{"id":185222,"date":"2025-01-22T09:10:15","date_gmt":"2025-01-22T09:10:15","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=185222"},"modified":"2025-01-22T09:10:18","modified_gmt":"2025-01-22T09:10:18","slug":"bobs-utility-function-over-good-x-and-y-is-uxy10x5y","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/22\/bobs-utility-function-over-good-x-and-y-is-uxy10x5y\/","title":{"rendered":"Bob\u2019s utility function over good X and Y is U(X,Y)=10X+5Y"},"content":{"rendered":"\n

(21 points) Bob\u2019s utility function over good X and Y is U(X,Y)=10X+5Y. His income is 100 and the price of X is 2 and price of Y is 5.<\/p>\n\n\n\n

(a) Calculate the marginal rate of substitution (MRS) between X and Y (6 points).<\/p>\n\n\n\n

(b) How much of X and Y will Bob buy? (6 points)<\/p>\n\n\n\n

(c) Would Bob\u2019s decision change, if Bob\u2019s utility function is now<\/p>\n\n\n\n

5 (28 points, 7 for each) Ann consumes two goods X and Y, his utility function is U(X, Y ) = 2XY2<\/sup> . Suppose the price of X is $10, while the price of Y is $15. Ann\u2019s income is $500.<\/p>\n\n\n\n

(a) Write the expression for indifference curve when Ann gets utility level 40. And along the indifference curve you found, calculate out the numbers of consumption of X when Y=4.<\/p>\n\n\n\n

(b) Write the expression for Ann\u2019s budget constraint, graph the budget constraint and determine its slope.<\/p>\n\n\n\n

(c) Determine the X, Y combination which maximizes Ann\u2019s utility, given her budget constraint. And figure out what\u2019s marginal rate of substitution (MRS) between two goods at that maximization point.<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

Let’s tackle the problems step by step:<\/p>\n\n\n\n

\n\n\n\n

Part 1: Bob’s Utility Problem<\/h3>\n\n\n\n

Utility function:<\/h4>\n\n\n\n

[ U(X, Y) = 10X + 5Y ]
Income: ( I = 100 )
Price of ( X ): ( P_X = 2 )
Price of ( Y ): ( P_Y = 5 )<\/p>\n\n\n\n

\n\n\n\n

(a) Marginal Rate of Substitution (MRS)<\/h4>\n\n\n\n

The MRS is the rate at which Bob is willing to trade ( Y ) for ( X ), calculated as:
[
MRS = \\frac{\\text{MU}_X}{\\text{MU}_Y}
]
Where:
[
\\text{MU}_X = \\frac{\\partial U}{\\partial X} = 10, \\quad \\text{MU}_Y = \\frac{\\partial U}{\\partial Y} = 5
]
Thus:
[
MRS = \\frac{10}{5} = 2
]<\/p>\n\n\n\n

\n\n\n\n

(b) Optimal Consumption of ( X ) and ( Y )<\/h4>\n\n\n\n

Bob maximizes utility by allocating income to maximize:
[
\\frac{\\text{MU}_X}{P_X} = \\frac{\\text{MU}_Y}{P_Y}
]
[
\\frac{10}{2} = \\frac{5}{5} \\implies 5 = 1
]
This condition doesn\u2019t hold here. Thus, Bob will spend his income on the good with the highest marginal utility per dollar.
Since:
[
\\frac{10}{2} > \\frac{5}{5}
]
Bob will spend all his income on ( X ):
[
\\frac{100}{2} = 50
]
So:
[
X = 50, \\quad Y = 0
]<\/p>\n\n\n\n

\n\n\n\n

Part 2: Ann’s Utility Problem<\/h3>\n\n\n\n

Utility function:<\/h4>\n\n\n\n

[ U(X, Y) = 2XY^2 ]
Price of ( X ): ( P_X = 10 )
Price of ( Y ): ( P_Y = 15 )
Income: ( I = 500 )<\/p>\n\n\n\n

\n\n\n\n

(a) Indifference Curve at ( U = 40 )<\/h4>\n\n\n\n

The indifference curve is derived from:
[
2XY^2 = 40 \\implies XY^2 = 20 \\implies X = \\frac{20}{Y^2}
]
If ( Y = 4 ):
[
X = \\frac{20}{4^2} = \\frac{20}{16} = 1.25
]
Thus, when ( Y = 4 ), ( X = 1.25 ).<\/p>\n\n\n\n

\n\n\n\n

(b) Budget Constraint<\/h4>\n\n\n\n

The budget constraint is:
[
P_X X + P_Y Y = I \\implies 10X + 15Y = 500
]
Simplify to slope-intercept form:
[
Y = \\frac{500 – 10X}{15}
]
The slope is:
[
-\\frac{P_X}{P_Y} = -\\frac{10}{15} = -\\frac{2}{3}
]<\/p>\n\n\n\n

Graph: The budget line intercepts are:<\/p>\n\n\n\n