# Consumers are willing to pay p(x)=100+x+.001x^2 dollars per item when the quantity is x, with...

## Question:

Consumers are willing to pay p(*x*) = 100 - *x* + .001*x*{eq}^2 {/eq} dollars per item when the quantity is *x*, where 0 < *x* < 112.

Assume the market price is $10/item.

A. What is the quantity sold?

B. Write an integral representing the consumer surplus.

C. Evaluate the integral from B.

## Consumer Surplus:

In economics, the consumer surplus is a measure of the consumer welfare gained from the consumption of goods. The consumer surplus from a unit of a consumption good is the difference between the marginal willingness to pay for the good and the actual price paid.

## Answer and Explanation:

**A.** The quantity sold is 100.

The consumer will buy until the marginal willingness to pay is the same as the market price.

For example,

{eq}p(x) = 100 - x + 0.001x^2 = 10 {/eq},

which yields

{eq}x = 100{/eq}.

**B.** The consumer surplus is the sum of the consumer surplus generated from each unit sold.

Expressed as an integral, the consumer surplus is:

{eq}\displaystyle \int_{0}^{100}{(p(x) - 10)dx} {/eq}.

**C.** The total consumer surplus is $4,333.33, calculated as:

{eq}\displaystyle \int_{0}^{100}{(100 - x + 0.001x^2 - 10)dx} \\ = \displaystyle \int_{0}^{100}{(90 - x + 0.001x^2)dx}\\ = \displaystyle (90x - \frac{x^2}{2} + \frac{0.001x^3}{3}) \bigg\vert_0^{100}\\ = \displaystyle 90*100 - \frac{100^2}{2} + \frac{0.001*100^3}{3}\\ = 4333.33 {/eq}

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