maximize profit. What happens if a1 - a2 - γ ≤ 0?
(c) Suppose the Chilean government imposes a tax on wine exports of τ per case. Rewrite firm's profit and solve for optimal q in this case.
(d) Imagine the Chilean government only cares about tax revenue. What is the optimal τ?
2. A consumer has $m and a utility function defined over two goods u(c1, c2). Prices of goods 1 and 2 are p1 and p2 respectively.
(a) Describe the consumer's budget constraint in terms of the inner product of two vectors.
(b) Let u(c1, c2) = α1c 1 α1 1 + α2c 1 α2 2 . Describe the utility function as a set.
3. Consider the market for cheese. Denote the quantity traded q and the price p. Supply of cheese is determined by optimal firm behaviour and characterized by ps = q. Consumer demand is pd = 16 - 9q + q 2 .
(a) Solve for the potential equilibria in this market. Illustrate with a diagram.
(b) Are all solutions found in part (a) "reasonable"? Discuss briefly.
4. Solve the following:
(a).
(b) Solve AB and BA, where
(c) Compute (A + B) T , for A and B below:
Check that (A + B) T = AT + BT .
5. Suppose a firm produces three types of output, using two types of input. Its output quantities are given by the column vector
q = [15, 000 27, 000 13, 000]
and the unit prices of these are given by the row vector
p = [ 10 12 5] .
The amounts of inputs it uses are given by the column vector
z = [11, 000, 30, 000]
and the input prices by the row vector
w = [ 20, 8]
Write an expression for and solve firm profits.
6. The market for tea is described by the following supply and demand functions:
Dt = 100 - 5pt + 3pc
St = -10 + 2p
and the market for coffee by:
Dc = 120 - 8pc + 2pt
Sc = -20 + 5pc,
where pt and pc are the prices of tea and coffee respectively. Set up the problem using matrix notation and solve for the equilibrium prices of tea and coffee.
7. Find A-1
A = [ 1 2 3 0 1 -1 1 2 1 ]
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