(21 points) Bob’s 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.
(a) Calculate the marginal rate of substitution (MRS) between X and Y (6 points).
(b) How much of X and Y will Bob buy? (6 points)
(c) Would Bob’s decision change, if Bob’s utility function is now
5 (28 points, 7 for each) Ann consumes two goods X and Y, his utility function is U(X, Y ) = 2XY2 . Suppose the price of X is $10, while the price of Y is $15. Ann’s income is $500.
(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.
(b) Write the expression for Ann’s budget constraint, graph the budget constraint and determine its slope.
(c) Determine the X, Y combination which maximizes Ann’s utility, given her budget constraint. And figure out what’s marginal rate of substitution (MRS) between two goods at that maximization point.
The Correct Answer and Explanation is :
Let’s tackle the problems step by step:
Part 1: Bob’s Utility Problem
Utility function:
[ U(X, Y) = 10X + 5Y ]
Income: ( I = 100 )
Price of ( X ): ( P_X = 2 )
Price of ( Y ): ( P_Y = 5 )
(a) Marginal Rate of Substitution (MRS)
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
]
(b) Optimal Consumption of ( X ) and ( Y )
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’t 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
]
Part 2: Ann’s Utility Problem
Utility function:
[ U(X, Y) = 2XY^2 ]
Price of ( X ): ( P_X = 10 )
Price of ( Y ): ( P_Y = 15 )
Income: ( I = 500 )
(a) Indifference Curve at ( U = 40 )
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 ).
(b) Budget Constraint
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}
]
Graph: The budget line intercepts are:
- ( X )-intercept: ( X = \frac{500}{10} = 50 )
- ( Y )-intercept: ( Y = \frac{500}{15} \approx 33.33 )
(c) Optimal Combination and MRS
The optimal combination satisfies:
[
\frac{\text{MU}_X}{P_X} = \frac{\text{MU}_Y}{P_Y}
]
Where:
[
\text{MU}_X = 2Y^2, \quad \text{MU}_Y = 4XY
]
Thus:
[
\frac{2Y^2}{10} = \frac{4XY}{15}
]
Simplify:
[
15Y^2 = 40XY \implies Y = \frac{40X}{15} = \frac{8X}{3}
]
Substitute ( Y = \frac{8X}{3} ) into the budget constraint:
[
10X + 15\left(\frac{8X}{3}\right) = 500 \implies 10X + 40X = 500 \implies 50X = 500 \implies X = 10
]
If ( X = 10 ):
[
Y = \frac{8(10)}{3} = \frac{80}{3} \approx 26.67
]
At the optimal point, the MRS is:
[
MRS = \frac{\text{MU}_X}{\text{MU}_Y} = \frac{2Y^2}{4XY} = \frac{Y}{2X} = \frac{\frac{80}{3}}{2(10)} = \frac{80}{60} = \frac{4}{3}
]
Explanation (300 Words)
Bob’s utility maximization is straightforward since his utility function is linear. This implies constant marginal utilities for ( X ) and ( Y ). The decision depends solely on the marginal utility per dollar for each good. Given ( \frac{\text{MU}_X}{P_X} > \frac{\text{MU}_Y}{P_Y} ), Bob allocates his entire income to ( X ). The MRS of 2 reflects his willingness to give up 2 units of ( Y ) for 1 unit of ( X ), but this doesn’t affect his decision since only ( X ) is purchased.
Ann’s problem involves a nonlinear utility function, leading to diminishing marginal utility. The indifference curve at ( U = 40 ) shows the trade-off between ( X ) and ( Y ). The budget constraint limits her choices, and its slope reflects the opportunity cost of consuming one good over the other.
At the optimal point, the condition ( \frac{\text{MU}_X}{P_X} = \frac{\text{MU}_Y}{P_Y} ) ensures the best allocation of income. Solving the system of equations yields ( X = 10 ), ( Y \approx 26.67 ). The MRS at this point, ( \frac{4}{3} ), aligns with the slope of the budget constraint, confirming utility maximization.