Question

Treat Bob and Joe as the same individual and having the same utility function as provided at the beginning of the question. Looking for the solutions to part e and f.  

Indifference curves and utility: Consider the utility function U (qi,%)-2q1/2 + q2 that describes Joe's preferences. For the following, think of q1 as the variable you would graph on the horizontal axis. 3. a. Derive an expression for his marginal utility (U) from a small increase in qi holding q2 fixed. Also, find b. What is Joe's marginal rate of substitution (MRS)? Give a brief (2 sentences maximum) intuitive c. Given your answer in part (b), find Joe's MRS when q1 4 and q2-4. If one unit of good 1 was taken U2. description of what MRS represents. from Joe, how much of good 2 would he need in compensation to maintain his original level of utility? Fractional units of each good are possible. Graph Joe's indifference curve for a utility level of 10. At a minimum, use the 3 bundles of the goods that have the following levels of good 2 (0, 2, & 4) to plot the indifference curve. Does Joe have a diminishing marginal utility for good one? How about for good two? d. e. Is the fact that Bob's indifference curve intersects with the q axis a violation of any of the 5 properties of indifference curves? Briefly support your answer. After considering the preferences represented by this utility function, provide an example of goods that might be reasonably represented by qi and q2 (Hint: think about how the utility Bob derives from consuming good 2 differs from that of good 1). f.

Answer 1

e) Bob's indifference curve intersects the horizontal axis, as a result, an important property gets violated. One basic assumption is that the consumer consumes both goods. When the indifference curve intersects the horizontal axis at point X (say), then at that point the consumer is consuming only good q₁. Therefore, the indifference curve shouldn't cross either axes.

f) The utility function given above is a quasilinear function. Consider the scenario in which Bob is a farmer and the two good he consumes are corn (q2) and leisure (q1). He would value corn at a constant rate which is relative to his leisure time. As a result, the Indifference curve would only depend on leisure time. To show: MRS = MU (q₁)/MU (q₂) = 1/q₁ is only dependent on q1.