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.

To find the Marginal Rate of Substitution (MRS), we first need to find the marginal utility of x and y. The marginal utility is the partial derivative of the utility function with respect to x and y. \(M_U_x = \frac{\partial U(x, y)}{\partial x} = 1\) \(M_U_y = \frac{\partial U(x, y)}{\partial y} = \frac{1}{y}\)

The MRS is the ratio of the marginal utility of x to the marginal utility of y. \(MRS = -\frac{M_U_x}{M_U_y} = -y\) Interpretation: The MRS represents the rate at which a consumer is willing to trade one good (x) for another (y) while maintaining the same level of utility. In this case, the MRS is -y, which means the consumer is willing to give up y units of good Y to gain one more unit of good X. #b. Confirm the function is quasi-concave#

A function is quasi-concave if and only if its Hessian matrix is negative semi-definite. The Hessian matrix is a square matrix of second-order partial derivatives. First, calculate the second-order partial derivatives: \(\frac{\partial^2 U}{\partial x^2} = 0\) \(\frac{\partial^2 U}{\partial x\partial y} = \frac{\partial^2 U}{\partial y\partial x} = 0\) \(\frac{\partial^2 U}{\partial y^2} = -\frac{1}{y^2}\) Now, form the Hessian matrix: \(H = \begin{pmatrix} 0 & 0 \\ 0 & -\frac{1}{y^2} \end{pmatrix}\)

A matrix is negative semi-definite if the determinant is non-negative, and all its leading principal minors are non-positive: Determinant: \(det(H) = 0 \times (-\frac{1}{y^2}) - 0 \times 0 = 0\) Leading principal minors: \(det \begin{pmatrix} 0 \end{pmatrix} = 0\) \(det \begin{pmatrix} -\frac{1}{y^2} \end{pmatrix} = -\frac{1}{y^2}\), which is non-positive since y > 0. As both conditions are satisfied, the Hessian matrix is negative semi-definite, and the function is quasi-concave. #c. Find the equation for an indifference curve#

An indifference curve represents all the combinations of goods x and y which provide the same level of utility. Let this constant level of utility be k. \(U(x, y)=x + \ln(y) = k\)

Rearrange the equation to solve for y in terms of x and k: \(y = e^{k - x}\) The equation \(y = e^{k - x}\) represents an indifference curve for the utility function \(U(x, y) = x + \ln(y)\). #d. Compare the marginal utility of x and y# We have calculated the marginal utility of x and y previously: \(M_U_x = \frac{\partial U(x, y)}{\partial x} = 1\) \(M_U_y = \frac{\partial U(x, y)}{\partial y} = \frac{1}{y}\) Comparison: The marginal utility of x is constant, which means the additional utility gained from consuming one more unit of good X is always the same. The marginal utility of y decreases as the consumption of good Y increases, which implies diminishing marginal utility for good Y. Interpretation: Consumers will try to balance the marginal utilities of x and y while maximizing their total utility. As their income increases, they would consume more of both goods, but they might consume more of good X since its marginal utility remains constant, unlike good Y, which decreases with an increase in consumption. #e. Describe situations where the utility function might be useful# Situation 1: A consumer prefers a combination of two goods, one with a constant marginal utility (e.g., money) and the other with a diminishing marginal utility (e.g., leisure time). The utility function U(x, y) = x + ln(y) accurately represents their preferences. Situation 2: An economist wants to study the income effect on consumers' willingness to trade different goods. The utility function helps analyze both goods' marginal utilities and MRS, leading to a better understanding of how consumers alter consumption patterns with increasing income. Situation 3: A company wants to study consumer preferences for two of its products, one which has constant satisfaction levels (e.g., a subscription service) and another with diminishing satisfaction (e.g., physical goods). The utility function U(x, y) = x + ln(y) may help the company understand consumption patterns and make relevant pricing or marketing decisions.