Consider the function U(x, y) = x + lny. This is a function that is used relatively frequently in economic modeling as it has some useful properties.a. Find the MRS 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.
Given- the function U(x, y) = x + lny. a) MRS = MU1/MU2 = 1/1/y = y We observe that MRS only depends on y, and not on x. This means that the indifference curves are vertical parallel shifts of each other, with X on vertical axis and y on horizontal axis. As a consequence, preferences are convex if and only if v(x1) is a concave function [Here, ln y is strictly increasing and concave function], so the marginal utility of x decreases in x. b) A utility function that satisfies u(tx + (1 t)y) Minu(x),u(y) for all t [0, 1]is called quasiconcave. So, our function is quasi-concave as the above property is satisfied by the quasilinear utility function. This can be seen from the graph of utility function also. c) Let K be a constant level of utility, then equation of a Indifference curve is- K = x + lny. d) MUx=1 The increase in utility from x from extra consumption is constant….
= 1/y The marginal utility of y is decreasing in number of units of y. If the income of consumer increases, then the consumer will increase the consumption of x and will keep the consumption of y same as marginal utility of y is decreasing with increasing number of units. e) These preferences are often used to analyze goods which constitute a small part of an agents income; here the utility of a consumer increases with the increase in consumption of x. The consumer keeps the level of consumption of y constant even after increase in income because he can increase his utility more by consuming x