Universal Widget produces high-quality widgets at its plant...

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 $=0.25 q^{2}$ Widgets are demanded only in Australia (where the demand curve is given by $q_{A}=100-2 P_{A}$ ) and Lapland (where the demand curve is given by $q_{L}=100-4 P_{L}$ ); 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?

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 $=0.25 q^{2}$
Widgets are demanded only in Australia (where the demand curve is given by $q_{A}=100-2 P_{A}$ ) and Lapland (where the demand curve is given by $q_{L}=100-4 P_{L}$ ); 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?

Key Concepts

Cost Function

A cost function represents the total cost of producing a given number of units of a good. In economic analysis, it is crucial for determining how production volume affects costs and, ultimately, profits. Understanding the cost function helps firms identify the production levels at which costs are minimized and profits are maximized.

Demand Curve

The demand curve illustrates the relationship between the quantity demanded of a good and its price, capturing consumer behavior in relation to market price changes. It is essential for determining how price adjustments affect sales quantities and for analyzing market equilibrium conditions.

Price Discrimination

Price discrimination involves charging different prices to different customer segments or markets based on their willingness to pay. This strategy is used by firms with market power to maximize profits by capturing more consumer surplus in each market segment while considering different demand elasticities.

Profit Maximization

Profit maximization is the process by which a firm determines the production and pricing strategy that yields the highest possible profit. This involves setting marginal revenue equal to marginal cost in each market segment and allocating products in a way that leverages differences in demand to achieve optimal pricing.

Marginal Analysis

Marginal analysis involves examining the additional benefits and costs associated with producing one more unit of a good. In the context of pricing and production decisions, it is used to determine the optimal allocation of resources by equating marginal revenue and marginal cost, ensuring that each unit produced contributes positively to overall profit.

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