Kim’s utility function is given by
𝑈
=
𝑋
𝑌
. For this utility function,
𝑀
𝑈
𝑋
=
𝑌
and
𝑀
𝑈
𝑌
=
𝑋
. If good
𝑋
costs $6, and good
𝑌
costs $3, consider the utility-maximizing bundle of
𝑋
and
𝑌
. If the price of
𝑌
rises to $4, how does the ratio of
𝑌
to
𝑋
change? How many units of
𝑌
will Kim buy for each unit of
𝑋
?

Question

Kim’s utility function is given by 
𝑈
=
𝑋
𝑌
. For this utility function, 
𝑀
𝑈
𝑋
=
𝑌
and 
𝑀
𝑈
𝑌
=
𝑋
. If good 
𝑋
costs $6, and good 
𝑌
costs $3, consider the utility-maximizing bundle of 
𝑋
and 
𝑌
. If the price of 
𝑌
rises to $4, how does the ratio of 
𝑌
to 
𝑋
change? How many units of 
𝑌
will Kim buy for each unit of 
𝑋
?

BlackTom AI · Accepted Answer

First, restating the setup: Kim’s utility is U = X × Y, so the marginal utilities are MUX = ∂U/∂X = Y and MUY = ∂U/∂Y = X. The budget constraint initially uses Px = 6 and Py = 3, and later Py rises to 4. To find the utility-maximizing bundle, we use the Lagrangian or the standard tangency condition for a Cobb-Douglas XY utility, which yields Y/6 = X/3 when Py = 3, and Y/6 = X/4 when Py = 4, after equating the multipliers.

Option A: 1 unit of Y for each unit of X.
- If the ratio were 1, that would mean Y = X. Our calculations under Py = 3 gave Y = 2X, not Y = X. Under Py = 4, solving Y/6 = X/4 gives Y = (6/4)X = 1.5X, which is still not Y = X. Thus this option misstates the ratio both before and after the price change.

Option B: 0.5 units of Y for each unit of X.
- A ratio of 0.5 would correspond to Y = 0.5X. Neither of our derived relationships matches this: initial condition yields Y = 2X, and after Py rises to 4 we get Y = 1.5X. Therefore, this option describes a ratio that is too small relative to the optimal demands in both price scenarios.

Option C: 2 units of Y for each unit of X.
- This option matches the ratio we obtain for the initial prices (before the price increase): with Py = 3, the condition Y/6 = X/3 gives Y = 2X, i.e., 2 units of Y per unit of X. However, after Py rises to 4, the new interior solution changes to Y = 1.5X, so the ratio would no longer be 2 under the higher price of Y. This option correctly describes the initial ratio but not the post-change ratio.

Option D: 1.5 units of Y for each unit of X.
- With Py increased to 4, solving Y/6 = X/4 yields Y = 1.5X, which translates to 1.5 units of Y per unit of X. This option accurately reflects the post-change optimal ratio given the new price of Y. It also contradicts the initial ratio, which was 2, showing the effect of the price rise on the allocation between X and Y.

Summary of the reasoning path: the utility maximization with U = XY under a linear budget constraint sets a proportionality between Y and X that depends on the relative prices. When Py = 3, the tangency condition gives Y = 2X. After Py rises to 4, the condition changes to Y = 1.5X. Each option was evaluated against these derived relationships, identifying which ratio is consistent with the price scenario described.

ECON3200001.1251 Homework Assignment 4

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