Question

could you please answer this please on paper if possible been asked numerous times before but not best presented and not clear to understand thanks!!

2. A profit-maximising monopolist produces strawberry jam (J) and marmalade (M). These goods are produced according to a joint cost function C = (qj+ qu)°, and the inverse demand functions are pj= 10 - qj and pM = 20 - 2 qM respectively. Why might the goods be related in production? What do the conditions (a) for profit maximisation look like in this case? Hence, find the output levels and prices. (b) How might a constant marginal tax on the output of jam affect the outputs of both jam and marmalade? Explain. (c)

Answer 1

(a)

Total profit is maximized when MR = MC for each product. Because Total Cost is a joint cost function, Marginal Cost (= dTC/dQ) is also a joint cost function. Hence price and quantity for each product depends on the quantity of the other product and hencem the goods are related in production.

For Jam,

Total revenue (TRJ) = pJ x qJ = 10qJ - qJ²

For Marmalade,

Total revenue (TRM) = pM x qM = 20qM - 2qM²

Overall revenue (TR) = TRJ + TRM = 10qJ - qJ² + 20qM - 2qM²

Profit (Z) = TR - TC = 10qJ - qJ² + 20qM - 2qM² - (qJ + qM)²

Z = 10qJ - qJ² + 20qM - 2qM² - (qJ² + qM² + 2qJqM)

Z = 10qJ - qJ² + 20qM - 2qM² - qJ² - qM² - 2qJqM

Z = 10qJ - 2qJ² + 20qM - 3qM² - 2qJqM

Profit-maximizing conditions are:

Z/qJ = 0 (i) and

Z/qM = 0 (ii).

Z/qJ = 10 - 4qJ - 2qM = 0

4qJ + 2qM = 10

2qJ + qM = 5............(1)

Z/qM = 20 - 6qM - 2qJ = 0

2qJ + 6qM = 20........(2)

(b)

Equilibrium is achieved by solving equations (1) and (2). Subtracting (1) from (2),

5qM = 15

qM = 3

qJ = (5 - qM)/2 [from (1)] = (5 - 2)/2 = 2/2 = 1

pM = 20 - (2 x 3) = 20 - 6 = 14

pJ = 10 - 1 = 9

(c)

A marginal unit tax on jam will increase the production cost for jam, so production of jam will fall. Hence, quantity of marmalade produced will increase.