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

Above case is the case of perfect substitutes.In these kind of utility functions consumers consume only 1 particular commodity depending on the fact whether their MRS is greater or less than price ratio. In the given question clearly it is clearly stated that consumer sam initially purchases only x1 as x2 is very costly.Clearly in this situation MRS=2 > Price ratio.However when sam takes the membership he can buy good2 for only $1 per unit.Clearly in that situation Sam would buy only good2 as then MRS would become < price ratio. 2<4/1.

MRS of 2 implies that Sam is ready to give up 2 units of good2 for 1 unit of good1.

Thus sacrificing 50 units of good1 is equivalent to sacrificing 100 units of good2.

Hence by taking the membership Sam is gaining an amount equal to (200 - 100) * $1 =$100.Thus $100 is the maximum Sam is willing to pay for the membership of the club.

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