Compensation Variation

### Background This is a scenario in which at time B the consumer can no longer purchase product x and all of his budget is directed only to product y. In this scenario, the blockage of product x usually detracts from the benefit he derived from the budget at time A. In order to keep the consumer at the same level of utility, we must allow him to purchase a sufficient amount of y that will yield him the same level of utility as at time A, which means increasing his budget. ### The question is how much he should be compensated. The compensating change refers to a change in the budget. ### Example 1 **The utility function** is `U(x,y)=sqrt(x)+sqrt(y)` **Market data**: – 50 NIS `I =`
– 5 NIS `P_x =`
– 5 NIS `P_y =` **The composition of the selected basket and its utility at date A** The selected basket at date A satisfies 2 equations: 1. `5*x+5*y=50` (the budget equation).
2. `y/x=(5/5)^(2)=1` (Equality of cost-benefit in the 2 products) `u_(x)/u_(y)=P_(x)/P_(y)=>(1/root(2)(x))/(1/root(2)(y))=sqrt(y)/sqrt(x)=5/5` Solving the 2 equations yields: – `5 = x` units
– `5 = y` units **The utility of the basket** is 4.47 `(sqrt(5)+sqrt(5)=)`. **The composition of the chosen basket and its utility at time B** At time B, the consumer can only purchase 10 units of y with his budget `(50/5=)` and his utility is only 3.16 `(sqrt(0)+sqrt(10)=)`. To reach a utility of 4.47, he must purchase 20 units of y `(sqrt(y)=4.47=>y=20)`. To purchase 20 units of y, he needs a budget of 100 NIS, so he must compensate him with 50 NIS (=50 NIS-100 NIS).