Question

Example 3.9 shows that the MRS for the Cobb-Douglas function \\[ U(X, Y)=X^{\alpha} Y^{\beta} \\] is given by \\[ M R S=\frac{\alpha}{\beta}(Y / X) \\] 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 \(X_{0}, Y_{c}\). In this case, \\[ U(X, Y)=\left(X-X_{4}\right)^{\alpha}\left(Y-Y_{s}\right)^{\beta} \\].

Short answer

Question: Analyze the given Cobb-Douglas utility function, its marginal rate of substitution (MRS), and the relationship between α, β, and MRS when Y=X. Discuss the relevance of α+β to the choice theory and find MRS for a modified utility function with minimal subsistence levels. Answer: The MRS for the given utility function is only dependent on the ratio of α to β, and not directly dependent on their sum. The sum of α and β is related to the choice theory in terms of constant, increasing, or decreasing returns to scale, but does not directly impact the MRS. When Y=X, the MRS is simply the ratio of α to β. If α>β, MRS>1, and the consumer values good X more than good Y. For the modified utility function with minimal subsistence levels, the MRS takes into account the actual consumption levels of both goods and is given by \\[MRS = \frac{\alpha\left(X-X_{4}\right)^{\alpha-1}}{\beta\left(Y-Y_{s}\right)^{\beta-1}} \\].

Step-by-step solution

MRS relationship with α+β

Remember that the given MRS formula is: \\[ M R S =\frac{\alpha}{\beta}(Y / X) \\] The MRS does not directly depend on the sum of α and β, as it is only based on the ratio of α to β. Regarding the relevance to the theory of choice, if α+β=1, the utility function represents constant returns to scale, which means that the individual is indifferent between consuming more or less as long as the proportions of X and Y remain the same. On the other hand, if α+β>1, the function represents increasing returns to scale, and if α+β<1, it represents decreasing returns to scale. The sum α+β helps us understand the consumer's preferences for allocating their income but does not directly impact the MRS. #b. MRS when Y=X and α, β Relationship#

Analyze MRS when Y=X

Since we need to find MRS when Y=X, we will substitute Y with X in the MRS formula: \\[ M R S=\frac{\alpha}{\beta}(X / X) = \frac{\alpha}{\beta} \\] We observe that the MRS in this case is only dependent on α and β.

Intuitive Explanation for MRS>1 when α>β

When α>β, the formula becomes: \\[ \frac{\alpha}{\beta} > 1 \\] This means that the consumer values good X more than good Y, and they are willing to give up more than one unit of Y to acquire an additional unit of X to keep their utility constant. The intuition is that when the exponent α is greater than β, the utility function is more sensitive to changes in X than in Y, making the consumer prefer X over Y.

Illustrate with a graph

To illustrate this idea with a graph, plot the indifference curve of the Cobb-Douglas utility function for different values of α and β. The indifference curve will have an elasticity equal to the MRS. You can compare how the curve becomes more steep (i.e., MRS>1) when α>β, indicating that the consumer is willing to trade more of good Y for an additional unit of good X. #c. MRS for utility function with minimal subsistence levels#

Find MRS of a modified utility function

We are given a modified Cobb-Douglas utility function: \\[ U(X, Y) = \left(X-X_{4}\right)^{\alpha} \left(Y-Y_{s}\right)^{\beta} \\] To find its MRS, we need to compute the first-order partial derivatives with respect to X and Y: \\[ \frac{\partial U}{\partial X} = \alpha \left(X-X_{4}\right)^{\alpha-1} \left(Y-Y_{s}\right)^{\beta} \\] \\[ \frac{\partial U}{\partial Y} = \beta \left(X-X_{4}\right)^{\alpha} \left(Y-Y_{s}\right)^{\beta-1} \\] MRS is the negative ratio of the marginal utilities, so: \\[ M R S = \frac{-\frac{\partial U}{\partial X} }{ \frac{\partial U}{\partial Y}} = \frac{\alpha\left(X-X_{4}\right)^{\alpha-1}}{\beta\left(Y-Y_{s}\right)^{\beta-1}} \\] Here, we can observe that the MRS in this case takes into account the minimal subsistence levels X4 and Ys, making it dependent on the actual consumption levels of both goods as well.

Key concepts

Cobb-Douglas Utility Function

The Cobb-Douglas utility function is a widely used form of utility function in economics represented as \( U(X, Y) = X^{\alpha} Y^{\beta} \), where \(X\) and \(Y\) denote the quantities of two commodities consumed, and \(\alpha\) and \(\beta\) are parameters that measure the relative importance of the commodities to the consumer. What sets the Cobb-Douglas apart is its flexibility to represent various types of preferences and its ease of use in economic calculations.

For instance, when analyzing the marginal rate of substitution (MRS), which is the rate at which a consumer is willing to substitute one good for another while keeping the same level of utility, the Cobb-Douglas function simplifies to \( MRS = \frac{\alpha}{\beta}(Y / X) \). This elegant form allows for straightforward understanding of the consumer’s willingness to trade off goods. The MRS declines as the consumer substitutes \(X\) for \(Y\), illustrating the principle of diminishing marginal utility.

Additionally, the parameters \(\alpha\) and \(\beta\) provide an intuitive grasp of consumer behavior. If \(\alpha > \beta\), it means the consumer places higher importance on \(X\) than on \(Y\), giving us a deeper insight into their preferences.

Theory of Choice

The theory of choice is a fundamental concept in microeconomics that analyzes how consumers allocate their limited resources to maximize utility, which is a measure of satisfaction or happiness derived from consuming goods and services. The Cobb-Douglas utility function plays a crucial role here, helping us understand the choices a consumer makes based on their preferences.

Central to this theory is the concept of budget constraints and the indifference curve analysis, where the MRS reflects the slope of these curves. Consumers move along indifference curves, trading off goods to find their optimal consumption bundle or the point where their budget line is tangent to the highest possible indifference curve.

When the sum of the exponents in a Cobb-Douglas function \(\alpha + \beta\) equals one, it represents constant returns to scale in the utility function. In practical terms, this implies that the consumer will find any combination of \(X\) and \(Y\) along a straight-line indifference curve equally satisfying, as long as the ratio of \(X\) to \(Y\) is maintained. This condition provides essential insights into consumer behavior and the nature of their preferences, influencing their choice and the allocation of resources between different goods.

Returns to Scale

Returns to scale is a concept in production theory related to how the output of a production process changes as the scale of inputs is increased. It is analogous to the concept of returns to scale in utility with respect to the consumption of goods.

In the context of the Cobb-Douglas utility function, the sum of the exponents \(\alpha + \beta\) can indicate the type of returns to scale a consumer experiences in their utility function. If the sum equals one, it signifies constant returns to scale, meaning the proportionate increase in consumption of \(X\) and \(Y\) leads to a proportionate increase in utility. A sum greater than one suggests increasing returns to scale, where utility increases by a greater proportion than consumption. Conversely, a sum less than one indicates decreasing returns to scale, where utility grows by a smaller proportion relative to the increase in consumption.

Understanding these nuances helps in analyzing how changes in consumption levels affect utility, and this forms the basis for consumers' decisions on how much of each good to consume. It also provides an insight into how the MRS changes with variations in the scale of consumption, as it affects the rate at which consumers are willing to substitute between goods to maintain a constant level of satisfaction.