## Cosumers' Surplus (CS) and Producers' Surplus(PS)

### Provided formulae: CS \(= \displaystyle{ \int_0^{\bar{x}} D(x)\ \mathrm{d}x -
\bar{p}\bar{x}} \),
PS \(= \displaystyle{\bar{p} \bar{x} - \int_0^{\bar{x}}
S(x)\ \mathrm{d}x }\).

### 1. A Company has determined that the quantity demanded \(x\) of their new product per week is given by the demand function \(p=-0.2x^{2}+80\) where \(p\) is the unit price in dollars, and \(x\) is the quantity demanded in units of a hundred. The company is willing to make their product available in the market by the supply function given by \(p=0.1x^{2}+x+40\). If the equilibrium point is \((10,
60)\), find (a) the consumers' surplus, and (b) the producers' surplus. solution

### 2. The demand function for widgets is given by
\(\displaystyle{p=200-x^2},\) where \(p\) is the unit price in US dollars
and \(x\) is the quantity of widgets demanded each week. If the supply
function is \( p=x^2+38,\) find (a) the consumers' surplus, and
(b) the producers' surplus. Given that \(\bar{x} = 9. \)
solution

### 3. The management of the Titan Tire Company has determined that
the quantity demanded x of their Super Titan tires per week is related to the
unit price p by the relation \(\displaystyle{p=144-x^2}\) where p is measured in
dollars and x is measured in units of a thousand. Titan will make x units of the
tires available in the market if the unit price is \(p=48+.5x^2\) dollars. Determine the consumers' surplus and the producers'
surplus when the market unit price is set at the equilibrium price. solution

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