Question 1: The demand function for a product is given by Q = 100 - 2P, where Q is the quantity demanded and P is the price. The cost function for the firm producing the product is given by TC = 50Q + 1000. Find the profitmaximizing price and quantity.
Solution: To find the profit-maximizing price and quantity, we need to determine the price and quantity that maximize the firm's profit. Profit (π) is given by the equation π = TRTC, where TR is total revenue and TC is total cost.
Total revenue is calculated by multiplying the price (P) by the quantity (Q). So, TR = P * Q.
Substituting the demand function into the total revenue equation, we have TR = P * (100 - 2P) = 100P - 2P^2.
Total cost is given by the cost function, TC = 50Q + 1000. Substituting the demand function into the cost equation, we have TC = 50(100 - 2P) + 1000 = 5000 - 100P. Now, we can express profit as a function of price: π = TRTC = (100P - 2P^2) - (5000 - 100P) = 200P - 2P^2 - 5000. To find the profit-maximizing price and quantity, we take the derivative of the profit function with respect to price, set it equal to zero, and solve for P.
dπ/dP = 200 - 4P = 0 4P = 200 P = 50
Substituting P = 50 back into the demand function, we can find the corresponding quantity demanded: Q = 100 - 2P = 100 - 2(50) = 100 - 100 = 0 Therefore, the profit-maximizing price is $50 and the corresponding quantity is 0.
Question 2: A firm has a production function given by Q = 10L^0.5K^0.5, where Q is the quantity produced, L is the labor input, and K is the capital input. The firm faces a wage rate of $20 per hour and a rental rate for capital of $50 per hour. If the firm wants to produce 100 units of output, how much labor and capital should it employ to minimize its cost?
Solution: To minimize its cost, the firm needs to determine the combination of labor and capital inputs that can produce 100 units of output at the lowest cost. The cost function is given by the equation C = wL + rK, where C is the total cost, w is the wage rate, L is the labor input, r is the rental rate for capital, and K is the capital input.
We can express the cost function as a function of labor, C(L), by substituting the production function into the cost function and solving for C(L):
C(L) = wL + rK = 20L + 50K
Since Q = 100, we can substitute this value into the production function and solve for L in terms of K: 100 = 10L^0.5K^0.5 10 = L^0.5K^0.5 100 = LK
Solving for L in terms of K: L = 100/K
Now, substitute L = 100/K into the cost function:
C(K) = 20(100/K) + 50K = 2000/K + 50K
To minimize the cost, we take the derivative of the cost function with respect to K, set it equal to zero, and solve for K: dC/dK =
√40 ≈ 6.32
Substituting K = 6.32 back into the production function, we can find the corresponding value of L: L = 100/K = 100/6.32 ≈ 15.82
Therefore, to produce 100 units of output at the lowest cost, the firm should employ approximately 15.82 units of labor and 6.32 units of capital.
Question 3: A monopolist faces a demand function given by P = 100 - Q, where P is the price and Q is the quantity. The monopolist also faces a constant marginal cost of $20 per unit. Find the profit-maximizing price, quantity, and profit level.
Solution: To find the profit-maximizing price and quantity, we need to determine the price and quantity that maximize the monopolist's profit. Profit (π) is given by the equation π = TR - TC, where TR is total revenue and TC is total cost.
Total revenue is calculated by multiplying the price (P) by the quantity (Q). So, TR = P * Q.
Substituting the demand function into the total revenue equation, we have TR = (100 - Q) * Q = 100Q - Q^2.
Total cost is given by the equation TC = MC * Q, where MC is the marginal cost. Since the marginal cost is constant at $20, TC = 20Q.
Now, we can express profit as a function of quantity: π = TR - TC = (100Q - Q^2) - 20Q = 80Q - Q^2.
To find the profit-maximizing quantity, we take the derivative of the profit function with respect to quantity, set it equal to zero, and solve for Q.
dπ/dQ = 80 - 2Q = 0 2Q = 80 Q = 40
Substituting Q = 40 back into the demand function, we can find the corresponding price: P = 100 - Q = 100 - 40 = 60
To calculate the profit level, substitute the profit-maximizing price and quantity into the profit function: π = 80Q - Q^2 = 80(40) - (40)^2 = 3200 - 1600 = 1600
Therefore, the profit-maximizing price is $60, the corresponding quantity is 40 units, and the profit level is $1600.