Question

Intermediate microeconomic theory courses, with topics production, cost, and supply

The book used in this course is Intermediate Microeconomics and Its Application, by walter nicholson

The question is :

1. It is known that the total revenue function is ?(?) = 45,75? − ?²
short-run cost function ??? (?) = ?³ - 12? ² + 60? + 40

A. Prove that: Income rises as price increases when the price elasticity of demand is inelastic.
Income rises if price falls when the price elasticity of demand is elastic. How about profit?

B. Calculate consumer surplus and producer surplus in the above problem? Note that the price of equilibrium goods when Qd = Qs

Answer 1

A.

When the elasticity of demand is inelastic, the producer enjoys a monopoly in the market and he can increase the price as he wishes and the consumer is bound to purchase the product from him as he enjoys the sole power in the market and the consumer has no other option and hence the income of the producer rises. Hence, with an increase in price the income of the producer rises when demand is inelastic. It is propagated theory of monopoly markets where the producers increase prices to raise their own margin and income.

When the elasticity of demand is elastic, the producer does not raise the price as he knows the responsiveness of demand is huge and the fall in demand will be proportionately greater than the rise in price and that will eventually lead to a decline of revenue and income and hence he prefers to reduce the price to increase the demand by a proportionately greater amount and that will eventually lead to a rise in revenue and profits. Hence, it is proved that when the demand is elastic the producer reduces prices to raise their own margin and income. It is a propagated theory of free markets and monopolistic competition that the producers prefer to set lower prices to earn more income.

B.

For calculating the consumer surplus we first need to solve the optimisation problem for optimising our profits (maximising our profits) and find the equilibrium quantity for the producer:-

The optimisation problem is as under:-

The aforesaid equation can be solved using the Sridhar Acharya's Method:-

q = (-b ^{+_{- (}}b² - 4ac)) / 2a

where a= 3; b= -22 and c= -4515.

Hence, q= (22 ⁺_-(-22)² - (4 * 3 * -4515)) / 2 * 3

= 35.301 or -42.634

Hence, equilibrium quantity cannot be a negative amount so it is 35.301.

The demand curve is the AR curve which is calculated as R(q)/ q = 4575-q.

Accordingly the equilibrium price is calculated as 4539.699.

The Consumer Surplus can be calculated as under:-

Area (P*MN) is the consumer surplus.

Area (P*MN) = 1/2 * base * height

= 1/2 *35.301 * (4575 - 4539.699) [ since the intercrept is 4575 so the length OM = 4575]

=623.080.

Consumer Surplus = 623.80

For understanding the producer surplus we need to delve into the concept of deriving the supply curve. The rising portion of the marginal cost curve is the supply curve so we simply differentiate the cost function to acquire the marginal cost curve as under. The supply curve has the same equation as the MC curve as it is a part of the latter.

In the aforesaid diagram, the shaded area is the producer surplus.

Hence, the producer surplus is 1,03,429.33209.