A market has demand function Q^D = 100 - P + 2 I, where I is income. Supply is Q

Question

A market has demand function Q^D = 100 - P + 2 I, where I is income. Supply is Q^S = P. Solve for the equilibrium price P* and quantity Q*. Find CS and PS. Now assume that a per unit tax t is imposed on the producers, so that P^D = P^S + t. Solve for the equilibrium prices and quantity as a function of t. Find P^D/ t and P^S/ t. Compute the new CS, PS, and DWL. Find the tax revenue. Graph the equilibrium after the tax showing Consumer Surplus, Producer Surplus, tax revenue and the Deadweight Loss.

Explanation / Answer

a) equilibrium price and quantity can be acheived by equating demand and supply functions:

100-P+2I = P

2P = 100+2I

P = 50+I

The above is equilibrium price P*. Lets put the above value into demand function

Q = 100-50-I+2I = 50+I

the above is equilibrium quantity Q*.

b) Consumer surplus = 1/2 * Q* * (maximum price that cosnumers are willing to pay - P*)

= 1/2 * (50+I) (100+2I - 50 - I) = 1/2(50+I)(50+I) = 1/2(2500+50I+10I+I2) = 0.5I2 + 30I + 1250

Producers surplus = 1/2 * Q* ( P*-minimum amount producer is willing to receive)

=1/2 * (50+I) (50+I - 0) = 1/2(2500+50I+50I+I2) = 0.5I2+100I+1250

c) When tax is imposed on producers, supply function will change as:

Qs = P+t

now again equating demand and supply:

100-P+2I = P+t

P = 50+I-0.5t

the above is new price P*

and Q = 100-(50+I-0.5t)+2I = 50+0.5t+I

the above is new quantity.

d) New CS = 1/2 * 50+0.5t+I * (100+2I - 50-I+0.5t) = 1/2(50+0.5t+I)(50+I+0.5t)

New PS = 1/2 * 50+0.5t+I * (50+I-0.5t - t) = 1/2 * (50+0.5t+I) (50+I-1.5t)

Tax revenue = Q* * T = (50+0.5t+I) * t

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