Suppose you are an aide to a U.S. Senator who is concerned about the impact of a recently proposed excise tax on the welfare of her constituents. You explained to the Senator that one way of measuring the impact on her constituents is to determine how the tax change affects the level of consumer surplus enjoyed by the constituents. Based on your arguments, you are given the go-ahead to conduct a formal analysis, and obtain the following estimates of demand and supply:
Qd=500-5P
Qs-2P-60
(a) What are the equilibrium quantity and equilibrium price? Graph your solution.
(b) If a $2 excise tax is levied on this good, what will happen to the equilibrium price and quantity? Show the changes in your graph from part (a).
(c) How much tax revenue does the government earn with the $2 tax?

Answer:

(a) P = 80 and Q= 100

(b) P = 80.57 and Q= 97.15

(c) Tax revenue = 194.3

Explanation:

Qd= 500 - 5P

Qs = 2P - 60

(a)

In equilibrium

[tex]Qd = Qs \\500 - 5P = 2P - 60 \\7P = 560 \\P = 80 \\[/tex]

Putting this value of P back into the Qd or Qs equation

[tex]Qd = 500 - 5p\\Q = 500 - 5 (80) \\Q = 500 - 400 \\Q = 100[/tex]

Thus, equilibrium price is 80 and equilibrium quantity is 100

(b)

When a tax is imposed the supply curve shifts up to the left by the amount of the tax. The new supply curve is given by

[tex]Qs = 2(P-2) - 60 \\Qs = 2p - 4 - 60 \\Qs = 2P - 64[/tex]

The new equilibrium is

[tex]Qd = Qs \\500 - 5P = 2P - 64 \\7P = 564\\P = 80.57 \\[/tex]

Substitute it into Qs or Qd we get

[tex]Q = 500 - 5 (80.57 ) \\Q = 97.15[/tex]

(c)

[tex]Tax revenue = Tax rate * Quantity \\ = 2 * 97.15\\ = 194.3[/tex]

a. At equilibrium the quantity demanded is equal to the quantity that was supplied.

Qd = Qs

500 - 5p = 2p - 60

We collect like terms from here

500+60 = 2p+5p

560 = 7p

p = 560/7

p = 80 dollars.

Therefore the equilibrium price is 80 dollars.

The equilibrium quantity

Qd = 500 - 5p

= 500-5*80

= 500-400

= 100

The equilibrium quantity is 100

b. Qs = 2p+60

2p = Qs + 60

divide through by 2

p = 0.5Qs + 30

P =  0.5Qs + 30 + t

where we have tax = t = 2

=  0.5Qs + 30 + 2

= 0.5Qs + 32

p - 32 = 0.5Qs

divide through by 0.5

Qs = 2p - 64

The demand function is still the same at Qd = 500 - 5P.

At equilibrium: Qd = Qs

2P- 64 = 500-5P

collect like terms

7P = 500+64

7P = 564

divide through by 7

P = 564/7

P = $80.57

When we put this in the demand function

Q = 500-5P

Q = 500-5*80.57

Q = 97.14

This is the equilibrium quantity

500-5*80.57

= 400-402.85

= 97.15 dollars

c. the tax revenue = 2x97.15

= 194.3 dollars

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