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Homework answers / question archive / The following shows the cost function of firm X: C(q) = a + bq + 0
The following shows the cost function of firm X: C(q) = a + bq + 0.5cq2 where q denotes the quantity of output. a) Derive the fixed cost and marginal cost of firm X respectively. b) Suppose firm X is a price-taking firm, what is the minimum price that it would produce output? Find the quantity of output q* that the firm would choose to produce, if the price has induced the firm to produce. If firm X is actually in a market with only one player, with market demand: p= d - 0.5eq where d and e are positive number c) Find the optimal amount of output for the monopolist to produce, as well as the corresponding price charged p and marginal cost ?. Compare q* and ? and explain the difference intuitively. The government is going to set a maximum price control on the market at ?. d) Explain the effect of the output level, if i > ii. >c 111. ?
Cost Function
C(q) = a+bq+0.5cq2
a) Fixed Cost and Marginal cost of firm X
Fixed Cost = a
Marginal Cost = d C(q)/ dq
=b+1cq
b) Firm X is a price taking firm, the minimum price at which it would produce output is at a price equal to the minimum of Average variable cost. So,
TVC = bq+0.5cq2
AVC= TVC/q = b + 0.5cq
P =MC=AVC
b+cq =b+0.5cq
0.5cq = 0
q= 0
P=b
So for any price greater than b, the firm would produce.
Market demand p=d-0.5eq; d &e >0
c) The optimal amount for a monopolist is the one where MR=MC
Total Revenue = P*q
=dq-0.5eq2
MR = d-eq
The q for which MR=MC
d-eq = b+cq
q ? = (d-b)/(c+e)
p ? = d-0.5e (d-b)/(c+e) = (dc-0.5de+0.5be) /(c+e)
MC c? = b+c(d-b)/(c+e) = (be+cd)/ (c+e)
q ? is less than q* . This happens because monopoly produce less output and charge a higher price to extract profit.
d) i. If the price set by government is above monopoly price, its called a price ceiling. At this price there would be surplus i.e., more supply than demand . The quantity sold would be equal to quantity demanded and less than q ?.
ii. Price set by government is greater than c?. The monopolist would only produce if the price is equal to the price corresponding to MR=MC.
