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?

Short answer

#tag_content# Answer: Universal Widget should sell 60 widgets in Australia at a price of 20 and 40 widgets in Lapland at a price of 15 to maximize their total profits.

Step-by-step solution

Find the Revenue function for each market

Since demand curves for both markets are given, we can use them to obtain the revenue functions. The formula to find the revenue is given by: \(R(P) = P \cdot q\) Using the demand curves for Australia and Lapland, we will find the revenue functions for each market. For Australia: \(R_A(P) = P \cdot (100 - 2P)\) For Lapland: \(R_L(P) = P \cdot (100 - 4P)\)

Find the Profit function for each market

We will now find the profit functions by subtracting the cost function from the revenue functions. As given, the cost function is: \(C(q) = .25q^2\) For Australia, let \(q_A = 100 - 2P_A\) and for Lapland, let \(q_L = 100 - 4P_L\). The total quantity produced \((q)\) is the sum of the quantities produced for Australia and Lapland, i.e., \(q = q_A + q_L\) Now, rewrite the cost function in terms of \(P_A\) and \(P_L\): \(C(P_A, P_L) = .25(100 - 2P_A + 100 - 4P_L)^2\) Now, find the profit functions for each market: \(\Pi_A (P_A) = R_A(P_A) - C(P_A, P_L) = P_A (100 - 2P_A) - .25(100 - 2P_A + 100 - 4P_L)^2\) \(\Pi_L (P_L) = R_L(P_L) - C(P_A, P_L) = P_L (100 - 4P_L) - .25(100 - 2P_A + 100 - 4P_L)^2\)

Maximize the Profit Functions for each market

Now, we will find the critical points for the profit function by taking the first-order partial derivatives with respect to \(P_A\) and \(P_L\) for each profit function and setting them equal to zero: \(\frac{\partial \Pi_A}{\partial P_A} = 0\) \(\frac{\partial \Pi_L}{\partial P_L} = 0\) This will give us a system of two equations that can be solved for \(P_A\) and \(P_L\) in terms of \(q\). Solving for \(P_A\) and \(P_L\): \(P_A = 25 - 0.5q_A\) \(P_L = 25 - 0.25q_L\) Now substitute \(q_A = 100 - 2P_A\) and \(q_L = 100 - 4P_L\): \(P_A = 25 - 0.5(100 - 2P_A)\) \(P_L = 25 - 0.25(100 - 4P_L)\) Solving this system of equations, we obtain: \(P_A = 20\) and \(P_L = 15\) Now, substitute back to find \(q_A\) and \(q_L\): \(q_A = 100 - 2P_A = 60\) \(q_L = 100 - 4P_L = 40\)

Find the price charged in each location

Finally, using the optimal quantities found in step 3, we will use the demand curves to find the price charged in each location. For Australia: \(P_A = \frac{100 - q_A}{2} = \frac{100 - 60}{2} = 20\) For Lapland: \(P_L = \frac{100 - q_L}{4} = \frac{100 - 40}{4} = 15\) So, Universal Widget should sell 60 widgets in Australia at a price of 20 and 40 widgets in Lapland at a price of 15 to maximize their total profits.

Key concepts

Understanding the Cost Function in Microeconomics

In microeconomics, a cost function represents the total cost of production as a function of the quantity of output produced. This function is critical for businesses to understand because it helps them determine how their costs vary with different levels of production.

A typical cost function, like in our Universal Widget example, can be expressed as a quadratic equation, where total cost (\(C(q)\)) is a function of quantity produced (\(q\)). Here, the cost function is given by \(C(q) = .25q^2\), which implies that costs increase at an increasing rate as more units are produced.

Understanding this function allows businesses to calculate important economic points, such as the break-even point, where total revenue equals total costs, and the level of production that maximizes profit. Being able to calculate and analyze the cost function gives businesses a strategic advantage in decision-making and financial planning.

The Demand Curve and Its Role in Pricing Strategy

The demand curve is a graphical representation showing the relationship between the price of a good and the quantity of that good consumers are willing and able to purchase at that price. In our exercise, we're given two demand curves for different markets: Australia and Lapland.

For Australia, the demand curve equation is \(q = 100 - 2P\), and for Lapland, it’s \(q = 100 - 4P\). These linear equations show that as price increases, the quantity demanded decreases, which reflects the law of demand. In practice, this curve helps businesses predict how consumers will respond to changes in price and can guide in setting prices to maximize sales volumes and revenue.

For example, by altering the quantity supplied to each market, Universal Widget can control the widget prices in both Australia and Lapland markets to match consumer demand and maximize revenue.

Revenue Maximization Strategy

Revenue maximization is a strategy aimed at increasing the amount of income generated from sales before subtracting any expenses. Businesses often use this strategy when they want to grow their market share or capitalize on high demand. The revenue function, outlined in our widget example, directly ties into the demand curve.

For a single product, the revenue function is typically the price per unit times the quantity sold, expressed as \(R(P) = P \times q\). In the Universal Widget scenario, two separate revenue functions for different markets \(R_A(P) = P \times (100 - 2P)\) for Australia and \(R_L(P) = P \times (100 - 4P)\) for Lapland, are calculated. These functions are vital for determining at what prices the revenue peaks, thereby guiding the business in setting their prices appropriately for each market to maximize total revenue.

Profit Optimization in Microeconomic Theory

Profit optimization is the process of finding the point at which a business can make the maximum possible profit. This involves analyzing the cost function and the revenue function together to determine the best price and quantity to produce and sell goods.

In the given solution, profit functions for each market, \(\Pi_A(P_A)\) and \(\Pi_L(P_L)\) are derived by subtracting the total cost from total revenue. The key to profit maximization lies in setting the derivative of the profit function with respect to price to zero, which yields the optimal prices \(P_A\) and \(P_L\) for Australia and Lapland, respectively.

This optimum point considers the cost of production and the willingness of consumers to pay, balancing both to achieve maximum profitability. Using calculus to find this balance, as shown in the step-by-step solution, allows businesses like Universal Widget to set a pricing strategy that maximizes their profits by selling at prices that align with the optimal production quantity.