As sales manager for Montevideo Productions, Inc., you are...

As sales manager for Montevideo Productions, Inc., you are planning to review the prices you charge clients for television advertisement development. You currently charge each client an hourly development fee of $$\$ 2,500$$. With this pricing structure, the demand, measured by the number of contracts Montevideo signs per month, is 15 contracts. This is down 5 contracts from the figure last year, when your company charged only $$\$ 2,000$$. a. Construct a linear demand equation giving the number of contracts $q$ as a function of the hourly fee $p$ Montevideo charges for development. b. On average, Montevideo bills for 50 hours of production time on each contract. Give a formula for the total revenue obtained by charging $$\$ p$$ per hour. c. The costs to Montevideo Productions are estimated as follows: $$ \begin{array}{ll} \text { Fixed costs: } & \$ 120,000 \text { per month } \\ \text { Variable costs: } & \$ 80,000 \text { per contract } \end{array} $$ Express Montevideo Productions' monthly cost (i) as a function of the number $q$ of contracts and (ii) as a function of the hourly production charge $p$. d. Express Montevideo Productions' monthly profit as a function of the hourly development fee $p$ and hence the price it should charge to maximize the profit.

As sales manager for Montevideo Productions, Inc., you are planning to review the prices you charge clients for television advertisement development. You currently charge each client an hourly development fee of $$\$ 2,500$$. With this pricing structure, the demand, measured by the number of contracts Montevideo signs per month, is 15 contracts. This is down 5 contracts from the figure last year, when your company charged only $$\$ 2,000$$.
a. Construct a linear demand equation giving the number of contracts $q$ as a function of the hourly fee $p$ Montevideo charges for development.
b. On average, Montevideo bills for 50 hours of production time on each contract. Give a formula for the total revenue obtained by charging $$\$ p$$ per hour.
c. The costs to Montevideo Productions are estimated as follows:
$$
\begin{array}{ll}
\text { Fixed costs: } & \$ 120,000 \text { per month } \\
\text { Variable costs: } & \$ 80,000 \text { per contract }
\end{array}
$$
Express Montevideo Productions' monthly cost (i) as a function of the number $q$ of contracts and (ii) as a function of the hourly production charge $p$.
d. Express Montevideo Productions' monthly profit as a function of the hourly development fee $p$ and hence the price it should charge to maximize the profit.

Key Concepts

Linear Demand Function

This concept involves modeling the relationship between price and quantity demanded with a straight line. In many economic scenarios, including advertising or service industries, a linear demand function can be expressed as q = a + b·p, where q represents the number of contracts or units, p is the price, and the slope (b) is typically negative. This function helps estimate changes in demand as prices are adjusted.

Revenue Function

The revenue function represents the total income generated from sales, typically calculated by multiplying the unit price by the quantity sold. When additional factors like hours per contract are involved, the revenue can be modeled as the product of the hourly price, the number of hours billed per contract, and the number of contracts obtained. This formulation is fundamental for understanding how pricing decisions impact overall income.

Cost Function

Cost functions in economic models consist of fixed and variable costs. Fixed costs are expenses that remain constant regardless of the number of units sold, while variable costs change in direct proportion to production or sales volume. Expressing the total cost as a function of quantity or price is crucial for evaluating the profitability of a business model and for making informed pricing decisions.

Profit Function

The profit function is derived by subtracting total costs from total revenue. It encapsulates both revenue and cost structures into a single expression, making it possible to analyze the financial return of different pricing strategies. This function is vital for assessing the performance of a business and identifying the break-even point and levels of profitability.

Profit Maximization

Profit maximization involves determining the price and quantity that yield the highest possible profit. In practical scenarios, this is typically achieved by setting up the profit function and finding its maximum value using optimization techniques, such as taking derivatives and setting them equal to zero. This analysis guides decisions on pricing to balance revenue against cost considerations effectively.

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