Question

how do you solve without lagrangians?

12 Pablo's utility function is U(x,y)= x+10y- y2/2, where x is the number of x's he consumes per week and y is the number of y's he consumes per week. Pablo has $200 a week to spend. The price of x is $1. The price ofy is currently $5 per unit. Pablo has received an invitation to join a club devoted to the consumption of y. If he joins the club, Pablo can get a discount on the purchase of y. If he belonged to the club, he could buy y for $1 a unit. How much is the most Pablo would be willing to pay to join this club? a. b. $8 $28 $36 $56 None of the above.

Answer 1

Initial MRS=-MU/MUy= -1/(10-y)

Slope of Budget constraint= -Px/Py= -1/5

optimal bundle:

-1/5=-1/(10-y)

10-y=5

y*= 5

Budget constraint: x+5y=200

x=200-5*5=175

x*=175

U= 175+10*5-5^2/2=175+50-12.5= 212.5

When Py=1

slope of budget constraint= -1/1=-1

-1=-1/(10-y)

10-y=1

y*= 9

x+y=200

x*=200-9= 192

U= 191+10*9-9^2/2= 240.5

The maximum price Pablo will be willing to pay= 240.5-212.5=$28

So, Answer=$28