Question

A competitive firm has a long-run total cost function c(y) = 3y^ 2 + 675 for y > 0 and c(0) = 0. Derive the equation or equations that would describe its long-run supply function.

Answer 1

Solution:-) C(y) = 3y^2 + 675 where y > 0 and C(0) = 0

Here, MC = Marginal Cost

AC = Average Cost

and, P = Price

Therefore, MC = dx/dy of C(y)

i.e, = dC(y) / dy

= 6y + 0

MC = 6y

Let MC = Price i.e, 'P'

Therefore, P = 6y

It implies, y = P/6 ....................1st equation

Now, AC(y) = C(y) / y

= (3y^2 + 675) / y

= (3y^2 / y) + (675 / y)

= 3y + (675 / y) .................... 2nd equation.

Then, we have to find dy/dx of AC

It implies, dAC / dy = 3 - (675 / y^2)

y = square root of (675 / 3)

y = 15

Min. AC = AC (15)

Put y = 15 in 2nd equation....

i.e, AC = 3(15) + (675 / 15)

= 45 + 45

= 90

We had seen in the 1st equation that y = P / 6 which means minimum P > 90

Therefore, y = 0 if P < 90

Therefore, the firm's long-run supply function is

y = P / 6 if P > 90, y = 0 if P < 90