Your company manufactures two models of speakers, the Ultra Mini and the Big Stack. Demand for...

Question:

Your company manufactures two models of speakers, the Ultra Mini and the Big Stack. Demand for each partly depends on the price of the other. If one is expensive, then more people will buy the other. If {eq}p_1 {/eq} is the price of he Ultra Mini, and {eq}p_2 {/eq} is the price of the Big Stack, demand for the Ultra mini is given by:

{eq}q_1 (p_1,p_2) = 100,000-700p_1+10p_2, {/eq}

where {eq}q_1 {/eq} represents the number of Ultra Minis that will be sold in a year. The demand for the Big Stack is given by

{eq}q_2(p_1,p_2)= 150,000+10p_1-700p_2. {/eq}

Find the prices for the Ultra Mini and the Big Stack that will maximize your total revenue.

Optimal Price:

{eq}\\ {/eq}

The demand function of two products can create an interdependency on the price of each other. In such cases, the revenue function is a multivariate {eq}f(x,y) {/eq} and at the critical point (a,b) of the function (maximum or minimum), the condition satisfied is:

{eq}f_x(a,b) = f_y(a,b) = 0 {/eq}

Answer and Explanation:

{eq}\\ {/eq}

Demand for ultra mini with a price {eq}p_1 {/eq} is given by: {eq}q_1(p_1, p_2) = 100,000-700p_1+10p_2 {/eq}

Demand for big stack with a price {eq}p_2 {/eq} is given by: {eq}q_2(p_1, p_2) = 150,000+10p_1-700p_2. {/eq}

The revenue generated by each model is:

{eq}\begin{align*} R_1 = q_1 \times p_1 &= 100000p_1-700p_1^2+10p_1p_2 && \dots \text{(ultra mini)} \\ R_2 = q_2 \times p_2 &= 150000p_2+10p_1p_2-700p_2^2 && \dots \text{(big stack)} \end{align*} {/eq}

The marginal reveunes for each product is given by:

{eq}\begin{align*} MR_1 &= \frac{\partial R_1}{\partial p_1} = 100000-1400p_1 + 10p_2 \\ MR_2 &= \frac{\partial R_2}{\partial p_2} = 150000-1400p_2 + 10p_1 \end{align*} {/eq}

At a critical point {eq}(p_1, p_2) {/eq}, where the revenues are maximum, we get:

{eq}\begin{align*} MR_1 = MR_2 = 0 \\ \implies 100000-1400p_1 + 10p_2 &= 0 && (1)\\ 150000-1400p_2 + 10p_1 &= 0 &&(2) \end{align*} {/eq}

(1) and (2) form a linear system of equations which can be easily solved to get: {eq}p_1 = 1415000/19599 \ \ \& \ \ p_2 = 2110000/19599 {/eq}

Thus, the optimum price point of the two speakers is: {eq}p_1 = $72.198 {/eq} and {eq}p_2 = $107.66 {/eq}


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Finding Critical Points in Calculus: Function & Graph

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