identical firm has a total variable cost 𝑇𝐶 = 1,000 + 1⁄ 3𝑄3 − 2𝑄2 + 6𝑄.
A
4. Wheat industry is produced in a perfectly competitive market. Each
firm’s fixed cost is entirely sunk. The estimated industry demand for and
𝑄𝑑 = 2900 − 125𝑃 𝑄𝑠 = 1460 + 115𝑃
supply of wheat are given.
a) Calculate the profit maximizing quantity of each firm.
TVC = 1/3(Q³)- 2Q²+ 6Q
So,
MC = d(TVC)/d(Q) = Q²- 4Q + 6
Now, to find the market price (P), we need the equilibrium price where
market demand equals market supply.
Set Qd = Qs:
2900 - 125P = 1460 + 115P
Let's solve for P:
2900 - 1460 = 115P + 125P
1440 = 240P
P = (1440)/(240) = 6
So, the equilibrium price is $6 per unit.
Now, set MC = P:
Q²- 4Q + 6 = 6
Subtract 6 from both sides:
Q² - 4Q = 0
Factor:
Q(Q - 4) = 0
So, Q = 0 or Q = 4.
Q = 0 would mean the firm produces nothing, which isn't profit-maximizing in
this context. So, the profit-maximizing quantity is Q = 4.
b) Calculate the maximum profit of the firm.
Profit is calculated as Total Revenue (TR) minus Total Cost (TC).
First, find TR:
TR = P × Q = 6 × 4 = 24
Next, find TC at Q = 4:
TC = 1000 + (1/3)(4)³ - 2(4)² + 6(4)
Calculate each term:
- (1/3)(64) = 64/3 ≈ 21.333
- -2(16) = -32
- +6(4) = 24
So,.
TC = 1000 + 21.333 - 32 + 24 = 1000 + (21.333 - 32 + 24) = 1000 +
13.333 = 1013.333
Now, Profit:
π= TR - TC = 24 - 1013.333 = -989.333.
The firm is making a loss. This is an economic loss, but it might still produce
in the short run as long as the price covers the average variable cost.
c) Calculate the price below which the firm will not produce any
output in the short run.
In the short run, a firm will shut down if the price falls below the minimum of
the Average Variable Cost (AVC). This is because if P < AVC, the firm can't
cover its variable costs and is better off shutting down (losing only fixed
costs).
First, find AVC:
AVC = TVC/Q = ((1/3)Q³ - 2Q² + 6Q)/(Q) = (1/3)Q² - 2Q + 6
To find the minimum AVC, take the derivative of AVC with respect to Q and
set it to zero:
d(AVC)/d(Q) = (2/3)Q - 2 = 0
Solve for Q:
(2/3)Q = 2
Q=3
Now, find AVC at Q=3:
AVC = (1/3)(3)² - 2(3) + 6 = (1/3)(9) - 6 + 6 = 3 - 6 + 6 = 3
So, the minimum AVC is 3. Therefore, if the price falls below $3, the firm will
not produce any output in the short run.
