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Chapter 3: Problem 1

Graph a typical indifference curve for the following utility functions, and determine whether they have convex indifference curves (i.e., whether the \(M R S\) declines as \(x\) increases). a. \(U(x, y)=3 x+y\) b. \(U(x, y)-\sqrt{x \cdot y}\) c. \(U(x, y)=\sqrt{x}+y\) \(\mathrm{d} U(x, y)=\sqrt{x^{2}-y^{2}}\) e. \(U(x, y)=\frac{x y}{x+y}\)

Short Answer

Expert verified
a) \(U(x, y) = 3x+y\) b) \(U(x, y) = \sqrt{x \cdot y}\) c) \(U(x, y) = \sqrt{x} + y\) d) \(U(x, y) = \sqrt{x^2 - y^2}\) e) \(U(x, y) = \frac{xy}{x + y}\) Answer: The utility functions b) and e) have convex indifference curves.

Step by step solution

01

Find an equation for the indifference curve

To find the equation for the indifference curve, set utility equal to a constant (\(U_0\)): \(U_0 = 3x + y\)
02

Determine the MRS

The MRS is the negative of the ratio of the marginal utilities: \(MRS = -\frac{MU_x}{MU_y} \), where \(MU_x = \frac{\partial U}{\partial x}\) and \(MU_y = \frac{\partial U}{\partial y}\). Find \(MU_x\) and \(MU_y\): \(MU_x = \frac{\partial U}{\partial x} = 3\) \(MU_y = \frac{\partial U}{\partial y} = 1\) Calculate MRS: \(MRS = -\frac{3}{1} = -3\)
03

Check convexity

Since MRS is constant and does not depend on \(x\), the indifference curve is not convex. ---------- ### Utility function b: \(U(x, y) = \sqrt{x \cdot y}\) ###
04

Find an equation for the indifference curve

Set utility equal to a constant (\(U_0\)): \(U_0 = \sqrt{x \cdot y}\)
05

Determine the MRS

Find \(MU_x\) and \(MU_y\): \(MU_x = \frac{\partial U}{\partial x} = \frac{1}{2\sqrt{xy}}\cdot y = \frac{y}{2\sqrt{xy}}\) \(MU_y = \frac{\partial U}{\partial y} = \frac{1}{2\sqrt{xy}}\cdot x = \frac{x}{2\sqrt{xy}}\) Calculate MRS: \(MRS = -\frac{\frac{y}{2\sqrt{xy}}}{\frac{x}{2\sqrt{xy}}} = -\frac{y}{x}\)
06

Check convexity

Since MRS depends on \(x\) and \(\frac{y}{x}\) declines as \(x\) increases, the indifference curve is convex. ---------- ### Utility function c: \(U(x, y) = \sqrt{x} + y\) ###
07

Find an equation for the indifference curve

Set utility equal to a constant (\(U_0\)): \(U_0 = \sqrt{x} + y\)
08

Determine the MRS

Find \(MU_x\) and \(MU_y\): \(MU_x = \frac{\partial U}{\partial x} = \frac{1}{2\sqrt{x}}\) \(MU_y = \frac{\partial U}{\partial y} = 1\) Calculate MRS: \(MRS = -\frac{\frac{1}{2\sqrt{x}}}{1} = -\frac{1}{2\sqrt{x}}\)
09

Check convexity

Since MRS depends on \(x\), and \(-\frac{1}{2\sqrt{x}}\) increases (in the negative direction) as \(x\) increases, the indifference curve is not convex. ---------- ### Utility function d: \(U(x, y) = \sqrt{x^2 - y^2}\) ###
10

Find an equation for the indifference curve

Set utility equal to a constant (\(U_0\)): \(U_0 = \sqrt{x^2 - y^2}\)
11

Determine the MRS

Find \(MU_x\) and \(MU_y\): \(MU_x = \frac{\partial U}{\partial x} = \frac{x}{\sqrt{x^2 - y^2}}\) \(MU_y = \frac{\partial U}{\partial y} = -\frac{y}{\sqrt{x^2 - y^2}}\) Calculate MRS: \(MRS = -\frac{\frac{x}{\sqrt{x^2 - y^2}}}{-\frac{y}{\sqrt{x^2 - y^2}}} = \frac{x}{y}\)
12

Check convexity

Since MRS depends on \(x\), and \(\frac{x}{y}\) grows as \(x\) increases, the indifference curve is not convex. ---------- ### Utility function e: \(U(x, y) = \frac{xy}{x + y}\) ###
13

Find an equation for the indifference curve

Set utility equal to a constant (\(U_0\)): \(U_0 = \frac{xy}{x + y}\)
14

Determine the MRS

Find \(MU_x\) and \(MU_y\): \(MU_x = \frac{\partial U}{\partial x} = \frac{y^2}{(x+y)^2}\) \(MU_y = \frac{\partial U}{\partial y} = \frac{x^2}{(x+y)^2}\) Calculate MRS: \(MRS = -\frac{\frac{y^2}{(x+y)^2}}{\frac{x^2}{(x+y)^2}} = -\frac{y^2}{x^2}\)
15

Check convexity

Since MRS depends on \(x\), and \(-\frac{y^2}{x^2}\) declines as \(x\) increases, the indifference curve is convex.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Utility Functions
Utility functions are fundamental concepts within microeconomics that are used to model preferences of consumers. Simply put, a utility function quantifies the satisfaction or happiness that a consumer derives from consuming various bundles of goods and services. Each function generates a numerical value representing utility, with higher numbers indicating preferred consumption bundles.

For example, consider the utility function U(x, y)=3x+y. Here, x and y could represent quantities of two different goods. This function suggests that for each additional unit of good x, the consumer receives three times more utility than from each additional unit of good y. This function helps us visualize and analyze how choices and trade-offs are being made between two goods.
Marginal Rate of Substitution (MRS)
The Marginal Rate of Substitution (MRS) is an important concept that is intricately linked to indifference curves. It measures the rate at which a consumer is willing to give up one good in exchange for another good while keeping the same level of utility. In mathematical terms, the MRS is the negative slope of an indifference curve and is calculated as the negative ratio of the marginal utilities of two goods.

For example, if the MRS between two goods x and y is 2, it means that a consumer would give up 2 units of y for 1 additional unit of x without undergoing any change in their overall utility. This ratio changes along the indifference curve, capturing the trade-offs that a consumer makes as they substitute one good for another.
Convexity of Indifference Curves
The shape of an indifference curve conveys powerful information about consumer preferences. Typically, indifference curves are convex to the origin. Convexity indicates that as the consumer substitutes one good for another, the willingness to exchange (MRS) decreases. In other words, as you consume more of good x and give up units of good y, you value the additional units of x less compared to the units of y you are losing.

However, not all indifference curves are convex. For instance, perfect substitutes and perfect complements produce straight-line indifference curves and L-shaped curves, respectively. Convexity can be checked by examining how MRS changes with the increase of x; in a convex curve, MRS decreases as x increases.
Marginal Utility
Marginal Utility (MU) represents the additional satisfaction a consumer gains from consuming an additional unit of a good or service. It is derived from the utility function, determined by taking the partial derivative with respect to one of the goods.

In our previous examples, the marginal utility of x from the utility function U(x, y)=3x+y is 3. This means that for every extra unit of x consumed, the utility increases by three units. Put differently, the marginal utility is a measure of the incremental utility that helps explain how consumers make decisions and achieve the equilibrium of utility maximization.

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Most popular questions from this chapter

The formal study of preferences uses a general vector notation. A bundle of \(n\) commoditics is denoted by the vector \(\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right),\) and a preference relation \((>)\) is defined over all potential bundles. The statement \(\mathbf{x}^{1}>\mathbf{x}^{2}\) means that bundle \(\mathbf{x}^{1}\) is preferred to bundle \(\mathbf{x}^{2}\). Indifference between two such bundles is denoted by \(\mathbf{x}^{1} \approx \mathbf{x}^{2}\) The preference relation is "complete" if for any two bundles the individual is able to state cither \(\mathbf{x}^{1}>\mathbf{x}^{2}, \mathbf{x}^{2}>\mathbf{x}^{1}\), or \(\mathbf{x}^{1} \approx\) \(\mathbf{x}^{2}\). The relation is "transitive" if \(\mathbf{x}^{1}>\mathbf{x}^{2}\) and \(\mathbf{x}^{2}>\mathbf{x}^{3}\) implies that \(\mathbf{x}^{1}>\mathbf{x}^{3}\), Finally, a preference relation is "continuous" if for any bundle \(y\) such that \(y>x\), any bundle suitably close to \(y\) will also be preferred to \(x\). Using these definitions, discuss whether each of the following preference relations is complete, transitive, and continuous. a Summation preferences: This preference relation assumes one can indeed add apples and oranges. Specifically, \(\mathbf{x}^{1}>\mathbf{x}^{2}\) if and only if \(\sum_{i=1}^{n} x_{i}^{1}>\sum_{i=1}^{n} x_{i}^{2} .\) If \(\sum_{i=1}^{n} x_{i}^{1}=\sum_{i=1}^{n} x_{i}^{2}, \mathbf{x}^{1} \approx \mathbf{x}^{2}\) b. Lexicographic preferences: In this case the preference relation is organized as a dictionary: If \(x_{1}^{1}>x_{1}^{2}, x^{1} \succ x^{2}\) (regardless of the amounts of the other \(n-1\) goods). If \(x_{1}^{1}=x_{1}^{2}\) and \(x_{2}^{1}>x_{2}^{2}, x^{1}>x^{2}\) (regardless of the amounts of the other \(n-2\) goods). The lexicographic preference relation then continues in this way throughout the entire list of goods. c. Preferences with satiation: For this preference relation there is assumed to be a consumption bundle (x") that provides complete "bliss." The ranking of all other bundles is determined by how close they are to \(\mathbf{x}^{*}\), That is, \(\mathbf{x}^{1}>\mathbf{x}^{2}\) if and only if \\[ \left|\mathbf{x}^{1}-\mathbf{x}^{*}\right|<\left|\mathbf{x}^{2}-\mathbf{x}^{\prime}\right| \text { where }\left|\mathbf{x}^{l}-\mathbf{x}^{*}\right|=\sqrt{\left(x_{1}^{\prime}-x_{1}^{*}\right)^{2}+\left(x_{2}^{\prime}-x_{x}^{*}\right)^{2}+\ldots+\left(x_{n}^{\prime}-x_{n}^{*}\right)^{2}} \\]

Find utility functions given each of the following indifference curves [defined by \(U(')=k]\) a \(z=\frac{k^{1 / 8}}{x^{a / b} y^{4 / 8}}\) b. \(y=0.5 \sqrt{x^{2}-4\left(x^{2}-k\right)}-0.5 x\) \(c_{1} z=\frac{\sqrt{y^{4}-4 x\left(x^{2} y-k\right)}}{2 x}-\frac{y^{2}}{2 x}\)

Consider the function \(U(x, y)=x+\ln y .\) This is a function that is used relatively frequently in economic modeling as it has some useful properties. a Find the \(M R S\) of the function. Now, interpret the result. b. Confirm that the function is quasi-concave. c. Find the equation for an indifference curve for this function. d. Compare the marginal utility of \(x\) and \(y\). How do you interpret these functions? How might consumers choose between \(x\) and \(y\) as they try to increase their utility by, for example, consuming more when their income increases? (We will look at this "income effect" in detail in the Chapter 5 problems.) e. Considering how the utility changes as the quantities of the two goods increase, describe some situations where this function might be useful.

Consider the following utility functions: a \(U(x, y)=x y\) \(U(x, y)=x^{2} y^{2}\) \(c(x, y)=\ln x+\ln y\) Show that each of these has a diminishing \(M R S\) but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?

In a 1992 article David G. Luenberger introduced what he termed the benefit function as a way of incorporating some degree of cardinal measurement into utility theory." The author asks us to specify a certain elementary consumption bundle and then measure how many replications of this bundle would need to be provided to an individual to raise his or her utility level to a particular target. Suppose there are only two goods and that the utility target is given by \(U^{*}(x, y)\). Suppose also that the elementary consumption bundle is given by \(\left(x_{0}, y_{0}\right)\). Then the value of the benefit function, \(b\left(U^{*}\right)\), is that value of \(\alpha\) for which \(U\left(\alpha x_{0}, \alpha y_{0}\right)=U^{*}\) a Suppose utility is given by \(U(x, y)=x^{8} y^{1-\beta}\). Calculate the benefit function for \(x_{0}=y_{0}=1\) b. Using the utility function from part (a), calculate the benefit function for \(x_{0}=1, y_{0}=0 .\) Explain why your results differ from those in part (a). c. The benefit function can also be defined when an individual has initial endowments of the two goods. If these initial endowments are given by \(\bar{x}, \bar{y},\) then \(b\left(U^{*}, \bar{x}, \bar{y}\right)\) is given by that value of \(\alpha\) which satisfies the equation \(\left.U\left(x+\alpha x_{0}, y+\alpha y_{0}\right)=U^{*}, \text { In this situation the "benefit" can be either positive (when } U(x, y)U^{*}\right) .\) Develop a graphical description of these two possibilities, and explain how the nature of the elementary bundle may affect the benefit calculation. d. Consider two possible initial endowments, \(\bar{x}_{1}, \bar{y}_{1}\) and \(\bar{x}_{2}, \bar{y}_{2}\). Explain both graphically and intuitively why \(b\left(U^{*}, \frac{\bar{x}_{1}+\bar{x}_{2}}{2}, \frac{\bar{y}_{1}+\bar{y}_{2}}{2}\right)<0.5 b\left(U^{*}, \bar{x}_{1}, \bar{y}_{1}\right)+0.5 b\left(U^{*}, \bar{x}_{2}, \bar{y}_{2}\right) .\) (Note. This shows that the benefit function is concave in the initial endowments.
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