Chapter 5, Income and Substitution Effects Video Solutions, Microeconomic Theory: Basic Principles and Extensions

Problem 1

Thirsty Ed drinks only pure spring water, but he can purchase it in two different-sized containers -.75 liter and 2 liter. Because the water itself is identical, he regards these two "goods" as perfect substitutes.
a. Assuming Ed's utility depends only on the quantity of water consumed and that the containers themselves yield no utility, express this utility function in terms of quantities of $.75 \mathrm{L}$ containers $(X)$ and $2 \mathrm{L}$ containers $(Y)$
b. State Ed's demand function for $X$ in terms of $P_{x}, P_{x},$ and $I$
c. Graph the demand curve for $X$, holding $I$ and $P_{y}$ constant.
d. How do changes in $I$ and $P_{y}$ shift the demand curve for $X$ ?
e. What would the compensated demand curve for Xlook like in this situation?

Oluwadamilola Ameobi

Numerade Educator

19:17

Problem 2

David N. gets $\$ 3$ per week as an allowance to spend any way he pleases. Because he likes only peanut butter and jelly sandwiches, he spends the entire amount on peanut butter (at $\$ .05$ per ounce) and jelly (at $\$ .10$ per ounce). Bread is provided free of charge by a concerned neighbor. David is a particular eater and makes his sandwiches with exactly 1 ounce of jelly and 2 ounces of peanut butter. He is set in his ways and will never change these proportions.
a. How much peanut butter and jelly will David buy with his $\$ 3$ allowance in a week?
b. Suppose the price of jelly were to rise to $\$ .15$ an ounce. How much of each commodity would be bought?
c. By how much should David's allowance be increased to compensate for the rise in the price of jelly in part (b)?
d Graph your results in parts (a) to (c).
e. In what sense does this problem involve only a single commodity, peanut butter and jelly sandwiches? Graph the demand curve for this single commodity.
f. Discuss the results of this problem in terms of the income and substitution effects involved in the demand for jelly.

Oluwadamilola Ameobi

Numerade Educator

01:04

Problem 3

Suppose that, by law, a person is required to consume a fixed amount of good $X$, say $X$, As suming $X$ is a normal good, explain how this law reduces utility for both high- and lowincome people.

Daniel Cisneros

Numerade Educator

01:03

Problem 4

Show that if there are only two goods $(X \text { and } Y$ ) to choose from, both cannot be inferior goods. If $X$ is inferior, how do changes in income affect the demand for $Y ?$

Lucas Finney

Numerade Educator

04:42

Problem 5

As defined in Chapter $3,$ an indifference map is homothetic if any straight line through the origin cuts all indifference curves at points of equal slope: The $M R S$ depends on the ratio $Y / X$
a. Prove that in this case $\partial X / \partial I$ is constant.
b. Prove that if an individual's tastes can be represented by a homothetic indifference map, price and quantity must move in opposite directions; that is, prove that Giffen's paradox cannot occur.

Rashmi Sinha

Numerade Educator

02:19

Problem 6

Suppose that an inclividual's utility for $X$ and $Y$ is represented by the CES function (for
\[
\begin{array}{r}
\delta=-1): \\
\text { utility }=U(X, Y)=-\frac{1}{X}-\frac{1}{Y}
\end{array}
\]
a. Use the Lagrangian multiplier method to calculate the uncompensated demand functions for $X$ and $Y$ for this function.
b. Show that the demand functions calculated in part (a) are homogeneous of degree zero in $P_{x}, P_{y},$ and $I$
c. How do changes in $I$ or in $P_{y}$ shift the demand curve for good $X$ ?

Md.Daniyal Arshad

Numerade Educator

03:03

Problem 7

As in Example 5.1 , assume that utility is given by
\[
\text { utility }=U(X, Y)=X^{3} Y^{*}
\]
a. Use the uncompensated demand functions given in Example 5.1 to compute the indirect utility function and the expenditure function for this case.
b. Use the expenditure function calculated in part (a) together with Shephard's lemma (footnote 5 ) to compute the compensated demand function for good $X$
c. Use the results from part (b) together with the uncompensated demand function for good $X$ to show that the Slutsky equation holds for this case.

Breanna Ollech

Numerade Educator

19:40

Problem 8

Suppose the utility function for goods $X$ and $Y$ is given by
\[
\text { utility }=U(X, Y)=X Y+Y
\]
a. Calculate the uncompensated (Marshallian) demand functions for $X$ and $Y$ and describe how the demand curves for $X$ and $Y$ are shifted by changes in $I$ or in the price of the other good.
b. Calculate the expenditure function for $X$ and $Y$.
c. Use the expenditure function calculated in part (b) to compute the compensated demand functions for goods $X$ and $Y$. Describe how the compensated demand curves for $X$ and $Y$ are shifted by changes in income or by changes in the prices of the other good.

Oluwadamilola Ameobi

Numerade Educator

01:25

Problem 9

Over a three-year period, an individual exhibits the following consumption behavior:
$$\begin{array}{lcccc}
& \boldsymbol{P}_{X} & \boldsymbol{P}_{Y} & \boldsymbol{X} & \boldsymbol{Y} \\
\hline Y_{\text {ear}} 1 & 3 & 3 & 7 & 4 \\
\text {Year 2 } & 4 & 2 & 6 & 6 \\
\text {Year 3 } & 5 & 1 & 7 & 3
\end{array}$$
Is this behavior consistent with the strong axiom of revealed preference?

Lindsay Bur

Numerade Educator

01:40

Problem 10

Suppose the individual's utility function for three goods, $X_{1}, X_{2},$ and $X_{1},$ is "separable"; that is, assume that
\[
U\left(X_{1}, X_{2}, X_{3}\right)=U_{1}\left(X_{1}\right)+U_{z}\left(X_{2}\right)+U_{3}\left(X_{3}\right)
\]
and $\boldsymbol{U}_{i}^{\prime}>0 \quad \boldsymbol{U}_{i}^{m}<\mathbf{0} \quad$ for $i=\mathbf{1}, 2,$ or 3
Show that
a. None of the goods can be inferior;
b. $\quad \partial X_{i} / \partial P_{i}$ must be $<0$ In the Chapter 6 extensions we examine this separable utility case in more detail.

Joshua Eastwood

Numerade Educator