Question

Jerry derives utility from consuming pineapples (X1) and chocolates (X2). His utility function is given by U(X1, X2) = 10X1X2. Each pineapple costs $5 whereas each bar of chocolate costs $3. His allowance is $60. Suppose he is considering buying 5 pineapples and 5 chocolates.

Which of the following combinations of pineapples and chocolates will Jerry consider better than the combination of pineapples and chocolates (5, 5) he is currently considering?

At the combination that he is considering to buy (i.e., 5,5), how much additional utility would he get from increasing the number of pineapples he consumes by one (keeping the number of chocolates the same)?

At the combination that he is considering to buy, how much additional utility would he get from increasing the number of chocolates he consumes by one (keeping the number of pineapples the same)?

What is the numerical value of Jerry’s marginal rate of substitution at the combination (5, 5)?

What is the slope of Jerry’s budget line? Given Jerry’s budget constraint and preferences, will Jerry maximize his utility if he buys the combination (5 pineapples, 5 chocolates) when the price of pineapples is $5, the price of chocolates is $3, and his income is $60?

What is Jerry’s optimal combination of pineapples (X1) and chocolates (X2) that will maximize Jerry’s utility or satisfaction level given his income and the prices of the two goods?

Answer 1

U = 10x1x2

BC : 5x1 + 3x2 = 60

1) at (5,5)

U = 10*5*5 = 250

any and all bundles which satisfy 5x1 + 3x2 = 60 and give utility > 250 will be preferred over (5,5)

such as (5,6) ; (6.5) ; (6,6) ; (5,7) ; (7,6) ; (7,5) and so on..

2) If he increases x1 by 1

U1 = 10*6*5 = 300

additonal utility = 300 - 250 = 50

3) if increases x2 by 1

U1 = 10*5* 6= 300

additonal utility = 300 - 250 = 50

4) MRS = MUx1/MUx2 = 10x2/10x1 = x2/x1

at (5,5), MRS = x2/x1 = 5/5 = 1

5) Slope budget line = - 5/3

at optimal, MRS = Slope

but at (5,5) MRS < slope, hence (5,5) is not utility maximizing.

To find utulity max budle :

x2/x1 = 5/3

x2 = 5x1/3

5x1 + 3x2 = 60

5x1 + 3(5x1/3) = 60

x1 = 60/10 = 6

and x2 = 5*6/3 = 10

Optimal bundle is (6,10).