A store sells two brands of laptop sleeves. The store pays ...

A store sells two brands of laptop sleeves. The store pays $\$ 25$ for each brand $A$ sleeve and $\$ 30$ for each brand $B$ sleeve. $A$ consulting firm has estimated the daily demand equations for these two competitive products to be $x=130-4 p+q$ Demand equation for brand $A$ $y=115+2 p-3 q$ Demand equation for brand $B$ where $p$ is the selling price for brand $A$ and $q$ is the selling price for brand $B$. (A) Determine the demands $x$ and $y$ when $p=\$ 40$ and $q=\$ 50 ;$ when $p=\$ 45$ and $q=\$ 55$ (B) How should the store price each brand of sleeve to maximize daily profits? What is the maximum daily profit? [Hint: $C=25 x+30 y, \quad R=p x+q y,$ and $P=R-C .]$

A store sells two brands of laptop sleeves. The store pays $\$ 25$ for each brand $A$ sleeve and $\$ 30$ for each brand $B$ sleeve. $A$ consulting firm has estimated the daily demand equations for these two competitive products to be
$x=130-4 p+q$ Demand equation for brand $A$
$y=115+2 p-3 q$ Demand equation for brand $B$
where $p$ is the selling price for brand $A$ and $q$ is the selling price for brand $B$.
(A) Determine the demands $x$ and $y$ when $p=\$ 40$ and $q=\$ 50 ;$ when $p=\$ 45$ and $q=\$ 55$
(B) How should the store price each brand of sleeve to maximize daily profits? What is the maximum daily profit? [Hint: $C=25 x+30 y, \quad R=p x+q y,$ and $P=R-C .]$

Key Concepts

Optimization in Pricing Strategy

Optimization in pricing strategy involves determining the optimal price levels that maximize a firm's profit, given the relationships defined by the demand, revenue, and cost functions. This often requires applying mathematical techniques, such as calculus or other optimization methods, to find the price points where profit is at its maximum.

Cost Function

The cost function represents the total expense incurred in producing or procuring products. It typically includes fixed and variable costs and is used in conjunction with the revenue function to analyze profitability. Understanding the cost structure is essential for making pricing and production decisions.

Profit Function

The profit function is defined as the difference between the revenue function and the cost function. It encapsulates the financial gain of a business and is central to problems involving maximization of profit. The objective in many pricing and production problems is to choose prices or quantities that maximize this function.

Demand Function

A demand function is a mathematical representation that describes how the quantity demanded of a product is affected by its price (and sometimes the price of related products). It typically shows an inverse relationship between price and quantity demanded, capturing consumer behavior in response to price changes and is fundamental in analyzing market dynamics.

Revenue Function

The revenue function quantifies the total income generated from selling goods or services, calculated as the product of the price and the quantity sold. It is a crucial component in business analysis as it helps determine how changes in price or quantity impact the overall sales revenue.

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