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 $2,600. With this pricing structure, the demand, measured by the number of contracts Montevideo signs per month, is 24 contracts. This is down 2 contracts from the figure last year, when your company charged only $2,400.
(a) Construct a linear demand equation giving the number of contracts q as a function of the hourly fee p Montevideo charges for development.
q(p)=
(b) On average, Montevideo bills for 40 hours of production time on each contract. Give a formula for the total revenue obtained by charging $p per hour.
R(p)=
(c) The costs to Montevideo Productions are estimated as follows:
Fixed costs: $130,000 per month
Variable costs: $70,000 per contract
Express Montevideo Productions' monthly cost as a function of the number q of contracts.
C(q)=
Express Montevideo Productions' monthly cost as a function of the hourly production charge p.
C(p)=
(d) Express Montevideo Productions' monthly profit as a function of the hourly development fee p.
P(p)
Find the price it should charge to maximize the profit.
p = $___per hour
How should I solve this problem?
Thank you.
To solve this problem, you can use some basic principles of economics and linear equations.
(a) To construct a linear demand equation, we need to find the relationship between the number of contracts (q) and the hourly fee (p) charged by Montevideo. Since you have the information that at a fee of $2,600, the demand is 24 contracts, we can use this as a data point to create the equation.
We use the point-slope form of linear equation: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Given point: (p=2600, q=24)
To find the slope:
Last year's fee: p1 = $2,400
Last year's demand: q1 = 26 contracts
Slope (m) = (q2 - q1) / (p2 - p1) = (24 - 26) / (2600 - 2400) = -2 / 200 = -1/100
So the linear demand equation q(p) is:
q(p) = q1 + m(p - p1)
q(p) = 26 - (1/100)(p - 2400)
(b) To find the formula for total revenue obtained by charging $p per hour, we multiply the number of contracts (q) by the total hours billed per contract (40 hours). Therefore, R(p) can be represented as:
R(p) = q(p) * 40
Substituting the value of q(p) from part (a):
R(p) = (26 - (1/100)(p - 2400)) * 40
(c) To express Montevideo's monthly cost as a function of the number of contracts (q), we consider fixed costs and variable costs. Fixed costs are $130,000 per month, and variable costs are $70,000 per contract.
So, C(q) = Fixed costs + Variable costs per contract * q
C(q) = $130,000 + $70,000 * q
To express Montevideo's monthly cost as a function of the hourly production charge (p), we need to consider the number of contracts (q) and total hours worked per contract (40 hours).
C(p) = C(q) * Number of hours per contract
C(p) = ($130,000 + $70,000 * q) * 40
(d) To express Montevideo's monthly profit as a function of the hourly development fee (p), we subtract the monthly costs (C(p)) from the monthly revenue (R(p)):
P(p) = R(p) - C(p)
To find the price Montevideo should charge to maximize profit, you can use calculus techniques, such as finding the critical points or maximum, by differentiating P(p) with respect to p and setting it equal to zero. Then solve for p.