Question

1: C_{1}\left(Q_{1}\right)=10 Q_{1}^{2} \\\ \text { Factory } \\# 2: C_{2}… # 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 \(Q_{1}, Q_{2}, Q,\) 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 profit-maximizing output for each factory are: Q1=0.25, Q2=0.125, total output Q=0.375, and the price P=681.25. With increased labor costs in Factory 1, output should decrease in Factory 1, stay unchanged in Factory 2, total output will decrease, and price will increase.

Step-by-step solution

Determine the marginal cost (MC)

The marginal cost function for the two factories can be derived from the cost functions using calculus. For Factory 1: MC1 = dC1/dQ1 = 20Q1. For Factory 2: MC2 = dC2/dQ2 = 40Q2.

Determine the average (AR) and marginal revenue (MR)

Since total output Q is equal to Q1 + Q2, we can substitute Q into the price function to get AR = P = 700 - 5Q. By taking the derivative of AR with respect to Q, the MR is obtained as MR = d(AR)/dQ = -5.

Plot the curves and determine the profit-maximizing output

On a graph plot the MC, AR and MR curves. Also plot the total MC which equals to MC1 + MC2. The point where MR cuts MC is the profit-maximizing output for each factory, total output and price.

Calculation of Q1, Q2, Q, and P

To maximize profit, set MR = MC. When MR = MC1, we get Q1 = (MR/20) = 5/20 = 0.25. When MR = MC2, we get Q2 = (MR/40) = 5/40 = 0.125. So, total output Q = Q1 + Q2 = 0.375. Substituting Q in the price function, P = 700 - 5Q = 681.25.

Understand the adjustments when the labor cost increases

If the labor cost increases in Factory 1, MC1 will increase. To maintain profit-maximizing conditions, the firm should reduce output in Factory 1. There is no change in Factory 2, so it should continue producing at the same level. The total output will decrease and due to the law of supply, the price will increase.

Key concepts

Marginal Cost

Marginal cost is a crucial concept in understanding how businesses operate efficiently. It refers to the additional cost incurred by producing one more unit of a good or service. In our given problem, we have two factories whose cost functions depend on the quantity they produce.
To find the marginal cost for each factory, calculus is used to derive the costs per additional unit produced:

• For Factory 1, the cost function is given as \(C_1(Q_1) = 10Q_1^2\). The marginal cost is the derivative of the cost function with respect to \(Q_1\), which computes to \(MC_1 = 20Q_1\).
• For Factory 2, the cost function is \(C_2(Q_2) = 20Q_2^2\), resulting in a marginal cost of \(MC_2 = 40Q_2\).

The marginal cost informs businesses on the cost of increasing production and helps determine the optimal output level.

Marginal Revenue

Marginal revenue is the additional income earned by selling one more unit of a product or service. It provides insights into the revenue dynamics of a firm and is essential in decision-making for the optimal level of production.
In this problem, the demand curve is given by \(P = 700 - 5Q\), which also represents the average revenue per unit sold. The marginal revenue is found by differentiating the average revenue with respect to total output \(Q\):

• The derivative yields \(MR = -5\), indicating that for each unit increase in output, revenue decreases by 5.

Understanding marginal revenue helps firms balance production costs and revenue to maximize profitability.

Profit Maximization

Profit maximization is the process through which a firm determines the price and output level that returns the maximum profit. The concept involves equating marginal revenue (MR) with marginal cost (MC). This equation helps identify the point beyond which producing additional units would result in decreased overall profits.
In this problem, profit maximization for the combined output of both factories occurs where the total marginal cost equals the marginal revenue. After calculations:

• Factory 1 should produce when \(MR = MC_1\), giving \(Q_1 = 0.25\).
• Factory 2 should operate at \(MR = MC_2\), resulting in \(Q_2 = 0.125\).
• The total output, \(Q\), is \(Q_1 + Q_2 = 0.375\), and at this output level, the price \(P\) is \(681.25\).

This approach assists businesses in strategizing to achieve the best financial outcomes.

Supply and Demand

Supply and demand are foundational principles of economics that determine the price and quantity of goods and services in a market. Demand refers to how much consumers want a product, while supply is the quantity that producers are willing and able to sell. The interaction of supply and demand dictates market prices and production levels.
In our given scenario, changes in the costs at Factory 1 alter its marginal cost function due to increased labor costs. Consequently, this affects the firm's supply curve:

• If the marginal cost of production rises in one factory, that factory should reduce output to maintain economic efficiency.
• Since Factory 2's costs remain unchanged, its output can stay constant.
• The overall supply from the firm decreases, pushing prices up, aligning with the supply and demand principle that reduced supply leads to higher prices.

Understanding supply and demand dynamics provides businesses with critical information for making production decisions and setting prices.