Utility Function: Questions
PROBLEM 1 Max has the utility function U(x, y) = x(y + 1). The price of x is $2 and the price of y is $1. Income is $10. How much x does Max demand? How much y? If his income doubles and prices stay unchanged, will Max’s demand for both goods double?
To set his MRS equal to the price ratio, Max sets (y+1)/x = 2. His budget constraint is 2x + y = 10. Solve these two equations to find that x=11/4 and y=9/2.
If his income doubles and prices stay unchanged, his demand for both goods does not double. A quick way to see this is to note that if quantities of both goods doubled, the MRS would not stay the same and hence would not equal the price ratio, which has stayed constant. PROBLEM 3
3. With some services, e.g., checking accounts, phone service, or pay TV, a consumer is offered a choice of two or more payment plans. One can either pay a high entry fee and get a low price per unit of service or pay a low entry fee and a high price per unit of service. Suppose you have an income of $100. There are two plans. Plan A has an entry fee of $20 with a price of $2 per unit. Plan B has an entry fee of $40 with a price of $1 per unit for using the service. Let x be expenditure on other goods and y be consumption of the service. (a) Write down the budget equation that you would have after you paid the entry fee for each of the two plans.
For Plan A, after paying the entry fee, income is $80, px = 1 and py = 2.
Then the budget equation for Plan A is x + 2y = 80.
For Plan B, after paying the entry fee, income is $60, px = 1 and py = 1.
Then the budget equation for Plan B is x + y = 60. (b) If your utility function is xy, how much y would you choose in each case?
Assume that we are dealing with rational individuals who will maximize their utility. For Plan A
For Plan B
Utility Function
U(x,y)=xy
U(x,y) = xy
Demand Function x=80-2y X=60-y
Utility Function
U(y)=(80-2y) y=80y-2y^2
U(y) = (60 - 2y) y = 60y – y^2
Derivative (FOC) dU/dy = 80 – 4y = 0
To set his MRS equal to the price ratio, Max sets (y+1)/x = 2. His budget constraint is 2x + y = 10. Solve these two equations to find that x=11/4 and y=9/2.
If his income doubles and prices stay unchanged, his demand for both goods does not double. A quick way to see this is to note that if quantities of both goods doubled, the MRS would not stay the same and hence would not equal the price ratio, which has stayed constant. PROBLEM 3
3. With some services, e.g., checking accounts, phone service, or pay TV, a consumer is offered a choice of two or more payment plans. One can either pay a high entry fee and get a low price per unit of service or pay a low entry fee and a high price per unit of service. Suppose you have an income of $100. There are two plans. Plan A has an entry fee of $20 with a price of $2 per unit. Plan B has an entry fee of $40 with a price of $1 per unit for using the service. Let x be expenditure on other goods and y be consumption of the service. (a) Write down the budget equation that you would have after you paid the entry fee for each of the two plans.
For Plan A, after paying the entry fee, income is $80, px = 1 and py = 2.
Then the budget equation for Plan A is x + 2y = 80.
For Plan B, after paying the entry fee, income is $60, px = 1 and py = 1.
Then the budget equation for Plan B is x + y = 60. (b) If your utility function is xy, how much y would you choose in each case?
Assume that we are dealing with rational individuals who will maximize their utility. For Plan A
For Plan B
Utility Function
U(x,y)=xy
U(x,y) = xy
Demand Function x=80-2y X=60-y
Utility Function
U(y)=(80-2y) y=80y-2y^2
U(y) = (60 - 2y) y = 60y – y^2
Derivative (FOC) dU/dy = 80 – 4y = 0