# The producer of a certain commodity determines that to protect profits, the price p should...

## Question:

The producer of a certain commodity determines that to protect profits, the price {eq}p {/eq} should decrease at a rate equal to half the inventory surplus {eq}S-D {/eq}, where {eq}S {/eq} and {eq}D {/eq} are respectively the supply and demand for the commodity. Suppose the supply and demand vary with price in such a way that {eq}S(p)=80+3p {/eq} and {eq}D(p)=150- 2p, {/eq} and that the price is 3 dollars per unit when {eq}t=0 {/eq}. Determine {eq}p(t) {/eq}.

## First Order Differential Equations:

This question involves forming a first order differential equation and then solving it by using the given constraints, a topic related to the field of quantitative economics. In order to write the differential equation, we need information about both the supply and demand, as well as the relationship between the rate of change of the price, and supply and demand.

## Answer and Explanation:

We are given the fact that the price {eq}p {/eq} should decrease at a rate equal to half the inventory surplus {eq}S-D {/eq}, and {eq}S(p) = 80 +...

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View this answerWe are given the fact that the price {eq}p {/eq} should decrease at a rate equal to half the inventory surplus {eq}S-D {/eq}, and {eq}S(p) = 80 + 3p {/eq} an d{eq}D(p) = 150 - 2p. {/eq} Therefore

{eq}\begin{eqnarray*}p'(t) & = & \displaystyle\frac12 (S(p) - D(p)) \\ & =& \displaystyle\frac12 ((80 + 3p) - (150 - 2p)) \\ & = & \displaystyle\frac52 p - 35 \end{eqnarray*} {/eq}

This is a separable differential equation:

{eq}\begin{eqnarray*}2 \: \displaystyle\frac{dp}{dt} & = & 5p - 70 \\ \displaystyle\frac{2 \: dp}{5p - 70} & = & dt \\ \displaystyle\int \frac{2 \: dp}{5p - 70} & =& \int dt \\ \displaystyle\frac25 \ln |5p - 70| & = & t + C \\ \ln |5p - 70| & = & \displaystyle\frac{5t}2 + C \\ 5p - 70 & = & C_1 e^{5t/2} \\ p(t) & = & \displaystyle\frac15 (C_1 e^{5t/2} + 70) \end{eqnarray*} {/eq}

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from Business 102: Principles of Marketing

Chapter 11 / Lesson 6