Answer to Question #227419 in Microeconomics for Ewura-Esi

Question

Suppose the demand and supply functions for milk are given by the equations:

Q_D=200/P

30=-Q_S+5P

1. Calculate the equilibrium price and quantity of milk
2. Suppose the government introduces a per unit tax of $12, calculate the new equilibrium price and quantity
3. Produce a relevant sketch of answer (1) and (2)
4. calculate the share of tax born by the producer and the consumer
5. How much is the tax yield?
6. what is the fraction of tax borne by the consumer and the producer?
7. calculate the arc price elasticity of demand for (1) and (2) and interpret the results. What pricing policy is recommended?

Answer 1

Solution:

1.). At equilibrium: Q_D = Q_S

Q_D = $\frac{200}{P}$

Q_S = -30 + 5P

$\frac{200}{P} = -30 + 5P$

Multiply both sides by P:

200 = -30P + 5P²

Solve for P using the quadratic formula:

P = 10

Equilibrium price = 10

Substitute in the Q_D function to derive quantity:

Q_D = $\frac{200}{P}$

Q_D = $\frac{200}{10} =20$

Equilibrium quantity = 20

2.). Q_D = $\frac{200}{P}$

Q_S = -30 + 5P

The tax of $12 added will only affect the supply curve:

New supply curve after tax:

Q_S = -30 + 5P

Q_S = -30 + 5(P – 12)

Q_S = -30 + 5P – 60

Q_S = -90 + 5P

New equilibrium after tax:

Q_D = Q_S

$\frac{200}{P}$ = -90 + 5P

Multiply both sides by P:

200 = -90P + 5P²

Solve for P using the quadratic formula:

P = 20

New equilibrium price = 20

Substitute in the Q_D function to derive quantity:

Q_D = $\frac{200}{P}$

Q_D = $\frac{200}{20}$ = 10

New equilibrium quantity = 10

3.). A relevant sketch of 1 and 2 is as below:

 

 

4.). The tax borne by the producer and consumer:

Tax borne by the producer = (10 – 8) $\times$ 10 = 2 $\times$ 10 = 20

Tax borne by the consumer = (20 – 10)$\times$ 10 = 10 $\times$ 10 = 100

 

5.). Tax yield = Tax borne by the producer + tax borne by the consumer

Tax yield = 20 + 100 = 120

 

6.). The fraction of the tax borne by the consumer is = $\frac{100}{120} = \frac{5}{6}\; or \; 83.3\%$

The fraction of the tax borne by the producer is = $\frac{20}{120} = \frac{1}{6}\; or \; 16.7\%$

 

7.). The arc price elasticity of demand formula = $=\frac{\%\;change\; in\; quantity\; demanded}{\%\; change\; in\; price}$

=$=\frac{Q_{2} -Q_{1}}{(Q_{2}+Q_{1})/2 } \div \frac{P_{2} -P_{1}}{(P_{2}+P_{1})/2 }$

Q₁ = 20

P₁ = 10

Q₂ = 10

P₂ = 20

$\frac{10 -20}{(10+20)/2 } \div \frac{20 -10}{(20+10)/2 } = \frac{-0.67}{0.67} = 1$

The arc price elasticity of demand = 1, which means that it is unit elastic. This is a situation where a change in price results in an equally proportionate change in quantity demanded.

 

The recommended price policy is to charge high prices and reduce costs. This is because any change in price will result in an equally proportionate change in quantity demanded. Increasing prices will result in an increase in total revenues.