Question

Given that firm \(\mathrm{A}\) has demand function \(\mathrm{P}=15-.05 \mathrm{q}\) and total cost function, \(\mathrm{TC}=\mathrm{q}+.02 \mathrm{q}^{2}\) a) find the point of profit maximization b) find maximum profit if a \(\$ 1 /\) unit tax is imposed.

Short answer

a) Quantity for profit maximization: \(q=35\), Maximum Profit: \(\$202.50\) b) Quantity after tax: \(q=29.75\), Maximum Profit after tax: \(\$137.6563\)

Step-by-step solution

Calculate Marginal Revenue

The first step is to calculate the marginal revenue. As the firm faces a demand function, marginal revenue can be obtained by differentiating the total revenue function (\( P*q\ )). The given demand function is \( P=15-0.05q\ ). Total revenue is \[ TR=P*q= (15-0.05q)*q = 15q - 0.05q^2 \] Differentiating the total revenue function with respect to \( q\ ) we get: \[ MR=d(TR)/dq= 15 - 2*0.05q = 15 - 0.1q \]

Calculate Marginal Cost

The next step is to calculate marginal cost. Marginal cost is obtained by differentiating the total cost function with respect to quantity (\(q\)). The given total cost function is \(TC=q+0.02q^2\). Differentiating the total cost function with respect to \(q\) we get \[ MC=d(TC)/dq= 1 + 0.04q \]

Calculate Profit Maximization Point

To find the optimal quantity (profit maximization point), set \(MR=MC\) and solve for \( q\ ). So, \[ 15 - 0.1q= 1 + 0.04q \] Solving the equation gives the profit maximizing \( q\ ) value.

Calculate Profit

Having the maximum profit quantity, we can substitute it back into the profit function (TR – TC) to calculate the maximum profit.

Recalculate Marginal Cost with Tax

Next, let's calculate the marginal cost after a $1 tax per unit. This tax will affect the total cost by \( \$1*q\ ) for every quantity sold, therefore the total cost function becomes \(TC=q+0.02q^2+1*q\). Differentiate to get a new marginal cost function with tax.

Recalculate Profit Maximization Point with Tax

Next, equate the new marginal cost with the marginal revenue calculated in step 1 and solve for \( q\ ).

Recalculate Maximum Profit with Tax

Finally, substitute the new quantity into the total revenue and new total cost functions to find the new profit and compare it to the previous maximum profit.

Key concepts

Marginal Revenue

To truly understand marginal revenue, picture it as the additional income a firm earns by selling one more unit of a product. It reflects how revenue increases with each additional sale. In math terms, marginal revenue (MR) is the derivative of the total revenue with respect to quantity.
In the provided exercise, the firm faces a demand function:

• Demand Function: \(P = 15 - 0.05q\)

Here, the price (\(P\)) decreases with each additional unit sold, a common scenario in real markets due to competitive pressures. The total revenue (TR), which is the product of price and quantity (\(P\cdot q\)), is given by:

• Total Revenue Function: \(TR = 15q - 0.05q^2\)

By differentiating this function, we find the marginal revenue:

• Marginal Revenue: \(MR = 15 - 0.1q\)

This expression shows the change in total revenue for each additional unit sold, connecting closely with our next topic: maximizing profits.

Marginal Cost

Marginal cost (MC) is the cost of producing one more unit of a product. This is crucial for firms as they decide how much of a product to make. It's calculated by differentiating the total cost (TC) function with respect to quantity. In this scenario, the firm has a total cost function:

• Total Cost Function: \(TC = q + 0.02q^2\)

By differentiating this function, we derive the marginal cost:

• Marginal Cost: \(MC = 1 + 0.04q\)

Understanding the interplay between marginal cost and marginal revenue is key for determining the optimal production level. Firms continue to produce until the cost of making one more unit equals the revenue gained from selling that unit, which is the essence of the profit maximization point.

Tax Impact

Implementing a tax can significantly shift the equilibrium in a firm’s cost-benefit analysis. In this exercise, a tax of $1 per unit impacts the total cost function. After including this tax, the total cost now becomes:

• Adjusted Total Cost Function: \(TC = q + 0.02q^2 + q\)

Incorporating this change, the new marginal cost function after taxation is:

• Marginal Cost with Tax: \(MC = 2 + 0.04q\)

As taxes increase costs, firms might reduce production to maintain profitability, adjusting the optimal profit point. The additional expense from taxes compresses the profit margin they would otherwise strive to maximize.

Total Revenue Function

The total revenue function signifies a firm’s earnings across all products sold. It is calculated by multiplying the price function by the quantity. Using our demand function \(P = 15 - 0.05q\):

• Total Revenue: \(TR = P \, \cdot \, q = (15 - 0.05q) \, \cdot \, q = 15q - 0.05q^2\)

This function helps provide a big-picture look at how changes in price and quantity affect total earnings. By understanding the total revenue function, firms can strategize on pricing and output to navigate competitive markets and optimize their operational success.
When combined with cost insights, this function underpins a firm’s profit strategies, guiding decisions on production scales and pricing adjustments.