In: Economics
An industry's inverse demand was PD = 20 - 0.1Q and its inverse supply was PS = 4 + 0.1Q.
a. Calculate the consumer surplus, producer surplus, government revenue and deadweight loss for taxes of $4, $8, $12 and $16 per unit sold.
b. Graph government revenue and deadweight loss as functions of these tax rates.
c. What tax maximizes government revenue?
Inverse demand function, P = 20 - 0.1Q
Inverse supply function, P = 4 + 0.01Q
At equilibrium supply and demand are equal.
20 - 0.1Q = 4 + 0.1Q
=> 0.1Q + 0.1Q = 20 - 4
=> 0.2Q = 16
=> Q = 80 units
P = 20 - 0.1× 80= $ 12 per unit
When tax = 0
Consumer surplus =(1/2)*(20-12)*80 = $ 320
Producer surplus = (1/2)*(12-4)*80 = $ 320
Government revenue = 0
Dead weight loss = 0
When government imposes a tax of T then the price paid by buyer will be greater than price received by sellers. Let P be the price paid by buyer and Ps price received by seller. Then, Ps = P -T.
When a tax of $ 4 is imposed
Supply curve , P - 4 = 4 + 0.1Q
P = 8 + 0.1Q
Equate it to demand curve
P = 20 - 0.1×60 = $ 14
CS = (1/2)*(20-14)*60 = $ 180
PS =(1/2)*(10-4)*60 = $ 180
Government revenue = 4 × 60= $ 240
Dead weight loss = (1/2)*(4)*(80-60) = $ 40
When tax = $ 8
New supply curve P = 12 + 0.1Q
Price paid by buyer, P = 20 - 0.1×40 = $ 16 / unit
CS=0.5×(20-16)×40 = $ 80
PS = 0.5×(8-4)×40= $ 80
Government revenue =8×40 =$ 320
Dead weight loss =0.5×8×(80 - 40) =$ 160
Now when tax = $ 12
New supply curve P = 16 + 0.1Q
Price paid by buyers, P = 20 - 0.1×20 = $ 18
Price received by sellers = 18 - 12 =.$ 6
CS = 0.5×(20-18)×20 = $ 20
PS=0.5 ×(6 -4) × 20 = $ 20
Government revenue = 12 × 20 = $ 240
Dead weight loss= 0.5× 12 × (80-20) = $ 360
Now when tax is $ 16
Q = 0
CS = 0
PS = 0
Government revenue = 0
Deadweight loss = 0.5×16 × 80 = $ 640
Now refer the graph for deadweight loss
C. Refer the graph for government revenue the tax is maximized at a tax of $ 8 per unit.
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