Science:Math Exam Resources/Courses/MATH105/April 2011/Question 04

MATH105 April 2011

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Question 04
The unit price of a certain commodity can fluctuate anywhere between $2 and $6.50. If the unit price is $\$p$, the supplier produces $(p-2)$ units of it, while the consumers demand $65p^{-1}-10$ units. Find the market equilibrium and the consumer surplus.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

The market equilibrium is defined to be the intersection point of the graphs of the demand function and the supply functions. If $D(q)$ denotes the demand function as a function in the quantity $q$ and if the market equilibrium is given by the point $(q_{e},p_{e})$, then the consumer surplus is given by the formula $\int _{0}^{q_{e}}\left(D(q)-p_{e}\right)\,dq$ See the wikipedia page on Economic surplus for more information about the market equilibrium and the consumer surplus.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let us write the supply and demand function as functions in the price $p$. The text says that at a price $p$ the supplier produces $p-2$ units. Thus, the supply function must be $\displaystyle p=S(q)=q+2$ At a price $p$ the consumer demands $65/p-10$ units. In terms of the demand function this translates into $p=D(q)={\frac {65}{q+10}}$ By definition, the market equilibrium is the intersection point of the graph of $S(q)$ and $D(q)$. Let us denote this point by $(q_{e},p_{e})$. To compute that point, we set both functions equal and compute a solution for $q$: ${\begin{aligned}S(q)&=D(q)\\q+2&={\frac {65}{q+10}}\end{aligned}}$ And rearranging we obtain the equation ${\begin{aligned}(q+2)(q+10)&=65\\q^{2}+12q-45&=0\end{aligned}}$ This is a quadratic equation and can be solved with the help of the $abc$-formula: the solutions to a quadratic equation of the form $ax^{2}+bx+c=0$ are given by $x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$ Note, that in our problem the variable $x$ is replaced by the variable $q$ and we have $a=1$, $b=12$ and $x=-45$. Hence, $q={\frac {-12\pm {\sqrt {144+180}}}{2}}=-6\pm 9$ The solution $-6-9$ is not feasible, because quantities are always nonnegative. Hence, we arrive at the unique solution $q_{e}=-6+9=3$. We can calculate the associated price by using $D(q)$ or $S(q)$. Since $S(q)$ looks simpler than $D(q)$ we use $S(q)$ and get $p=S(q_{e})=S(3)=3+2=5$. Note, that this price is in between the range given in the problem. Let us summarize: The market equilibrium is attained at $(q_{e},p_{e})=(3,5)$ The second part of the question asks us to compute the consumer surplus. By definition, the consumer surplus is given by the formula ${\text{consumer surplus}}=\int _{0}^{q_{e}}(D(q)-p_{e})\,dq$ The following steps compute this integral: ${\begin{aligned}\int _{0}^{q_{e}}(D(q)-p_{e})\,dq&=\int _{0}^{3}\left({\frac {65}{q+10}}-5\right)\,dq\\&=65\int _{0}^{3}{\frac {1}{q+10}}\,dq-\int _{0}^{3}5\,dq\\&=65\ln(q+10){\Bigr |}_{0}^{3}-5q{\Bigr |}_{0}^{3}\\&=65(\ln(13)-\ln(10))-15\\&=65\ln(13/10)-15\\&\approx 2.05\end{aligned}}$ The consumer surplus is approximately $2.05

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