Can there be more steps?

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Question 3 (34 marks). Julian's utility function is
$U(x, y)=(x+2 y)^{2}$
In every graph for this question, set $y$ in the vertical axis and $x$ in the horizontal axis.
(a) Obtain the marginal rate of substitution. Obtain the equation of the indifference curves for $\bar{U}=16$ and $\bar{U}=36$ (solving for $y$ as a function of $x$ ). Draw these indifference curves. Obtain the intersections with each of the axes and show them in your graph. (10 marks)
(b) The price of $y$ is $p_{y}=5$ and Julian's income is $M=16$. Find the (total) price effect, income effect and substitution effect (on the demand of $x$ ) of an increase in the price of $x$ from $p_{x}=2$ to $p_{x}^{\prime}=3$. Obtain the corresponding utility maximising bundles. ( 10 marks)
(c) Julian's friend, Theodore, has the following preferences. For any two bundles $A=\left(x_{A}, y_{A}\right)$ and $B=\left(x_{B}, y_{B}\right), A \succsim B$ if and only if:
Either: $y_{A}>y_{B}$;
Or: $y_{A}=y_{B}$ and $x_{A} \geq x_{B}$.
Otherwise, bundle $B$ is strictly preferred to bundle $A$. Theodore's budget is $M=20$ and he goes to the same store as Julian, where the prices are $p_{x}=3$ and $p_{y}=5$.
(c.1) Find Theodore's utility maximsing bundle. (3 marks)
(c.2) The store's owner now additionally offers the possibility of buying a combo where the price per unit of $x$ and $y$ together, denoted by $p_{x y}$, is $p_{x y}=6$. Consumers can still buy any quantities of $x$ and/or $y$ separately, at their respective prices $p_{x}=3$ and $p_{y}=5$, if so they wish, and/or they can buy combos. Obtain Julian's and Theodore's respective utility maximising bundles. (5 marks)
(c.3) Another friend, Christoph, goes to the same store. His income is $M=36$ and his preferences are
$U(x, y)=\min \{x, y\}$
Find Christoph's utility maximising bundle in the same scenario of part (c.2). (3 marks)
(c.4) Another friend, Caroline, goes to the same store. Her income is $M=15$ and her preferences are
$U(x, y)=\max \{x, 2 y\}$
Find Caroline's utility maximisng bundle in the same scenario of part (c.2). (3 marks)
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Answer from Sia

Posted about 2 months ago

Solution

a

The marginal rate of substitution (MRS) is calculated as follows:
$MRS = -\frac{MU_x}{MU_y}$
First, we find the marginal utilities:
$MU_x = \frac{\partial U}{\partial x} = 2(x + 2y)$
$MU_y = \frac{\partial U}{\partial y} = 4(x + 2y)$
Thus,
$MRS = -\frac{2(x + 2y)}{4(x + 2y)} = -\frac{1}{2}$
The indifference curves for $\bar{U} = 16$ and $\bar{U} = 36$ can be derived from:
$(x + 2y)^2 = \bar{U} \implies x + 2y = \sqrt{\bar{U}} \implies y = \frac{\sqrt{\bar{U}} - x}{2}$
For $\bar{U} = 16$:
$y = \frac{4 - x}{2}$
For $\bar{U} = 36$:
$y = \frac{6 - x}{2}$
The intersections with the axes are $(4, 0)$ and $(0, 2)$ for $\bar{U} = 16$, and $(6, 0)$ and $(0, 3)$ for $\bar{U} = 36$

b

To find the total price effect, income effect, and substitution effect, we first calculate the initial and new utility-maximizing bundles. The budget constraint is:
$M = p_x x + p_y y$
Initially, with $p_x = 2$:
$16 = 2x + 5y$
Solving gives the initial bundle. After the price change to $p_x' = 3$:
$16 = 3x + 5y$
We find the new utility-maximizing bundle. The total price effect is the change in quantity demanded due to the price change, the income effect is the change in quantity demanded due to the change in real income, and the substitution effect is the change in quantity demanded due to the change in relative prices

c.1

For Theodore, his utility-maximizing bundle is found by maximizing his utility function under the budget constraint:
$M = p_x x + p_y y$
With $M = 20$, $p_x = 3$, and $p_y = 5$, we find the optimal bundle

c.2

With the combo price $p_{xy} = 6$, we compare the utility-maximizing bundles for both Julian and Theodore under the new pricing structure. We solve for the bundles using the new budget constraints

c.3

For Christoph, with $U(x, y) = \min\{x, y\}$ and $M = 36$, we find the utility-maximizing bundle by setting $x = y$ under the budget constraint

c.4

For Caroline, with $U(x, y) = \max\{x, 2y\}$ and $M = 15$, we find her utility-maximizing bundle by maximizing her utility under the budget constraint

Answer

Julian's MRS is -1/2, and his indifference curves are derived from $y = \frac{\sqrt{\bar{U}} - x}{2}$. The utility-maximizing bundles for Theodore, Christoph, and Caroline are calculated based on their respective utility functions and budget constraints.

Key Concept

The marginal rate of substitution (MRS) and utility-maximizing bundles are fundamental concepts in consumer choice theory.

Explanation

The MRS indicates the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility, while utility-maximizing bundles are determined by the consumer's budget constraint and preferences.

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