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 “shutdown 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 “shutdown 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

Ans)- Given TC = a + bq + cq²

a) In this case

Fixed cost (FC) = a [fixed cost is a term in the total cost function, which is independent of q]

Variable cost(VC) = bq +cq² [because TC = FC +VC]

b) Average cost (AC) and marginal cost (MC)

Average cost (AC) = TC/q

AC = (a + bq + cq²)/q

[AC = a/q + b + cq]

Marginal cost (MC) = dTC/dq

i.e. to find MC differentiate TC function w.r.t. ‘q’

[MC = b+2cq]

c) In the long run the shutdown price will be minimum of AC(q) and the long-run supply curve (LSC) is a horizontal straight line (i.e., perfectly elastic) at the price, which is equal to the minimum average cost.

So, AC(q) = a/q + b + cq

Differentiate AC w.r.t. ‘q’

Put

So,

q² = a/c

Put the value of q in AC function

AC(q) = a/q +b+cq²

So, in the long-run the shutdown price will be.

And so, the long-run supply curve (LRSC) will be.

i.e. the horizontal line at

d) Long-run supply curve at a=10, b=4 and c=2

Put these values in Long-run supply curve (LRSC)

e) The short run supply curve of a firm in perfect competition is precisely its Marginal Cost Curve for all rates of output equal to or greater than the rate of output associated with minimum average variable cost.

Variable cost(VC) = bq +cq²

AVC = VC/q

AVC = (bq +cq²)/q

AVC = b + cq

To find minimum of AVC, differentiate AVC w.r.t. ‘q’

Put

So, [c=0]

Here shutdown price in the short-run will be [P=b] i.e. price below ‘b’ will cause the firm to shut-down.

Hence, in the short run the supply curve will be portion of marginal cost above the minimum of AVC i.e. above ‘b’

So, with given ,MC = b+2cq

Short run supply curve (SRSC) = 2cq [i.e. above AVC =b]

f) At a=10, b=4 and c=2

Short run supply curve (SRSC) = 2cq = 2×2q

[SRSC = 4q]----------------(upward sloping supply curve in short run)