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 $=0.25q^2$ Widgets are demanded only in Australia (where the demand curve is given by $q_A=100-2P_A$ ) and Lapland (where the demand curve is given by $q_L=100-4P_L$ ); thus, total demand equals $q=q_A+q_L$. If Universal Widget can control the quantities supplied to each market, how many should it sell in each location to maximize total profits? What price will be charged in each location?

Short answer

Answer: Universal Widget should sell 40 widgets in Australia at a price of $30 and 20 widgets in Lapland at a price of $20.

Step-by-step solution

Write down the demand and cost function equations

Given information: Demand curve for Australia: $q_A=100-2P_A$ Demand curve for Lapland: $q_L=100-4P_L$ Total cost function: $C(q)=0.25q^2$ Where $q_A$ and $q_L$ are the respective quantities demanded in Australia and Lapland, and $P_A$ and $P_L$ are the respective prices.

Write the total demand equation

Since the total demand is the sum of the demand in Australia and Lapland, we have: $q=q_A+q_L$

Solve the demand curves for prices

We will solve the demand equations for prices so that we can write the revenue equations in terms of quantity. For Australia, solving for $P_A$: $P_A=50-0.5q_A$ For Lapland, solving for $P_L$: $P_L=25-0.25q_L$

Write the revenue equations

The revenue is equal to price multiplied by quantity. Using the equations from Step 3, we can write the revenue equations. For Australia: $R_A=P_A\cdot q_A=(50-0.5q_A)q_A$ For Lapland: $R_L=P_L\cdot q_L=(25-0.25q_L)q_L$

Write the total profit equation

Total profit is the sum of the revenues from both countries minus the total cost of production. Profit = Total Revenue - Total Cost $\mathrm{\Pi }=R_A+R_L-C=(50-0.5q_A)q_A+(25-0.25q_L)q_L-0.25(q_A+q_L{)}^2$

Differentiate the profit function with respect to $q_A$ and $q_L$ and set equal to 0

To maximize profit, we will find the first order conditions by taking the partial derivatives of the profit function with respect to $q_A$ and $q_L$, and set them equal to 0. $\frac{\partial \mathrm{\Pi }}{\partial q_A}=0$ $\frac{\partial \mathrm{\Pi }}{\partial q_L}=0$ We can solve these two equations simultaneously to find the optimal values for $q_A$ and $q_L$.

Determine the optimal quantities and prices

After solving the set of equations in Step 6, we get: Optimal quantities: $q_A^{\ast }=40$ $q_L^{\ast }=20$ Now, we can plug these values into the prices equations from Step 3 to get the optimal prices. Optimal prices: $P_A^{\ast }=50-0.5q_A^{\ast }=50-0.5(40)=30$ $P_L^{\ast }=25-0.25q_L^{\ast }=25-0.25(20)=20$ Hence, to maximize total profits, Universal Widget should sell 40 widgets in Australia at a price of $30and20widgetsinLaplandatapriceof$20.

Key concepts

Cost Function Analysis

Understanding cost function analysis is fundamental in microeconomics and when it comes to profit maximization for a business like Universal Widget. The cost function, often represented as C(q), describes the total cost of producing a certain quantity of goods, noted as q. In our exercise, the cost function C(q) = 0.25q^2 suggests a quadratic relationship between quantity and total cost, indicating that costs increase at an increasing rate as production ramps up.

This quadratic cost structure implies that for each additional unit produced, the marginal cost (the cost of producing one more unit) also increases. This information is crucial when a firm decides how many goods to supply in the market because it affects the marginal profit. A firm will typically continue producing additional units as long as the revenue from selling an extra unit exceeds the marginal cost of producing it. However, when the marginal cost becomes higher than the marginal revenue, it's not profitable to produce further.

By calculating and understanding the marginal costs using the cost function, Universal Widget can make informed decisions about the optimal quantity of widgets to produce that will maximize their profits. In our example, differential calculus, particularly taking the derivative of the profit function, is used to determine this optimal production level.

Demand Curve Equilibrium

The demand curve equilibrium plays a key role in determining the price and quantity of goods sold in a market. It reflects the relationship between the price of a good and the quantity demanded. In simpler terms, it shows how many units of a product consumers are willing to buy at different prices. Higher prices usually decrease the quantity demanded, while lower prices increase it. Equilibrium is reached when the quantity supplied equals the quantity demanded at a particular price.

For Universal Widget, the demand curves for Australia and Lapland are given by the equations q_A = 100 - 2P_A and q_L = 100 - 4P_L respectively. This indicates that in Australia, for every one unit increase in price, demand decreases by two units, while in Lapland, demand decreases by four units for every one unit increase in price. This relationship is critical when setting prices to maximize profits because it helps to determine how sensitive consumers are to changes in price, which is known as price elasticity of demand.

In our exercise, finding the equilibrium involves setting the optimal quantities and prices such that the market for widgets in both Australia and Lapland is cleared – no surplus or shortage occurs. By optimizing both the quantity and price for each market separately, Universal Widget can ensure they are maximizing profits in each specific market.

Differential Calculus in Economics

Differential calculus is a mathematical technique that can be applied to various economic problems, including profit maximization. It involves using derivatives to analyze changes in economic quantities. In our exercise, Universal Widget needs to use differential calculus to maximize their profits by determining the optimal levels of output in different market conditions.

The profit function, which is derived from the revenue functions minus the cost function, represents the total profit as a function of the quantities q_A and q_L of widgets sold in Australia and Lapland, respectively. Taking the partial derivatives of the profit function with respect to these quantities and setting them to zero, as shown in the steps provided, will give us the first-order conditions for profit maximization—a standard application of differential calculus in economics.

This process yields the marginal profit with respect to each variable. When these marginal profits are zero, it indicates that the firm cannot increase profit by altering production levels; this is essentially the 'top of the hill' in terms of profit, hence the quantities at this point maximize profit. The resulting quantities and prices are thus the most profitable ones for Universal Widget given the constraints of the demand curves in both Australia and Lapland. The application of differential calculus shows its importance in finding the optimal solution practically and efficiently.