Question

Example 3.3 shows that the MRS for the Cobb-Douglas function \[ U(X, Y)=X^{n} Y^{\wedge} \] is given by \[ M R S=_{P}^{\wedge}(Y / X) \] a Does this result depend on whether $a+(3=1?$ Does this sum have any relevance to the theory of choice? b. For commodity bundles for which $Y=\mathrm{X}$, how does the $MRS$ depend on the values of $a$ and $/3?$ Develop an intuitive explanation of why if $a>(3,MRS>1$. Illustrate your argument with a graph. c. Suppose an individual obtains utility only from amounts of Xand Fthat exceed minimal subsistence levels given by ${\mathrm{X}}_c,{\mathrm{F}}_{\mathrm{o}.}$ In this case, $U(X,Y)=(X-Xo{)}^{\circ }(Y-Y_o)e.$ Is this function homothetic? (For a further discussion, see the extensions to Chapter $4.)$

Short answer

Explain your answer. Answer: No, the MRS result does not depend on the sum a + β. The MRS depends only on the ratio of X and Y, and not the sum. Question: How does the MRS depend on the values of a and β when Y = X? Provide an intuitive explanation. Answer: When Y = X, the MRS depends solely on the ratio of a and β. If a > β, the individual gives more importance to good X over good Y, leading to an indifference curve that is convex to the origin. Question: Is the modified Cobb-Douglas function with subsistence levels homothetic? Explain your answer. Answer: No, the modified Cobb-Douglas function with subsistence levels is not homothetic. The MRS is not constant along rays since it depends on (X-Xo) and (Y-Yo), which indicates that the utility function is not homothetic.

Step-by-step solution

a. Does the MRS result depend on the sum (a + β)?

First, let's find the partial derivatives of the utility function with respect to $X$ and $Y$: $U_x=n{X}^{n-1}{Y}^{\beta }$ $U_y=\beta X^n{Y}^{\beta -1}$ Now, let's find the MRS, which is the ratio of the marginal utilities: $MRS=\frac{U_x}{U_y}=\frac{n{X}^{n-1}{Y}^{\beta }}{\beta X^n{Y}^{\beta -1}}=\frac{n}{\beta }\frac{Y}{X}$ From this MRS formula, we can see that the MRS result does not depend on the sum $a+\beta$. This sum has no direct relevance to the theory of choice since the MRS depends only on the ratio of $X$ and $Y$ and not the sum.

b. How does the MRS depend on the values of a and β when Y = X?

Since $Y=X$, we can rewrite the MRS formula as follows: $MRS=\frac{n}{\beta }\frac{Y}{X}=\frac{n}{\beta }$ From here, we can gather that the MRS solely depends on the ratio of $n$ and $\beta$ when $Y=X$. Intuitively, if $n>\beta$, the MRS will be greater than $1$. This means that the individual is willing to give up more than one unit of good $Y$ for every additional unit of good $X$. Such scenario occurs if the individual gives more importance to good $X$ over good $Y$. To illustrate this point, let's graph an indifference curve. When $n>\beta$, the indifference curve will be convex to the origin, indicating that the individual has a strong preference for good $X$ and is willing to trade more than one unit of good $Y$ for a unit of good $X$.

c. Is the modified Cobb-Douglas function with subsistence levels homothetic?

Now, let's examine the modified Cobb-Douglas utility function with subsistence levels: $U(X,Y)=(X-X_o{)}^n(Y-Y_o{)}^{\beta }$ Remember that a function is homothetic if the marginal rate of substitution (MRS) is constant along rays emanating from the origin. To verify if this function is homothetic, we will find its MRS and check if it is constant along rays. First, find the partial derivatives: $U_x=n(X-X_o{)}^{n-1}(Y-Y_o{)}^{\beta }$ $U_y=\beta (X-X_o{)}^n(Y-Y_o{)}^{\beta -1}$ Now, let's find the MRS: $MRS=\frac{U_x}{U_y}=\frac{n(X-X_o{)}^{n-1}(Y-Y_o{)}^{\beta }}{\beta (X-X_o{)}^n(Y-Y_o{)}^{\beta -1}}$ After simplifying, we get: $MRS=\frac{n}{\beta }\frac{(X-X_o)}{(Y-Y_o)}$ As we can see, the MRS is not constant along rays since it depends on $(X-X_o)$ and $(Y-Y_o)$. Therefore, the modified Cobb-Douglas utility function with subsistence levels is not homothetic.

Key concepts

Cobb-Douglas Utility Function

The Cobb-Douglas utility function is a widely used representation of consumer preferences in economics. It is expressed as

$$U(X,Y)=X^a{Y}^b$$
where $X$ and $Y$ are quantities of two goods, and $a$ and $b$ are positive constants that reflect the consumer's preferences for those goods. One of the remarkable features of this utility function is its simplicity and flexibility in illustrating different degrees of substitutability between goods. Its form allows us to calculate the Marginal Rate of Substitution (MRS), which is the rate at which a consumer is willing to trade off one good for another while maintaining the same level of utility.

To provide further clarity, MRS in the context of a Cobb-Douglas utility function is defined by the ratio of the marginal utilities of both goods, showing how many units of one good a consumer would give up for an additional unit of the other. A graphical representation of this can be depicted on an indifference curve, which is a plot that reveals all the different combinations of two goods that provide equal utility to the consumer.

Indifference Curves

Indifference curves are graphical representations of a consumer's preferences. Each curve shows combinations of goods between which a consumer is indifferent, i.e., which provide the same level of satisfaction or utility. For the Cobb-Douglas utility function, indifference curves take on a characteristic convex shape towards the origin.

This convexity signifies diminishing MRS; as the consumer has more of one good, they are less willing to give up units of the other good to obtain even more. This property can be visually elaborated by noting that the slope of the indifference curve, which represents the MRS, becomes flatter as we move along the curve from left to right, hence indicating that more of good $Y$ is required to compensate for a loss of a unit of good $X$.

Understanding MRS with Indifference Curves

To understand the nature of the MRS in the context of indifference curves for the Cobb-Douglas utility function, consider a point where $Y=X$. At this point, the slope of the tangent line, which is the MRS, depends on the exponents $a$ and $b$ in the utility function. As logically indicated in the exercise solutions, when $a>b$, it means that the consumer prefers good $X$ over good $Y$, and thus the indifference curve would show a steeper slope at any given bundle where $Y=X$. This implies that the consumer would require more than one unit of good $Y$ to be taken away for them to willing accept an additional unit of good $X$, demonstrating an MRS greater than 1.

Homothetic Utility Functions

Homothetic utility functions are a special class of preferences where the MRS between any two goods is the same for all bundles that are scalar multiples of one another. This property means that the shape of indifference curves is consistent when they are scaled up or down, depicting proportional trade-offs between goods regardless of the quantity.

In the case of the modified Cobb-Douglas function discussed in the exercise, with subsistence levels involved, the utility function takes on the form

$$U(X,Y)=(X-X_o{)}^n(Y-Y_o{)}^{\beta }$$
This function incorporates minimum required levels of consumption, denoted by $X_o$ and $Y_o$, below which utility is not defined. Here, the MRS depends on the consumption of goods relative to these subsistence levels, changing as the consumer moves away from these minimum levels. Since MRS changes due to the subtraction of the subsistence levels, the utility function is not homothetic.

Implication of Non-Homothetic Preferences

In the non-homothetic case, the proportion by which a consumer is willing to substitute between two goods changes as their overall level of consumption changes. This has important implications for consumer behavior, particularly in analyzing how demand patterns evolve with income changes, reflecting the realistic aspect that preferences do not always scale evenly as consumers become wealthier or poorer.