Science:Math Exam Resources/Courses/MATH105/April 2011/Question 04
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The unit price of a certain commodity can fluctuate anywhere between $2 and $6.50. If the unit price is , the supplier produces units of it, while the consumers demand units. Find the market equilibrium and the consumer surplus.
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The market equilibrium is defined to be the intersection point of the graphs of the demand function and the supply functions.
If denotes the demand function as a function in the quantity and if the market equilibrium is given by the point , then the consumer surplus is given by the formula
See the wikipedia page on Economic surplus for more information about the market equilibrium and the consumer surplus.
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Let us write the supply and demand function as functions in the price . The text says that at a price the supplier produces units. Thus, the supply function must be
At a price the consumer demands units. In terms of the demand function this translates into
By definition, the market equilibrium is the intersection point of the graph of and . Let us denote this point by . To compute that point, we set both functions equal and compute a solution for :
And rearranging we obtain the equation
This is a quadratic equation and can be solved with the help of the -formula: the solutions to a quadratic equation of the form are given by
Note, that in our problem the variable is replaced by the variable and we have , and . Hence,
The solution is not feasible, because quantities are always nonnegative. Hence, we arrive at the unique solution . We can calculate the associated price by using or . Since looks simpler than we use and get . Note, that this price is in between the range given in the problem. Let us summarize:
The market equilibrium is attained at
The second part of the question asks us to compute the consumer surplus. By definition, the consumer surplus is given by the formula
The following steps compute this integral:
The consumer surplus is approximately $2.05