Question

A firm faces a demand curve given by \\[ q=\mathbf{1 0 0}-2 R \\] Marginal and average costs for the firm are constant at \(\$ 10\) per unit a. What output level should the firm produce to maximize profits? What are profits at that output level? b. What output level should the firm produce to maximize revenues? What are profits at that output level? c. Suppose the firm wishes to maximize revenues subject to the constraint that it earn \(\$ 12\) in profits for each of the 64 machines it employs. What level of output should it produce? d. Graph your results.

Short answer

**Short Answer** The firm maximizes its profit when it produces an output level of 40 units and sells them at a price of $30 per unit, resulting in a profit of $800. When the firm focuses on maximizing revenue, it increases the output level to 50 units and sells them at a price of $25 per unit, leading to a profit of only $750. When the firm must maximize revenue subject to a profit constraint of $768, it produces approximately 43.5 units and sells them at a price of around $28.25 per unit.

Step-by-step solution

Inverse demand curve and total revenue

The demand curve is given by: \\[q = 100 - 2R.\\] To find the inverse demand curve, solve for R in terms of q: \\[R = 50 - \frac{1}{2}q.\\] Total revenue (TR) is given by the product of R and q: \\[TR(q) = Rq = \left(50 - \frac{1}{2}q\right)q.\\] The marginal and average costs are constant at \(10 per unit, so the total cost (TC) is \)10 times the number of units produced: \\[TC(q) = 10q.\\]

Maximize profit

To maximize profit, we need to find the output level (q) that maximizes the difference between total revenue and total cost. The profit function (π) is given by \\[\pi(q) = TR(q) - TC(q) = (50 - \frac{1}{2}q)q - 10q.\\] To find the maximum profit, take the first derivative of the profit function with respect to q and set it equal to zero: \\[\frac{d\pi(q)}{dq} = 50 - q - 10 = 0.\\]

Calculate optimal output level and profit

Solve for the optimal output level: \\[q^* = 50 - 10 = 40.\\] Plug this optimal output level (q^*) back into the inverse demand curve to find the price: \\[R^* = 50 - \frac{1}{2}(40) = 30.\\] Now, calculate the total revenue, total cost, and profit at the optimal output level: \\[TR^* = R^*q^* = 30 \times 40 = 1200,\\] \\[TC^* = 10q^* = 10 \times 40 = 400,\\] \\[\pi^* = TR^* - TC^* = 1200 - 400 = 800.\\] **Part b)**

Maximize revenue

To maximize total revenue, take the first derivative of the total revenue function with respect to q and set it equal to zero: \\[\frac{dTR(q)}{dq} = 50 - q = 0.\\]

Calculate output level and profit

Solve for the output level that maximizes revenue: \\[q^{**} = 50.\\] Plug this output level (q^{**}) back into the inverse demand curve to find the price: \\[R^{**} = 50 - \frac{1}{2}(50) = 25.\\] Now, calculate the total revenue, total cost, and profit at this output level: \\[TR^{**} = R^{**}q^{**} = 25 \times 50 = 1250,\\] \\[TC^{**} = 10q^{**} = 10 \times 50 = 500,\\] \\[\pi^{**} = TR^{**} - TC^{**} = 1250 - 500 = 750.\\] **Part c)**

Set up the constraint

The constraint is that the firm must earn \(12 in profit for each of the 64 machines it employs, so the profit must be equal to \)12 \times 64 = 768. The profit function is: \\[\pi(q) = (50 - \frac{1}{2}q)q - 10q.\\] Set this function equal to the constraint: \\[(50 - \frac{1}{2}q)q - 10q = 768.\\]

Calculate the output level

Solve for the output level that satisfies the constraint: \\[q^{***} \approx 43.5\\] (rounding to the nearest half unit for simplicity).

Calculate the price

Plug this output level (q^{***}) back into the inverse demand curve to find the price: \\[R^{***} \approx 50 - \frac{1}{2}(43.5) \approx 28.25.\\] **Part d)**

Graph the results

On a graph, plot the demand curve and the constant marginal and average cost lines. Mark the optimal output levels and prices for each of the three scenarios: 1. profit maximization: \\[q^* = 40, \hspace{2mm} R^* = 30,\\] 2. revenue maximization: \\[q^{**} = 50, \hspace{2mm} R^{**} = 25,\\] 3. maximizing revenue subject to profit constraint: \\[q^{***} \approx 43.5, \hspace{2mm} R^{***} \approx 28.25.\\]

Key concepts

Demand Curve Analysis

Understanding the demand curve is crucial for any firm aiming to determine the optimal level of production. The demand curve, in its simplest form, shows the relationship between the price of a good and the quantity demanded. It typically slopes downwards, indicating that as prices fall, demand rises.

In the exercise above, the demand curve is represented by the equation \( q = 100 - 2R \). This means that the quantity demanded \( q \) decreases as the price \( R \) increases. By rearranging this equation to express \( R \) as a function of \( q \), companies can better understand how changes in quantity impact pricing. This gives us the inverse demand curve: \( R = 50 - \frac{1}{2}q \).

Analyzing the demand curve helps a business to predict customer behavior, set strategical price points, and plan production accordingly, which is foundational for profit maximization.

Revenue Maximization

Revenue maximization involves finding the point where the firm's revenue is at its highest. It is essential to understand that maximizing revenue may not always result in maximizing profit because costs are not taken into account.

In the context of the provided exercise, total revenue (TR) as a function of output (q) is derived by multiplying the price (R) by the quantity (q):\[TR(q) = Rq = (50 - \frac{1}{2}q)q = 50q - \frac{1}{2}q^2.\]

To find the level of output that maximizes revenue, we take the derivative of the revenue function with respect to quantity and set it to zero: \[\frac{dTR(q)}{dq} = 50 - q = 0.\] Solving this gives us \( q^{**} = 50 \), indicating the output level that maximizes revenue. At this point, total revenue is \( 1250 \), but profits might not be optimal due to the high production costs involved.

Marginal Costs

Marginal cost represents the increase in total cost associated with producing one additional unit of output. In many industries, understanding and analyzing marginal costs is key to deciding the optimal level of production.

In our exercise, marginal costs are constant at $10 per unit. This simplifies calculations as any increase in output directly adds $10 to the total cost. The concept of constant marginal and average costs implies that each new unit produced costs the same as previous ones, which is often the case in highly efficient production settings.

By comparing marginal cost with marginal revenue, firms make informed decisions about how much to produce. If the marginal cost exceeds marginal revenue, producing more would result in a loss. Thus, the decision should be centered on producing where marginal cost equals marginal revenue for maximum profitability.

Cost-Benefit Analysis

Cost-benefit analysis is the process of comparing the estimated costs and expected benefits of a decision or project. In the realm of profit maximization, this analysis helps firms decide how much to produce to achieve the best financial outcome.

From the exercise, we can see cost-benefit principles applied in various scenarios like maximizing profit, maximizing revenue, and meeting profit constraints. For example, when firms are constrained by specific profit goals per machine or unit, such as needing to achieve a total profit of \( \$ 768 \), they must carefully choose production levels that meet these constraints.

The firm determines that by producing \( q^{***} \approx 43.5 \) units, they can align with this profit constraint. By comparing the costs and adjusting their output accordingly, businesses strike a balance between achieving maximum revenues and maintaining desired profit levels. This interplay is critical in strategic business decision-making.