Question

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. \(^{10}\) 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 \(U(x, y)=x^{\beta} y^{1-\beta}\). 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 \(a\), which satisfies the equation \(U\left(\bar{x}+\alpha x_{0}, \bar{y}+\alpha y_{0}\right)=U^{*} .\) In this situation the "bene- fit" can be either positive (when \(U(\bar{x}, \bar{y})U^{*}\) ). 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.)

Short answer

#Question# Based on the given analysis and solution, explain the impact of different initial endowments on the benefit function graph and the concavity of the benefit function. #Answer# The impact of different initial endowments on the benefit function graph can be classified into two groups. When the initial utility is less than the target utility, the benefit is positive, and individuals move to a higher indifference curve to increase their utility. In contrast, when the initial utility is greater than the target utility, the benefit is negative, and individuals move to a lower indifference curve to decrease their utility. The nature of the elementary consumption bundle also affects the benefit function calculation, as seen in the given examples. The concavity of the benefit function can be explained graphically by the change in the slope as individuals move along the curve of possible initial endowments. Utility changes more dramatically when moving from high utility to low utility than when moving from low utility to high utility. Intuitively, the concavity is due to the higher "resource cost" needed for an individual with a higher initial endowment to achieve the same increase in utility compared to an individual with a lower initial endowment.

Step-by-step solution

Write down the given utility function

We are given the utility function \(U(x, y) = x^{\beta} y^{1-\beta}\).

Set up the benefit function equation

We are given the elementary consumption bundle \((x_0, y_0) = (1,1)\). Now, we have to find the value of \(\alpha\) such that \(U(\alpha x_0, \alpha y_0) = U^{*}\). Substitute the elementary consumption bundle into the utility function: \(U^{*} = U(\alpha, \alpha)\)

Solve for \(\alpha\)

Plug the consumption bundle into the utility function and solve for \(\alpha\): \(U^{*} = \alpha^{\beta}\alpha^{1-\beta}\) \(U^{*} = \alpha^{\beta} \alpha^{1-\beta}\) \(U^{*} = \alpha\) So, the benefit function \(b\left(U^{*}\right) = \alpha = U^{*}\). #b. Calculating the benefit function for x0=1, y0=0#

Set up the benefit function equation with the new elementary consumption bundle

We are given the elementary consumption bundle \((x_0, y_0) = (1,0)\). We need to find the value of \(\alpha\) such that \(U(\alpha x_0, \alpha y_0) = U^{*}\). Substitute the elementary consumption bundle into the utility function: \(U^{*} = U(\alpha, 0)\)

Solve for \(\alpha\)

Plug in the consumption bundle into the utility function: \(U^{*} = \alpha^{\beta} * 0^{1-\beta}\) Since \(0^{1-\beta}\) is always 0 (for non-zero \(\beta\)), any combinations of x and y won't satisfy the equation, as \(U^{*}\) should be non-zero. Therefore, the benefit function can't be calculated with this elementary consumption bundle. #c. Graphical explanation for different initial endowments#

Describe the positive benefit

When \(U(\bar{x}, \bar{y}) < U^{*}\), the benefit is positive and needs to be compensated by α. This can be seen on the indifference curve graph. The individual will move to a higher indifference curve that corresponds to \(U^{*}\), which means their utility level would increase.

Describe the negative benefit

When \(U(\bar{x}, \bar{y}) > U^{*}\), the benefit is negative and compensation is needed to lower the utility level. On the graph, the individual will move to a lower indifference curve that corresponds to \(U^{*}\), which means their utility level would decrease.

Explain the effect of an elementary bundle

The nature of the elementary bundle can affect the benefit function calculation. Depending on the values of \(x_0\) and \(y_0\), the benefit may vary, as seen in parts (a) and (b). The relative "weights" of x and y in the utility function and the chosen elementary bundle can play decisive roles in determining the benefit. #d. Explanation for the concavity of the benefit function#

Describe the graphical aspect

Graphically, as individuals move along the curve of possible initial endowments, the change in the slope of their benefit function is more substantial when they move from high utility to low utility rather than moving from low utility to high utility. This means that utility changes more dramatically in the former situation, therefore displaying a concave shape.

Describe the intuitive aspect

Intuitively, an individual who already has a relatively high initial endowment will need more additional endowment to increase their utility level by a fixed amount than an individual who starts with a relatively low endowment. The difference in the "resource cost" to achieve a certain increase in utility will cause the benefit function to be concave in the initial endowments.