Question

The inverse demand function of specific commodity is given by ??(??) = ?20?? + 1400, where ?? and ?? are the price and quantity of the commodity, respectively.

a- Determine the maximum consumption.

b- If the price is $500 per unit, calculate consumption, consumers’ gross surplus, consumers’ net surplus, and the revenue collected by the producers.

c- If the price increases 40%, calculate change in demand and lost surplus for the consumers.

d- What is the consumers’ elasticity? Is it elastic or inelastic?

e- If the price rises by 3%, the quantity demanded falls by 1.5%.

f- If the elasticity of demand for a commodity were estimated to be 1.5, then a decrease in price from $2.0 to $1.90 would be expected to increase daily sales by how much?

Answer 1

The inverse demand is given as .

(a) The maximum consumption will be where Q is large enough to be just feasible. For P=0, we have or . Hence, at zero price, quantity consumed will be 70. The consumption can not be more than this quantity, as negative price is not feasible.

(b) At P = $500, we have the demand as or .

Area of region ABOFEA can be divided into area of triangle ABE and area of rectangle BOFE. Area of triangle ABE will be or or dollars. Area of rectangle BOFE will be or or dollars.

The gross consumer surplus will be the area of ABOFEA, which is $20250 plus $22500 or $42750. The net consumer surplus, which is gross CS minus the total expenditure on those 45 units will be the area of triangle ABE, which is $20250. The revenue collected by producers will be price times quantity, ie the area of rectangle BOFE, which is $22500.

(c) If price increases by 40%, then price will be dollars. For $700, the demand would be or . Change in demand is (-)10 units.

Current net surplus will be or dollars. The current expenditure or producer's revenue is or dollars. The current gross revenue is hence $36750. Hence, decrease in gross CS is $6000, and decrease in net CS is $8000.

(d) The price elasticity is the ratio of percentage change in quantity and percentage change in price. The price elasticity of demand is or . The inverse demand was , and hence the demand function will be . The ealsticity is hence or or .

Checking for both points, we have at first ($500), , and at later on (P=$700), . Hence, the demand curve is said to be inelastic above price equal $700, while elastic below $700, and unitary elastic at $700.

(e) If price rises by 3% and quantity of demand falls by 1.5%, the elasticity would be . Equating it to the founded elasticity formula , we have or or . Putting in the demand function , we have or or , and hence . Hence, the incresae in P of 3% and decrease in Q of 1.5% occurs at where the price is $466.66 and quantity is 46.66.

(f) If the elasticity of demand for a commodity be , ie . A decrease in price from $2.0 to $1.90 would mean that the price decreased by or or , ie 5%. Hence, as , hence or , ie demand would decrease by 7.5 percent.