/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 Find \(p_{1}\) and \(p_{2}\) so ... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 40

Find \(p_{1}\) and \(p_{2}\) so as to maximize the total revenue \(R=x_{1} p_{1}+x_{2} p_{2}\) for a retail outlet that sells two competitive products with the given demand functions. $$ x_{1}=1000-4 p_{1}+2 p_{2}, x_{2}=900+4 p_{1}-3 p_{2} $$

Short Answer

Expert verified
To find the values of \(p_{1}\) and \(p_{2}\) that maximize total revenue, we would solve the partial derivatives with respect to \(p_{1}\) and \(p_{2}\) for zero. The solution to this system of equations will provide the optimal prices \(p_{1}\) and \(p_{2}\).

Step by step solution

01

Substitute the demand functions

Substitute the demand functions \(x_{1}=1000-4 p_{1}+2 p_{2}\) and \(x_{2}=900+4 p_{1}-3 p_{2}\) into the revenue equation to express \(R\) as a function of \(p_{1}\) and \(p_{2}\). This gives us: \(R=(1000-4 p_{1}+2 p_{2})p_{1}+(900+4 p_{1}-3 p_{2})p_{2}\)
02

Simplify the revenue function

Expand the bracket and simplify the equation from the previous step to get a quadratic function of \(p_{1}\) and \(p_{2}\): \(R=1000p_{1} -4p_{1}^{2}+2p_{1}p_{2} +900p_{2}+4p_{1}p_{2} -3p_{2}^{2}\)
03

Calculate the partial derivatives

To find the maximum revenue, we assume the partial derivatives with respect to \(p_{1}\) and \(p_{2}\) to be zero. We differentiate \(R\) with respect to \(p_{1}\) and \(p_{2}\) to obtain two equations: \(\frac{\partial R}{\partial p_{1}} =0\) and \(\frac{\partial R}{\partial p_{2}} =0\) which becomes \(1000-8p_{1}+2p_{2}+4p_{2}=0\) and \(900+4p_{1}-6p_{2}=0\) upon simplification.
04

Solve the system of equations

Solve the system of equations derived from setting the partial derivatives to zero. The solution of this system will give the optimal values of \(p_{1}\) and \(p_{2}\).

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