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Chapter 10: Problem 4

A firm faces the following average revenue (demand) curve: $$P=120-0.02 Q$$ where \(Q\) is weekly production and \(P\) is price, measured in cents per unit. The firm's cost function is given by \(C=60 Q+25,000 .\) Assume that the firm maximizes profits. a. What is the level of production, price, and total profit per week? b. If the government decides to levy a tax of 14 cents per unit on this product, what will be the new level of production, price, and profit?

Short Answer

Expert verified
In absence of the tax, optimal production is 1500 units, price is 90 cents per unit, and profit is \$500 weekly. After introducing the tax, optimal production reduces to 1375 units, price increases to 97 cents, and profit decreases to \$332.25 weekly.

Step by step solution

01

Define the Firm’s Profit Equation

The profit \(\Pi\) for a firm is defined as total revenues \(R\) minus total costs \(C\). In this case, the revenue is given by the price \(P\) times quantity \(Q\) and cost is given as \(C = 60Q + 25000\). So, we have the profit function as: \(\Pi = P \cdot Q - C = (120 - 0.02Q) \cdot Q - (60Q + 25000)\)
02

Find the Quantity That Maximizes Profit

In order to maxmise the profit, we take the derivative of the profit function with respect to the quantity \(Q\) and set it equal to zero. This gives us: \(\frac{d \Pi}{d Q} = 120 - 0.04Q - 60 = 0\). Solving this equation gives us the optimal quantity \(Q^* = (120 - 60) / 0.04 = 1500 \) units.
03

Calculate the Price and Profit at the Optimal Quantity

Plug the optimal quantity \(Q^*\) into the price and cost functions to find the price \(P^* = 120 - 0.02 \cdot 1500 = 90\) cents per unit, and total cost \(C^* = 60 \cdot 1500 + 25000 = 115000\) cents. Now, we can determine the total profit, which is \( \Pi^* = P^* \cdot Q^* - C^* = 90 \cdot 1500 - 115000 = 50000\) cents, or \$500.
04

Introduce a Unit Tax and Re-calculate

Because of the 14 cents per unit tax levied by the government, the cost function changes to \(C = 60Q + 1400 + 25000\). Now, we need to recalculate the optimal quantity, price, and profit. Following the same steps as before, we find the new quantity \(Q^{**} = 1375\) units, price \(P^{**} = 97\) cents per unit, and profit \( \Pi^{**} = P^{**} \cdot Q^{**} - C^{**} = 33225\) cents, or \$332.25.

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Most popular questions from this chapter

A certain town in the Midwest obtains all of its electricity from one company, Northstar Electric. Although the company is a monopoly, it is owned by the citizens of the town, all of whom split the profits equally at the end of each year. The CEO of the company claims that because all of the profits will be given back to the citizens, it makes economic sense to charge a monopoly price for electricity. True or false? Explain.

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