Intermediate Microeconomic Theory

Intermediate Microeconomic Theory

Please make sure that your problem set is legible and stapled. Please write your name and section number on your problem set. On your graphs, please label axes, curves, intercepts, and all points relevant to the question. Explain your work thoroughly in order to receive full credit. Possible points are given in parentheses.
1. (25) Consider the following utility function which specifies Eli’s preferences over two goods, Xylophones (X) and Yo-Yos (Y ). (Please note that we call this type of utility function quasi-linear.) U(X,Y ) = 10√X + 20Y
(a) (5) What is the marginal utility of X? Label your answer MUX and draw a box around your answer. What is the marginal utility of Y ? Label your answer MUY and draw a box around your answer. What is the marginal rate of substitution of X for Y ? Label your simplified answer MRSX,Y and draw a box around it.
(b) (5) What is the slope of the indifference curves when X = 4 and Y = 1? Label your answer slope and draw a box around your answer.
(c) (5) What is the slope of the indifference curves when X = 1 and Y = 1? Label your answer slope and draw a box around your answer.
(d) (10)Start at the bundle described in (c) where X = 1 and Y = 1. How does the slope of the indifference curves change when we increase X while holding Y fixed at 1? In two sentences or less, describe your answer in terms of Eli’s willingness to give up Y for X. Include an argument as to why this intuitively makes sense.
2. (25) Rebecca is just starting a two-day, fully-funded vacation. First thing this morning, she is given $1000. First thing tomorrow morning, she is given $500. This is all the money that Rebecca has access to. Rebecca has no access to credit and cannot borrow money. She can, however, save her money overnight in a savings account that pays 10% interest per day. She will spend all of her money while on vacation.
(a) (5) Plot and label Rebecca’s endowment (the bundle she starts off with) in the space of consumption today (C1) and consumption tomorrow (C2). Put C1 on the horizontal axis.
1 of 2
(b) (10) On a new set of axes, plot and label the point indicating the maximum amount of consumption that Rebecca can get today. Plot and label the maximum amount of consumption that Rebecca can get tomorrow. Use these points, along with the endowment from (a), to draw Rebecca’s budget line and label its slope.
(c) (5) Suppose that Rebecca’s preferences are given by the following utility function:
U(C1,C2) = C1 + 2C2
What is Rebecca’s marginal rate of substitution given these preferences? How does this marginal rate of substitution change as she gets more C1? Explain in two sentences or less.
(d) (5) Suppose that Rebecca’s preferences are instead given by the following utility function:
U(C1,C2) = C0.5 1 C0.5 2
What is Rebecca’s marginal rate of substitution given these preferences? How does this marginal rate of substitution change as she gets more C1? Explain in two sentences or less.