Question

Consider a perfectly competitive firm with total costs

?? = ? + ?? + ??2

a) Identify the fixed cost ??, and the variable cost of this firm, ??(?). (Each of them is just a part of the total cost.)

b) Find the average cost ??(?), and the marginal cost ??(?).

c) Long-run supply. Find the minimum of the ??(?) curve, which constitutes the “shut- down price” in a long-run setting. Use this “shut-down price” to describe the firm’s long-run supply curve. d) Evaluate the long-run supply curve at ? = 10, ? = 4, and ? = 2.

e) Short-run supply curve. Use your results from part (a) to find the average variable cost function. Find the minimum of the ???(?) curve, which constitutes the “shut- down price” in a short -run setting. Use this “shut-down price” to describe the firm’s short-run supply curve.

f) Evaluate the short-run supply curve at ? = 10, ? = 4, and ? = 2

Answer 1

The Total Cost of the Perfectly Competetive Firm is

(a) The Fixed Cost or FC is the part of TC which is independent of quantity produced (q).

Hence, Fixed Cost is

FC = q

The variable cost or VC is the part of TC which is the function of quantity produced (q).

Hence, Variable Cost is

(b) The Average Cost is defined as

AC(q) = TC(q)/q

or, AC(q) = a/q + b + c.q

And, the Marginal Cost is defined as

MC(q) = d[TC(q)]/dq

or, MC(q) = b + 2c.q

(c) Let us find the minimum of the ??(?) curve, which constitutes the “shut- down price” in a long-run setting.

AC = a/q + b + c.q

We will differentiate the AC(q) and set the derivative equals to zero. Hence, at minimum point of AC(q) we get

d[AC(q)]/dq = 0

or,

or, q* = √(a/c)

Hence, putting q* in AC(q) we get the long run "shut down price". Hence,

AC(q*) = a/q* + b + c.q*

or, AC(q*) = a.√(c/a) + b + c.√(a/c)

or, AC(q*) = √(ac) + b + √(ac)

or, AC(q*) = b + √(2.a.c)

Now, the long run shut down price is P* (say). Hence,

P* = AC(q*) = b + √(2ac)

This is the long run supply curve i.e. horizontal.

(d) Putting, ? = 10, ? = 4, and ? = 2, we get

P* = b + √(2ac)

or, P* = 4 + √(2×10×2)

or, P* = 4 + 2√10

This is the long-run supply curve.

(e) Now we will find the minimum of the ???(?) curve, which constitutes the “shut- down price” in a short-run setting.

AVC(q) = VC(q)/q

or, AVC(q) = b + c.q

Hence, we can see that, the AVC(q) is always increasing. Hence, we will get the minimum point where q=0.

Hence, Minimum AVC(q) i.e. the short run shut down price P° is

P° = b + c×0

or, P° = b

This is the minimum of AVC(q).

The short run supply curve is defined by the Marginal Cost or MC(q) curve.

Hence,

MC(q) = b + 2c.q

This is the short run supply curve which passes through the minimum point of AVC(q) i.e. at P°=b.

(f) Putting ? = 10, ? = 4, and ? = 2, we get

MC(q) = 4 + 2×2×q

or, MC(q) = 4.(1+q)

This is the short run supply curve which passes through the shut down price i.e. P°=b=4.

Hope the solutions are clear to you my friend.