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