Sam\'s utility function is U(x1, x2) = 2x1 + x2, where x1 is the number of units

Question

Sam's utility function is U(x1, x2) = 2x1 + x2, where x1 is the
number of units of good 1 he consumes per week and x2 is the number of units of good 2 he
consumes per week. Sam has $200 a week to spend. The price of x1 is $4. Sam currently
doesn't consume good 2, as it is way too expensive. Sam has received an invitation to join a
club devoted to the consumption of good 2. If he joins the club, Sam can get a discount on
the purchase of good 2. If he belonged to the club, he could buy good 2 for $1 a unit. How
much is the most Sam would be willing to pay to join this club?

Explanation / Answer

U = 2x1 + x2

Budget line: I = x. px + y.py

Current budget line is

200 = 4x1 + 0 = 4x1 [Since x2 = 0]

So, 4x1 = 200

x1 = 200/4 = 50

His current utility = 2x1 + 0 = 2 x 50 + 0 = 100

If he joins the club, he can buy x2 at a price of $1.

So, revised budget line will be:

200 = 4x1 + x2

Since x1 & x2 are substitutes, Sam will consumer either x1 or x2 but not both.

If he buys only x1, then x2 = 0.

From budget line, 200 = 4x1 & x1 = 50 as before.

If he consumes only x2, then from budget line.

200 = 0 + x2

x2 = 200 units

Utility = x2 = 200

So his utility will increase if he buys only x2 and no x2.

The maximum amount Sam will pay will be the difference in the utility derived, multiplied by price of good x2:

(200 - 100) x $1 = $100

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