Question

Example 3.3 shows that the \(M R S\) for the Cobb-Douglas function \\[ U(x, y)=x^{a} y^{\beta} \\] is given by \\[ M R S=\frac{\alpha}{\beta}\left(\frac{y}{x}\right) \\] a Does this result depend on whether \(\alpha+\beta=1 ?\) Does this sum have any relevance to the theory of choice? b. For commodity bundles for which \(y=x\), how does the \(M R S\) depend on the values of \(\alpha\) and \(\beta\) ? Develop an intuitive explanation of why, if \(\alpha>\beta, M R S>1 .\) Illustrate your argument with a graph. c. Suppose an individual obtains utility only from amounts of \(x\) and \(y\) that exceed minimal subsistence levels given by \(x_{0}, y_{0}\) In this case, \\[ U(x, y)=\left(x-x_{0}\right)^{a}\left(y-y_{0}\right)^{\beta} \\] Is this function homothetic? (For a further discussion, see the Extensions to Chapter \(4 .\) )

Short answer

Is the utility function accounting for minimal subsistence levels homothetic? Answer: The relevance of \(\alpha + \beta = 1\) in the MRS result is that it represents constant returns to scale and unitary elasticity of substitution, meaning that the consumer can replace one good with another at a constant rate while keeping their utility constant. When \(y=x\), MRS depends only on the ratio \(\frac{\alpha}{\beta}\), with the MRS being greater than \(1\) if \(\alpha>\beta\). This means that consumers derive more marginal utility from good \(x\) than good \(y\) and are willing to give up more than one unit of \(y\) for one additional unit of \(x\). The utility function accounting for minimal subsistence levels is not homothetic because it does not satisfy the homotheticity condition, implying that specific base amounts of goods are necessary for the utility function to be valid.

Step-by-step solution

a) Relevance of \(\alpha + \beta = 1\)

According to the MRS formula for the Cobb-Douglas function, the MRS does not directly depend on whether \(\alpha+\beta=1\). The condition \(\alpha+\beta=1\) represents constant returns to scale, which implies that the consumer has unitary elasticity of substitution. It means that the consumer can replace one good with another at a constant rate, keeping the utility constant.

b) MRS dependence on \(\alpha\) and \(\beta\) when \(y=x\)

When commodity bundles are such that \(y=x\), we need to determine the MRS and how it depends on the values of \(\alpha\) and \(\beta\). We substitute \(y=x\) into the MRS formula: \[MRS = \frac{\alpha}{\beta}\left(\frac{x}{x}\right) = \frac{\alpha}{\beta}\] The MRS depends only on the ratio \(\frac{\alpha}{\beta}\). Intuitively, if \(\alpha>\beta\), consumers derive more marginal utility from good \(x\) than good \(y\), and the MRS should be greater than \(1\). Indeed, with \(\frac{\alpha}{\beta} > 1\), consumers are willing to give up more than one unit of \(y\) for one additional unit of \(x\). On a graph, this can be illustrated with an indifference curve that is steeper for higher values of \(x\) and relatively flatter for higher values of \(y\).

c) Homotheticity of the modified utility function

To determine if the modified utility function is homothetic, we check if it satisfies the condition: \(U(\lambda x, \lambda y) = \lambda^nU(x, y)\), where \(\lambda>0\) is a scalar and \(n\) is a constant. So, we have: \[U(\lambda x, \lambda y) = (\lambda x - x_0)^a(\lambda y - y_0)^{\beta}\] On the contrary, we have: \[\lambda^nU(x, y) = \lambda^n (x - x_0)^a(y - y_0)^{\beta}\] We can conclude that the modified utility function is not homothetic because the given function does not satisfy the homotheticity condition. The introduction of minimal subsistence levels (\(x_0\) and \(y_0\)) makes the utility function specific to situations where certain base amounts of goods are necessary.

Key concepts

Marginal Rate of Substitution (MRS)

In the realm of consumer choice theory, the Marginal Rate of Substitution (MRS) is a crucial concept. It refers to the rate at which a consumer is willing to substitute one good for another while maintaining the same level of utility. In mathematical terms, for a Cobb-Douglas utility function \(U(x, y) = x^a y^\beta\), the MRS can be expressed as \(MRS = \frac{\alpha}{\beta}\left(\frac{y}{x}\right)\). This formula shows the proportional trade-off a consumer is willing to make between two goods, \(x\) and \(y\).

An interesting aspect of the MRS in a Cobb-Douglas utility function is that it simplifies under special conditions. When the quantities of the two goods are equal (i.e., \(y = x\)), the MRS boils down to just \(\frac{\alpha}{\beta}\). This tells us that the MRS depends solely on the ratio of the exponents \(\alpha\) and \(\beta\). Intuitively, if \(\alpha > \beta\), it indicates that good \(x\) is relatively more valuable to the consumer compared to good \(y\), leading to an MRS greater than 1. Graphically, this is represented by an indifference curve that is steeper, showing a higher willingness to give up \(y\) for \(x\).

Constant Returns to Scale

Constant returns to scale (CRS) is an important concept in economics, revealing how output responds to a proportional increase in all inputs. It suggests that doubling the input quantity leads to a doubling of the output. In the context of the Cobb-Douglas function, the condition \(\alpha + \beta = 1\) corresponds to constant returns to scale. This implies that the function is linear in its logarithmic form, which is a characteristic of unitary elasticity of substitution. This elasticity implies that consumers are willing to substitute goods at a constant rate while maintaining their level of utility.

While the MRS itself does not depend on whether \(\alpha + \beta = 1\), the condition indicates a balanced substitution capacity between goods. Essentially, it implies that changing the quantity of one good by a certain percentage could be completely offset by changing the quantity of the other good by the same percentage, keeping the utility constant.

Homothetic Preferences

Homothetic preferences describe a condition where the consumer's preference ranking remains the same regardless of changes in scale. Mathematically, for a utility function to be homothetic, it must satisfy \(U(\lambda x, \lambda y) = \lambda^nU(x, y)\). This pattern means that multiplying consumption bundles by a positive constant \(\lambda\) results in a proportionate scaling of utility.

A utility function with subsistence levels changes this scenario. For instance, the modified Cobb-Douglas function \(U(x, y) = (x - x_0)^a(y - y_0)^\beta\) adjusts for minimal subsistence levels \(x_0\) and \(y_0\). This means some base consumption levels are necessary for utility to start increasing. Because these baseline quantities cannot be scaled up proportionately with \(\lambda\), such a utility function is not considered homothetic. The underlying preference shifts as scale changes due to these necessary base levels.

Elasticity of Substitution

Elasticity of substitution relates to how easy it is to substitute one good for another in response to changes in price or utility. For the Cobb-Douglas utility, the elasticity of substitution is constant and equal to 1. This elasticity reflects a straightforward, constant proportion change between the two goods that does not vary, regardless of the specific quantities involved.

The constant elasticity of substitution in a Cobb-Douglas function implies that consumers adjust the quantities of the two goods in the same proportion for any relative price change. This characteristic simplifies consumer choice and results in uniform responses to variations in economic variables. It's central to understanding consumption behavior in environments modeled by Cobb-Douglas preferences, allowing for predictable and manageable adjustments.