Question

Universal Widget produces high-quality widgets at its plant in Gulch, Nevada, for sale throughout the world. The cost function for total widget production \((q)\) is given by total cost \(=.25 q^{2}\) . Widgets are demanded only in Australia (where the demand curve is given by \(q=100-2 P\) ) and Lapland (where the demand curve is given by \(q=100-4 P\) ). If Universal Widget can control the quantities supplied to each market, how many should it sell in each location in order to maximize total profits? What price will be charged in each location? The production function for a firm in the business of calculator assembly is given by \\[ q=2 \sqrt{L} \\] where \(q\) is finished calculator output and \(L\) represents hours of labor input. The firm is a price taker for both calculators (which sell for \(P\) ) and workers (which can be hired at a wage rate of \(w \text { per hour })\) a. What is the supply function for assembled calculators \([q=f(P, w)]\) ? b. Explain both algebraically and graphically why this supply function is homogeneous of degree zero in \(P\) and \(w\) and why profits are homogeneous of degree one in these variables. c. Show explicitly how changes in \(w\) shift the supply curve for this firm.

Short answer

#Answer#: To maximize total profits for Universal Widget in the Australia and Lapland markets, we must follow a series of steps: 1. Identify the required equations for Total Cost, and the demand curves for Australia and Lapland. 2. Formulate the Total Revenue equation. 3. Find expressions for \(P_{A}\) and \(P_{L}\) using the demand curve equations. 4. Substitute the expressions for \(P_{A}\) and \(P_{L}\) into the total revenue equation. 5. Calculate the profit function and set up the optimization problem. 6. Find the first-order partial derivatives with respect to \(q_{A}\) and \(q_{L}\), and set them equal to zero to solve for the optimal quantities. 7. Find the optimal prices for each market using the optimal quantities. By following these steps, we can determine the optimal quantities (\(q_{A}\) and \(q_{L}\)) and prices (\(P_{A}\) and \(P_{L}\)) for selling widgets in Australia and Lapland that maximize Universal Widget's total profits.

Step-by-step solution

Identify the required equations and profit function

To start, let's identify the equations in the problem: - Total Cost is given by \(TC = 0.25 q^{2}\). - Demand curve in Australia: \(q_{A} = 100 - 2 P_{A}\). - Demand curve in Lapland: \(q_{L} = 100 - 4 P_{L}\). Next, we will find the profit function for Universal Widget. The profit function is given by \(Profit = Total \thinspace Revenue - Total \thinspace Cost\). Additionally, we know that \(TR = P \cdot q\).

Formulate Total Revenue

Since Universal Widget can control the quantity supplied to each market separately, we can calculate the total revenue separately for each market. So, the Total Revenue for both markets can be calculated as: \(TR = P_{A} q_{A} + P_{L} q_{L}\).

Find expressions for \(P_{A}\) and \(P_{L}\) using the demand curve equations

Now, we will find the expressions for \(P_{A}\) and \(P_{L}\) in terms of \(q_{A}\) and \(q_{L}\) using the respective demand curves. From the demand curve in Australia: \(q_{A} = 100 - 2 P_{A}\) and we have, \[P_{A} = \frac{100 - q_{A}}{2}\] Similarly, from the demand curve in Lapland: \(q_{L} = 100 - 4 P_{L}\) and we get, \[P_{L} = \frac{100 - q_{L}}{4}\]

Substitute the expressions for \(P_{A}\) and \(P_{L}\) in the total revenue equation

Now, we will substitute the expressions for \(P_{A}\) and \(P_{L}\) found in step 3 in to the total revenue equation we found in step 2. \[TR = (\frac{100 - q_{A}}{2}) q_{A} + (\frac{100 - q_{L}}{4}) q_{L}\]

Calculate the profit function and set up the optimization problem

First, let's calculate the total cost using the given cost function: \(TC = 0.25 (q_{A} + q_{L})^{2}\). Now, we are going to calculate the profit function by subtracting the total cost from the total revenue: \(Profit = TR - TC\). Our optimization problem now becomes: \[\max _{q_{A}, q_{L}} \left((\frac{100 - q_{A}}{2}) q_{A} + (\frac{100 - q_{L}}{4}) q_{L} - 0.25 (q_{A} + q_{L})^{2} \right)\]

Find the partial derivatives and solve for the optimal quantities

To maximize the profit, we will find the first-order partial derivatives with respect to \(q_{A}\) and \(q_{L}\), and set them equal to zero: \[\frac{\partial Profit}{\partial q_{A}}=0\] \[\frac{\partial Profit}{\partial q_{L}}=0\] Now, solve for \(q_{A}\) and \(q_{L}\) simultaneously to obtain the optimal quantities that maximize profits.

Find the optimal prices for each market using the optimal quantities

When we find the optimal quantities for both \(q_{A}\) and \(q_{L}\), we can plug these back into the expressions for \(P_{A}\) and \(P_{L}\), from step 3, to find the optimal prices that Universal Widget should charge in each market to maximize their total profits. At the end of this process, we'll have the optimal quantities (\(q_{A}\) and \(q_{L}\)) and prices (\(P_{A}\) and \(P_{L}\)) for the widget sales in Australia and Lapland that maximize Universal Widget's total profits.

Key concepts

Cost Function

The concept of a cost function is fundamental in understanding how costs are related to the production volume of a company. In this exercise, the total cost function is defined based on the widget production volume ( q), and it is given by: \(TC = 0.25q^2\). This quadratic function implies that costs will increase with the square of production.

This means that as production volume increases, the cost of producing additional units rises faster. This can occur due to factors like the need for overtime, additional shifts, or more materials.

• Understanding the behavior of the cost function helps in decision-making about how much to produce.
• The cost function also helps in understanding the efficiency of the production process.

It is important to analyze the cost function carefully to ensure that production remains optimal and that costs do not outweigh revenues.

Demand Curve

The demand curve is a graphical representation of the relationship between the price of a product and the amount of it that consumers are willing to purchase. In this exercise, there are two markets with distinct demand curves: Australia and Lapland.

For Australia, the demand curve is expressed as:
\[ q_A = 100 - 2P_A \]

In Lapland, it is given by:
\[ q_L = 100 - 4P_L \]

• The slope of the demand curve indicates how sensitive the quantity demanded is to price changes.
• A steeper demand curve suggests consumers are less sensitive to price changes.
• The intercept (value where demand is zero) tells us the theoretical maximum quantity demanded.

Understanding these demand curves is crucial for setting prices optimally to maximize sales without losing customers to high prices.

Price Discrimination

Price discrimination occurs when a company sells the same product at different prices to different markets or segments. In this exercise, Universal Widget can decide the quantity of widgets each market receives, thus allowing them to engage in price discrimination by offering distinct prices in Australia and Lapland.

• Different prices can be set due to variations in demand curves, allowing the company to maximize total revenue from each market.
• This strategy helps the firm capture consumer surplus by charging prices closer to what each group is willing to pay.
• Understanding market segmentation and demand elasticity is key to successful price discrimination.

Price discrimination can significantly boost profits if done correctly, ensuring that the firm can cover costs efficiently while maximizing revenue.

Optimization Problem

An optimization problem involves finding the most efficient and effective solution under given constraints. In this exercise, the goal is to maximize profits by determining the optimal quantities of widgets to sell in each market.

To set up the optimization problem, calculate the total revenue from both markets and form the profit function by subtracting total costs from total revenue. This requires:

• Formulating revenue functions for each market using respective demand equations.
• Substituting demand equations into revenue calculations to express everything in terms of quantities.
• Setting up the overall profit function as \(\max_{q_{A}, q_{L}} \, \left[ \text{Revenue} - \text{Cost} \right] \).
• Using calculus, particularly derivatives, to find the quantity combinations that maximize profits.

The solution to this optimization gives us the best strategy for setting prices and quantities to maximize Universal Widget's profits in both Australia and Lapland.