Suppose that a competitive firm faces a total cost function c(q)=450+15q+2q^2, and the industry price is 115$ per unit of output.
A) Find the firm's equilibrium output, levels of profit and producer surplus?
B) Is this a short-run or log equilibrium?
C) Above which prices will this firm make a positive economic profit?
D) Suppose due to a lower industry demand, the price falls to 63$ per unit. What supply decision would this firm make?
all question must be done without calculus
A) To find the firm's equilibrium output, we need to set marginal cost (MC) equal to the industry price (P):
MC = c'(q) = 15 + 4q
115 = 15 + 4q
4q = 100
q = 25
So the equilibrium output is 25 units.
To find the firm's profit, we need to subtract total cost from total revenue:
total revenue = Pq = 115(25) = 2875
total cost = c(q) = 450 + 15(25) + 2(25)^2 = 1875
profit = total revenue - total cost = 2875 - 1875 = 1000
To find producer surplus, we need to find the area below the industry price curve and above the marginal cost curve up to the quantity produced:
producer surplus = (115 - 15)(25)/2 - (2)(25)^2/2 = 1375
B) This is a short-run equilibrium because we are given a specific cost function and not all inputs can be adjusted.
C) To find the minimum price at which the firm makes a positive economic profit, we need to set profit equal to zero and solve for P:
Pq = c(q)
P(25) = 450 + 15(25) + 2(25)^2
P = 125
So above a price of $125 per unit, the firm will make a positive economic profit.
D) If the price falls to $63 per unit, the firm's profit-maximizing output will be where MC = P:
MC = 15 + 4q = 63
4q = 48
q = 12
So the firm would reduce its output to 12 units.
how did you get this MC = c'(q) = 15 + 4q
To find the marginal cost (MC), we take the derivative of the total cost function (c(q)) with respect to q:
c'(q) = 15 + 4q
The first term in this expression comes from the constant term in the total cost function, which disappears when we take the derivative. The second term comes from the linear term in the total cost function, and the third term comes from the quadratic term in the total cost function. Taking the derivative with respect to q eliminates the quadratic term, as the derivative of q^2 with respect to q is 2q.
A) To find the firm's equilibrium output, we need to determine the quantity where the firm maximizes its profit.
First, we need to find the marginal cost (MC) of the firm, which is the derivative of the total cost (TC) function with respect to quantity (q). In this case, the total cost function is c(q) = 450 + 15q + 2q^2.
To find MC, we take the derivative of c(q) with respect to q:
MC = d(c(q))/dq = 15 + 4q
Next, we equate MC to the price per unit of output (P) to obtain the profit-maximizing condition:
MC = P
In this case, the industry price is given as $115 per unit, so we have:
15 + 4q = 115
Now, we can solve this equation to find the equilibrium quantity (q):
4q = 115 - 15
4q = 100
q = 25
So, the firm's equilibrium output is 25 units.
To find the levels of profit and producer surplus, we need to calculate the firm's total revenue (TR), total cost (TC), and subtract TC from TR.
The firm's total revenue (TR) is given by multiplying the equilibrium quantity (q) by the price (P):
TR = P * q = 115 * 25 = $2875
The firm's total cost (TC) is calculated by substituting the equilibrium quantity (q) into the total cost function (c(q)):
TC = 450 + 15q + 2q^2
TC = 450 + 15(25) + 2(25^2)
TC = 450 + 375 + 2(625)
TC = 450 + 375 + 1250
TC = $2075
Now, we can find the level of profit by subtracting the total cost (TC) from the total revenue (TR):
Profit = TR - TC
Profit = $2875 - $2075
Profit = $800
Producer surplus is the difference between the total revenue (TR) and the variable cost (VC), which represents the cost of producing the last unit of output:
Variable cost (VC) = 15q = 15 * 25 = $375
Producer surplus = TR - VC
Producer surplus = $2875 - $375
Producer surplus = $2500
Therefore, the firm's level of profit is $800, and the producer surplus is $2500.
B) To determine if this is a short-run or long-run equilibrium, we need additional information about the firm's cost structure and market conditions. Without this information, we cannot conclusively determine whether it is a short-run or long-run equilibrium.
C) To determine the prices at which the firm makes a positive economic profit, we need to compare the average variable cost (AVC) to the price (P).
The average variable cost is calculated by dividing the variable cost (VC) by the quantity (q):
AVC = VC/q = 15q/q = 15
If the price (P) is greater than the average variable cost (AVC), the firm makes a positive economic profit. In this case, if the price is higher than $15, the firm will make a positive economic profit.
D) If the price falls to $63 per unit due to lower industry demand, the firm will need to reassess its supply decision.
To determine the firm's supply decision, we need to compare the price (P) to the average variable cost (AVC). If the price is below the average variable cost, the firm will not be able to cover its variable costs and may consider shutting down in the short run. However, if the price is above the average variable cost, the firm will continue to produce at a quantity where price equals marginal cost (MC=P).
Since in this case, the average variable cost (AVC) is $15, which is higher than the price of $63, the firm would not be able to cover its variable costs. Therefore, the firm may choose to shut down in the short run.