1. A perfectly competitive industry comprises of 35 competitive, profit-maximizing firms, each of which
has short-run total costs and marginal costs of
SMC = 0.25q
STC = 20 + 0.125q(squared)
The market demand curve is P = 20 − 0.05Q
(a) Find P, q and Q
(b) Suppose a tax of $2 per unit of output is imposed on the firms in the industry, find the new P, q, and Q.
Industry supply curve is the horizontal sum of each firm's marginal cost curves. If SMC=q/4 then the supply curve will be P=Q/(4*35) = Q/140 = .007142Q
Demand is P=20-.05Q At equilibrium, supply=demand, so, solve for Q and P.
Hint: I get Q=350,P=2.5, which means q=10.
b) with the tax, raise the supply curve vertically by 2. So, supply = 2+.007142Q
To find P, q, and Q in a perfectly competitive industry and then determine the new values after imposing a tax, we can follow these steps:
(a) Finding P, q, and Q initially:
Step 1: Determine the industry's equilibrium condition by equating market supply and demand.
Market demand is given by: P = 20 − 0.05Q
Since there are 35 firms in the industry, each firm's output is q, so total industry output is given by Q = 35q.
Step 2: Calculate the industry's supply curve based on the firms' cost functions.
For the cost functions:
SMC = 0.25q (short-run marginal cost)
STC = 20 + 0.125q^2 (short-run total cost)
To calculate the short-run supply curve, we find the quantity at which the short-run marginal cost equals the market price. In a perfectly competitive industry, this is the profit-maximizing point.
Setting SMC equal to P:
0.25q = 20 − 0.05Q [Substituting the values of SMC and P]
0.25q + 0.05Q = 20 [Rearranging the equation]
0.3q = 20 [Combining like terms]
q = 20 / 0.3 = 66.67 [Dividing both sides by 0.3]
Step 3: Substitute the value of q into the market demand equation to find P:
P = 20 − 0.05Q [Substituting Q = 35q]
P = 20 − 0.05(35 * 66.67) [Substituting the value of q]
P = 20 − 116.675 [Calculating]
P ≈ -96.68 [Simplifying]
Note: The negative price indicates that there is no equilibrium in the market at this quantity.
Hence, there is no equilibrium price and quantity unless there is a mistake in the given data or equations.
(b) Finding P, q, and Q after imposing a tax:
To calculate the new equilibrium values after imposing a tax, we need to consider how the tax affects the firms' cost functions and subsequently the market price and quantity.
Given that a tax of $2 per unit of output is imposed, the new short-run total cost (STC) function becomes:
STC_new = STC + 2q
Therefore, the new equation for STC is:
STC_new = 20 + 0.125q^2 + 2q
We then proceed with the steps outlined in part (a) to find the new P, q, and Q using this updated cost function:
1. Set the new short-run marginal cost (SMC_new) equal to the new market price inclusive of tax (P_new):
SMC_new = P_new
0.25q = P_new [Substituting the value of SMC_new]
2. Calculate the new q by solving the equation:
0.25q = P − 2 [Substituting the value of P_new = P − 2]
0.25q = 20 − 2 [Substituting the value of P]
0.25q = 18 [Calculating]
q = 18 / 0.25 = 72 [Dividing both sides by 0.25]
3. Calculate the new Q using Q_new = 35q:
Q_new = 35 * 72 = 2520
4. Substitute the new q value into the market demand equation to find the new P:
P_new = 20 − 0.05Q_new [Substituting the value of Q_new]
P_new = 20 − 0.05(2520) [Calculating]
P_new ≈ 10.40 [Simplifying]
Hence, after imposing a tax of $2 per unit of output, the new equilibrium values are:
P_new ≈ $10.40
q_new ≈ 72
Q_new ≈ 2520