Question

1: } C_{1}\left(Q_{1}\right)=10 Q_{1}^{2} \\\ \text { Factory #2: } C_{2}\lef… # A firm has two factories, for which costs are given by: \\[ \begin{array}{l} \text { Factory #1: } C_{1}\left(Q_{1}\right)=10 Q_{1}^{2} \\ \text { Factory #2: } C_{2}\left(Q_{2}\right)=20 Q_{2}^{2} \end{array} \\] The firm faces the following demand curve: \\[ p=700-5 Q \\] where \(Q\) is total output-i.e., \(Q=Q_{1}+Q_{2}\) a. \(\mathrm{On}\) a diagram, draw the marginal cost curves for the two factories, the average and marginal revenue curves, and the total marginal cost curve (i.e., the marginal cost of producing \(Q=Q_{1}+Q_{2}\) ). Indicate the profit-maximizing output for each factory, total output, and price. b. Calculate the values of \(Q_{1^{\prime}} Q_{2^{\prime}} Q,\) and \(P\) that maximize profit c. Suppose that labor costs increase in Factory 1 but not in Factory \(2 .\) How should the firm adjust (i.e. raise, lower, or leave unchanged) the following: Output in Factory \(1 ?\) Output in Factory \(2 ?\) Total output? Price?

Short answer

The marginal cost curves for Factories 1 and 2 are \( MC_{1} = 20Q_{1} \) and \( MC_{2} = 40Q_{2} \) respectively. The profit-maximizing outputs corresponding to the quantities that make MC = MR are \( Q_{1^{\prime}} = 35 \) and \( Q_{2^{\prime}} = 17.5 \). The total output is \( Q = 52.5 \) and the corresponding price is \( P = 437.5 \). In case of an increase in labor costs in Factory 1, the output in Factory 1 should be reduced while that in Factory 2 should be increased. Total output should be reduced and the price will therefore go up.

Step-by-step solution

Understanding Factory Cost Equations

The cost functions for Factory 1 and Factory 2 are given by \( C_{1}(Q_{1}) = 10Q_{1}^{2} \) and \( C_{2}(Q_{2}) = 20Q_{2}^{2} \). The marginal cost for each factory is found by taking the derivative with respect to quantity. For Factory 1, the marginal cost is \( MC_{1} = 20Q_{1} \) and for Factory 2, it's \( MC_{2} = 40Q_{2} \).

Drawing Marginal Cost Curves

On graph paper, draw each of the marginal cost, average and marginal revenue, and the total marginal cost curve. Their axes are price on the y-axis and quantity on the x-axis. The curves are \( MC_{1} = 20Q_{1} \), \( MC_{2} = 40Q_{2} \), and \( MC = 20Q \). The marginal revenue curve is given by \( MR = 700 - 10Q \). Determine the price and quantities for Q1 and Q2 that makes MC = MR. Those points are the profit-maximizing outputs.

Calculating Values for Maximum Profit

By setting the marginal cost equal to the marginal revenue, we can solve for the quantities Q1 and Q2. For factory 1, \(20Q_{1} = 700 - 10Q \), which gives us \( Q_{1^{\prime}} = 35 \). Similarly for factory 2, \(40Q_{2} = 700 - 10Q \), which gives us \( Q_{2^{\prime}} = 17.5 \). The total output, \( Q \), is going to be the sum \( Q = Q_{1^{\prime}} + Q_{2^{\prime}} = 52.5 \). The price, \( P \), is given by substituting \( Q \) into the demand equation, \( P = 700 - 5*52.5 = 437.5 \). After those computations, substitute the quantity values into the cost functions to verify that Factory 1 and Factory 2 earn the maximum profit.

Dealing with Increased Labor Costs

Subsequently, when labor costs increase, the marginal cost of Factory 1 increases, but that of Factory 2 remains unchanged. Therefore, to reduce overall costs, the company should reduce the output of Factory 1 and increase the output of Factory 2, while the total output should be reduced to keep costs under control. As a result, with lower overall supply, the price increases according to the demand curve.

Key concepts

Profit Maximization

Every firm's primary goal is to maximize its profit, which is the difference between total revenue and total cost. In this context, the firm operates two factories with different cost structures and aims to determine the appropriate output level for each to maximize profit. The key to profit maximization lies in equating marginal revenue (MR) with marginal cost (MC) for each unit produced. This balancing act ensures that the cost of producing an additional unit does not exceed the revenue generated by selling it.

• In Factory 1, marginal cost is derived from the cost function and expressed as \( MC_1 = 20Q_1 \).
• For Factory 2, it's \( MC_2 = 40Q_2 \).
• The firm's strategy is to adjust production in both factories so that \( MC = MR \).

This process involves carefully calculating the output quantities at the intersection of marginal costs and marginal revenues for optimal profit.

Demand Curve

The demand curve reflects how much quantity of a good consumers are willing to purchase at different price points. It is crucial for determining the revenue side of the profit equation. In this case, the demand curve is linear and given by the equation \( p = 700 - 5Q \), where \( p \) represents the price and \( Q \) the total quantity produced.

• The slope of the demand curve indicates how sensitive quantity demanded is to price changes.
• Understanding the demand curve helps in predicting changes in sales volume with price adjustments.
• This equation allows the firm to calculate total revenue as a function of output.

Essentially, the demand curve guides how the firm should set its prices to align with consumer expectations and maximize profits.

Marginal Revenue

Marginal revenue (MR) is the additional revenue that a firm generates from selling one more unit of a product. It is a crucial concept because firms must match marginal revenue with marginal cost to achieve profit maximization. In our scenario, the marginal revenue can be derived from the demand curve, giving us \( MR = 700 - 10Q \).

• MR tends to decrease as output increases due to the negative slope of the demand curve.
• Comparing MR to MC helps determine the optimal production level.
• If MR exceeds MC, producing more can increase profit.

Thus, understanding how MR decreases with increased sales is pivotal in ensuring that production aligns with maximum profitability conditions.

Output Adjustment

Output adjustment involves changing the production levels in response to market conditions, changes in costs, or other factors. For firms facing varying costs across production sites, like the two factories in this problem, adjusting outputs based on cost efficiency is key. When external factors, such as an increase in labor costs, come into play, these decisions become even more crucial.

• If labor costs rise in Factory 1, raising the marginal cost, it makes sense to reduce its output.
• Conversely, Factory 2 might increase its production to offset reductions in Factory 1, given its unchanged costs.
• Total output and prices adjust based on overall costs and market supply-demand dynamics.

Adapting to changes quickly allows a firm to maintain profitability despite shifts in production costs or market conditions.