Problem on sellers utility function

Question

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

b ln x^B₁ + x^B₂

with b ≤ 10. The seller has the utility function

s ln x^S₁ + x^S₂

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?

Answer 1

Given: Sellers Utility function and endowment as:

SlnX₁^S + X₂^S (1,10)
Buyers as:  blnX₁^B + X₂^B (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  x₁^B*=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