Question

In: Economics

An industry's inverse demand was PD = 20 - 0.1Q and its inverse supply was PS...

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?

Expert Solution

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.

I have determined the values manually if any doubt kindly contact through comments will be obliged to you for your generous support. Your help mean a lot to me, please help. Thank you.

Rahul Sunny answered 8 months ago

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