Question

Assume that a firm operates with the total revenue (TR) and total cost (TC) functions: \(\mathrm{TR}=41.5 \mathrm{Q}-1.1 \mathrm{Q}^{2}\) \(\mathrm{TC}=150+10 \mathrm{Q}-0.5 \mathrm{Q}^{2}+0.02 \mathrm{Q}^{3}\) where \(\mathrm{Q}\) represents the quantity of output produced and sold. a) Determine the profit-maximizing output level for this firm via the \(\mathrm{TR}-\mathrm{TC}\) approach. b) Solve for the profit-maximizing output level by using the \(\mathrm{MR}=\mathrm{MC}\) approach

Short answer

Based on the given Total Revenue (TR) and Total Cost (TC) functions, we solved for the profit-maximizing output level, Q, using both the TR-TC approach and the MR=MC approach. We found the profit function, and after differentiating and solving the quadratic equation, we obtained the profit-maximizing output level, Q. The MR=MC approach led to the same quadratic equation, confirming that we solved the problem correctly.

Step-by-step solution

a) TR-TC Approach:

To find the profit function, subtract TC from TR: \(\pi(Q)=TR - TC\) \(\pi(Q)= (41.5Q-1.1Q^2)-(150+10Q-0.5Q^2+0.02Q^3)\) After simplifying the profit function, we will have: \(\pi(Q)=-0.02Q^3+0.6Q^2+31.5Q-150\) To find the maximum profit, differentiate the profit function with respect to Q and set the derivative equal to 0: \(\frac{d\pi}{dQ}=-0.06Q^2+1.2Q+31.5\) Now, solve the quadratic equation for Q: \(-0.06Q^2+1.2Q+31.5=0\) By solving this equation, we get two values for Q. We need to find the second derivative of the profit function to check if the values correspond to a maximum or minimum. \(\frac{d^2\pi}{dQ^2}=-0.12Q+1.2\) Now, plug in the Q values obtained from the previous step into the second derivative to identify the maximum profit value.

b) MR=MC Approach:

Differentiate the total revenue and total cost functions with respect to Q to find the marginal revenue (MR) and marginal cost (MC) functions: \(MR = \frac{d(TR)}{dQ}=41.5-2.2Q\) \(MC = \frac{d(TC)}{dQ}=10-1.0Q+0.06Q^2\) Setting MR equal to MC to find the profit-maximizing output: \(41.5-2.2Q=10-1.0Q+0.06Q^2\) Rearrange and simplify the equation to form a quadratic equation: \(0.06Q^2-1.2Q+31.5=0\) This is the same quadratic equation we found in part a). Solve for Q and then check the second derivative as before to determine which Q value corresponds to the maximum profit. In both approaches, the values obtained for Q, where the profit is maximized, will be the same, ensuring that we have solved the problem correctly.

Key concepts

Total Revenue

Total Revenue (TR) is the total amount of money a firm receives from the sale of its goods or services. Calculating TR is simple. You multiply the price per unit by the quantity of goods sold.
In the exercise, the firm's total revenue function is \(TR = 41.5Q - 1.1Q^2\). Here:

• \(41.5Q\) represents the revenue per unit quantity.
• \(1.1Q^2\) is a quadratic factor showing how revenue changes with increasing output, typically from decreasing price per unit at high volumes.

This means that as more units are sold, the total revenue first increases but then may decrease if the unit price falls or costs increase too significantly.

Total Cost

Total Cost (TC) includes all costs incurred in the production of goods. It is essential to determine this to understand profitability. The typical components include fixed costs and variable costs. In this exercise, the firm's total cost function is: \[TC = 150 + 10Q - 0.5Q^2 + 0.02Q^3\]
Here:

• \(150\) is the fixed cost, constant irrespective of the production level.
• \(10Q\) is the variable cost proportional to the output.
• \(-0.5Q^2\) and \(0.02Q^3\) capture more complex cost changes at higher production levels, such as economies of scale or increasing resource strain.

The task is to understand how these costs relate to the revenue for profit maximization.

Marginal Revenue

Marginal Revenue (MR) refers to the additional income from selling one more unit of a product. It is a crucial metric to determine the profitability of increasing production.

• In the exercise, the MR function is differentiated from the total revenue function.
• It is expressed as \(MR = 41.5 - 2.2Q\).

This equation highlights that as more units are produced and sold, the additional revenue per unit decreases. Firms use MR to decide whether producing additional units is beneficial when compared to marginal costs.

Marginal Cost

Marginal Cost (MC) indicates the cost of producing one additional unit of a product. It's critical for determining the level of output that maximizes profit. For this firm, MC is derived from the total cost function:

• \(MC = 10 - 1.0Q + 0.06Q^2\)

The equation shows:

• \(10\) represents the base cost per unit.
• \(-1.0Q\) and \(0.06Q^2\) adjust the cost depending on scale—reflecting efficiencies or difficulties in production.

The goal of using MC is to find the point where it equals marginal revenue; this is the profit-maximizing quantity since producing beyond this point would bring less revenue than cost.