Construct a short-run supply function for a firm whose short-run cost function is

C= = 0.04q^3 – 0.8q^2 +10q+5

3. The long run cost function for each firm is

C = q^3 – 4q^2 + 8q.

Find the industry’s long run supply curve. If the market demand curve is is D = 200 – 10p, find equilibrium price, total output, firm output and number of firms.

Tc=Q^4-12Q+100Q+36 find average total cost

To find the short-run supply function for a firm, we need to determine the optimal level of output that the firm should produce at different prices. In the short run, a firm's supply is influenced by its cost function.

Given the short-run cost function:
C = 0.04q^3 – 0.8q^2 + 10q + 5

To find the short-run supply function, we need to minimize the cost function with respect to quantity (q). This can be done by taking the derivative of the cost function with respect to q and setting it equal to zero:

dC/dq = 0
0.12q^2 - 1.6q + 10 = 0

Using the quadratic formula, we can solve for q:

q = (-(-1.6) ± sqrt((-1.6)^2 - 4*0.12*10)) / (2*0.12)
q ≈ (-(-1.6) ± sqrt(2.56 - 4.8)) / 0.24
q ≈ (1.6 ± sqrt(-2.24)) / 0.24

Since the square root of a negative number is not possible here, we can see that there is no real solution for q. This means that the firm's cost function does not have a minimum value and does not meet the necessary conditions for a short-run supply curve to exist.

Moving on to the long-run analysis, we have the long-run cost function for each firm:
C = q^3 – 4q^2 + 8q

To find the industry's long-run supply curve, we need to determine the number of firms that will be operating at different prices. In the long run, firms can enter or exit the market, so the industry supply curve is determined by the conditions of zero economic profits.

In the long run, economic profit is zero when price (p) equals the minimum average cost (AC) of production (since AC represents the cost per unit in the long run):

p = AC

Since AC = C/q, we can substitute the long-run cost function to find AC:

p = (q^3 – 4q^2 + 8q) / q
p = q^2 – 4q + 8

Now, we need to determine the equilibrium price and total output by finding the intersection of the market demand curve and the industry's long-run supply curve.

Given the market demand curve:
D = 200 – 10p

Setting D equal to p, we get:

200 – 10p = p
11p = 200
p ≈ 18.18

Substituting this equilibrium price into the industry's long-run supply curve equation, we can find the total output:
q = p^2 – 4p + 8
q ≈ (18.18)^2 – 4 * 18.18 + 8
q ≈ 201.82

To find the firm's output, we divide the total output by the number of firms. However, we still need to determine the number of firms. Since economic profits are zero in the long run, each firm will earn normal profits. Let's assume each firm earns a profit of 0. We can find the number of firms by substituting this profit value into the long-run cost function:

0 = q^3 – 4q^2 + 8q
q(q^2 – 4q + 8) = 0

Using the quadratic formula again, we can solve for q:

q = 0 or q ≈ 2 ± sqrt((-4)^2 - 4*1*8) / 2
q = 0 or q ≈ 2 ± sqrt(16 - 32) / 2
q ≈ 0 or q ≈ 2 ± sqrt(-16) / 2

Since the square root of a negative number is not possible here, we can see that there is no real solution for q. This means that there are no firms in the long-run equilibrium.

In conclusion, in the long-run equilibrium, the industry's long-run supply curve does not exist. The equilibrium price is approximately $18.18, the total output is approximately 201.82 units, the firm's output is zero, and there are no firms operating in the industry.