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Problem on sellers utility function

The economy consists of a single buyer and a single seller. The buyer has the utility function

b ln xB1 + xB2

with b ≤ 10. The seller has the utility function

s ln xS1 + xS2

The buyer is endowed with 0 units of good 1 and 10 units of good 2. The seller is endowed with 1 unit of good 1 and 10 units of good 2.

(i) Find the demand and supply, the inverse demand and the inverse supply functions.
(ii) Find the competitive equilibrium.
(iii) The government imposes a tax on good 1. The tax is t units of good 2 for every 1 unit of good 1 transacted. Find the after-tax competitive equilibrium. What is the effect of this tax on the price paid by the buyer?

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Given: Sellers Utility function and endowment as:

SlnX1S + X2S (1,10)
Buyers as:  blnX1B + X2B (0,10)

Budget constraint for buyer will be: p1x1 + p2x2= p1(0) +p2(10)
Plus at the optimal MRS= Price Ratio

b/x1= p1/p2

Put P2=1 (numeraire)

So  x1B*=B/p1 This is the inverse demand curve

Similarily we do it for seller and we get

s/x1= p1/p2

or  x1*s = s/p1.This is the Inverse  supply curve

b) For competitive equilibrium We know that  total supply In the economy for X1 is 1 that should be equal to demand implies x1*B= B/p1=1 and  p1*= B and P2*= 1

c) There is Only one good 1 in the economy so  there is a tax t for good two and price will be b+t for buyers now and p2=1

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