5. A monopolist’s demand function is P = 25 − 0.5Q The fixed costs of production are 7 and the variable costs are Q + 1 per unit. (a) Show that TR = 25Q − 0.5Q2 and TC = Q2 + Q + 7 and deduce the corresponding expressions for MR and MC. (b) Sketch the graphs of MR and MC on the same diagram and hence find the value of Q which maximizes profit.

(a) Given the supply and demand functions
P = 2Q2 S + 10QS + 10
P = −Q2 D − 5QD + 52
calculate the equilibrium price and quantity.

(b) If fixed costs are 25, variable costs per unit are 2 and the demand function is

P = 20 − Q
obtain an expression for π in terms of Q and hence sketch its graph.
(a) Find the levels of output which give a profit of 31.
(b) Find the maximum profit and the value of Q at which it is achieved.
2. Find an expression for the profit function given the demand function
2Q + P = 25
and the average cost function
AC = + 5
Find the values of Q for which the firm (a) breaks even (b) makes a loss of 432 units (c) maximizes profit.

3. (a) If the demand function is
P = 60 – Q
find an expression for TR in terms of Q.
Differentiate TR with respect to Q to find a general expression for MR in terms of Q. Hence write down the exact value of MR at Q = 50.
Calculate the value of TR when
(a) Q = 50 (b) Q = 51 and hence confirm that the 1 unit increase approach gives a reasonable approximation to the exact value of MR obtained in part (1)
(b) The consumption function is
C = 0.01Y2 + 0.8Y + 100
(a) Calculate the values of MPC and MPS when Y = 8.
(b) Use the fact that C + S = Y to obtain a formula for S in terms of Y.
By differentiating this expression find the value of MPS at Y = 8 and verify that this agrees with your answer to part (a).

4. (a) If the supply equation is
Q = 7 + 0.1P + 0.004P2
find the price elasticity of supply if the current price is 80.
(a) Is supply elastic, inelastic or unit elastic at this price? (b) Estimate the percentage change in supply if the price rises by 5%.

(b) A firm’s short-run production function is given by
Q = 300L2 − L4
where L denotes the number of workers. Find the size of the workforce that maximizes the average product of labour and verify that at this value of L MPL = APL.

5. A monopolist’s demand function is
P = 25 − 0.5Q
The fixed costs of production are 7 and the variable costs are Q + 1 per unit.
(a) Show that TR = 25Q − 0.5Q2 and TC = Q2 + Q + 7 and deduce the corresponding expressions for MR and MC.
(b) Sketch the graphs of MR and MC on the same diagram and hence find the value of Q which maximizes profit.

6. Solve the system of equations
4x1 + x2 + 3x3 = 8
−2x1 + 5x2 + x3 = 4
3x1 + 2x2 + 4x3 = 9
using Cramer’s rule to find x3.

7. A manufacturer of outdoor clothing makes wax jackets and trousers. Each jacket requires 1 hour to make, whereas each pair of trousers takes 40 minutes. The materials for a jacket cost $32 and those for a pair of trousers cost $40. The company can devote only 34 hours per week to the production of jackets and trousers, and the firm’s total weekly cost for materials must not exceed $1200. The company sells the jackets at a profit of $12 each and the trousers at a profit of $14 per pair. Market research indicates that the firm can sell all of the jackets that are produced, but that it can sell at most half as many pairs of trousers as jackets.
(a) How many jackets and trousers should the firm produce each week to maximize profit?
(b) Due to the changes in demand, the company has to change its profit margin on a pair of trousers. Assuming that the profit margin on a jacket remains at $12 and the manufacturing constraints are unchanged, find the minimum and maximum profit margins on a pair of trousers which the company can allow before it should change its strategy for optimum output.
8. (a) Use Lagrange multipliers to optimize 2x2 − xy
subject to x + y = 12
(b) A firm is a monopolistic producer of two goods G1 and G2. The prices are related to quantities Q1 and Q2 according to the demand equations
P1 = 50 − Q1
P2 = 95 − 3Q2
If the total cost function is
TC = Q21 + 3Q1Q2 + Q22
show that the firm’s profit function is π = 50Q1 − 2Q21 + 95Q2 − 4Q2 2 − 3Q1Q2
Hence find the values of Q1 and Q2 which maximize π and deduce the corresponding prices.