Supose a firme faces a demaned curve Q=5p-2 and its cost function is TC=Q. Fined Optimal Level of out Put (Q) ,Price (p),and Profit
To find the optimal level of output (Q), price (p), and profit, we need to consider the demand curve (Q = 5p - 2) and the cost function (TC = Q).
1. Optimal Level of Output (Q):
At the optimal level of output, the firm maximizes its profit. This occurs when the marginal revenue (MR) is equal to the marginal cost (MC). The marginal revenue represents the change in revenue resulting from selling one additional unit, while the marginal cost represents the cost of producing one additional unit.
Since the demand curve is given by Q = 5p - 2, we can find the marginal revenue by taking the derivative of the demand curve with respect to quantity (Q).
dQ/dp = 5 (derivative of Q with respect to p)
The marginal revenue (MR) is equal to the derivative of the demand curve multiplied by the price (p).
MR = dQ/dp * p = 5p
The marginal cost (MC) is given by the cost function TC = Q, which means that MC = dQ/dp.
Equating MR to MC, we have:
5p = MC = dQ/dp
Since MC is equal to derivative of Q with respect to p, we use the derivative of the demand curve:
5p = 5
Solving for p, we get:
p = 1
Substituting p = 1 into the demand curve Q = 5p - 2:
Q = 5(1) - 2
Q = 5 - 2
Q = 3
Therefore, the optimal level of output is Q = 3.
2. Price (p):
From the previous step, we found that the optimal price is p = 1.
3. Profit:
To find the profit, we need to calculate the total revenue (TR) and total cost (TC) using the optimal level of output (Q = 3).
Total Revenue (TR) can be calculated by multiplying the quantity (Q) by the price (p):
TR = Q * p = 3 * 1 = 3
Total Cost (TC) is given by the cost function TC = Q, so:
TC = Q = 3
Profit (π) is calculated by subtracting the total cost from the total revenue:
π = TR - TC = 3 - 3 = 0
Therefore, the profit at the optimal level of output is 0.