A competitive firm has a long-run total cost function c(y)= 2y2 + 288 for y 0 and c(0)= 0.Its long-run supply function is described as A)y = p/2 if p 46, y = 0 if p 51, y = 0 if p 48, y = 0 if p 51, y = 0 if p 53, y = 0 if p

Question

A competitive firm has a long-run total cost function c(y)= 2y2 + 288 for y > 0 and c(0)= 0.Its long-run supply function is described as A)y = p/2 if p > 46, y = 0 if p < 46. B)y = p/2 if p > 51, y = 0 if p < 54. C)y = p/4 if p > 48, y = 0 if p < 48. D)y = p/4 if p > 51, y = 0 if p < 51. E)y = p/2 if p > 53, y = 0 if p < 43.

Quizplus · Accepted Answer

To find the long-run supply function, we need to compare the minimum of the average cost curve with the price.

The average cost function is: 
ATC(y) = c(y)/y = (2y^2 + 288)/y = 2y + 288/y

To find the minimum point of the ATC curve: 
ATC'(y) = 2 - 288/y^2
Setting this equal to 0 gives us y = 12. Therefore, the minimum point is ATC(12) = 48.

Now, we can set up the long-run supply function: 
If p > 48: 
p = 2y + 288/y 
py = 2y^2 + 288 
2y^2 - py + 288 = 0 
Using the quadratic formula, we get y = (p +/- sqrt(p^2 - 4*2*288))/4 
Since y has to be positive, we take the positive root: 
y = (p + sqrt(p^2 - 2304))/4 
Therefore, y = p/4 if p > 48

If p < 48: 
The firm produces 0 output since the price is below the minimum point on the ATC curve.

Comparing the answer choices, we see that only answer choice C matches the derived long-run supply function.

A Competitive Firm Has a Long-Run Total Cost Function C(y)=