Problem Set 5: Foundations of Finance
Due Saturday, October 25 at 11pm on Gradescope
Patience
Nikolai has an income of $€2000$ this year, and he expects an income of $€1800$ next year. He can borrow money from his future self and save money for his future self at an interest rate of $20\%$. Suppose that consumption goods cost $€1$ each and that Nikolai has the utility function $u(c_1, c_2) = \ln c_1 + {1 \over 6}\ln c_2$.
What is the present value of Nikolai’s endowment? What is the equation of Nikolai’s intertemporal budget line?
Solve for Nikolai’s optimal consumption between the two periods. Will he save or borrow in the first period (and how much)?
Draw a graph with Nikolai’s endowment; intertemporal budget line; his indifference curve passing through his endowment; and his indifference curve passing through his optimal consumption point that you found in part (b).
Now, suppose that the interest rate at which Nikolai can both save and borrow decreases from $20\%$ to $10\%$. Would this change in the interest rate make Nikolai better or worse off? Why? (You don’t need to mathematically solve for his new optimal consumption bundle to answer.)
True or false: if the interest rate rose to $100\%$ — meaning that $€1$ saved today would pay back $€2$ in a year — Nikolai would choose to save some of his current income.
He’s here, he’s there, he’s…
Roy Kent is an aging football star playing his last season. His current salary is £800,000. Next year he expects to retire and take a job as a TV commentator, which will pay him £360,000. (As usual, for this question, assume we’re only thinking about a two-period model.) Any money he doesn’t spend this year will grow at an interest rate of 5%. If he wants to borrow money against his future income, he would have to pay 20% interest on any money he borrows.
Draw Roy’s budget constraint. Include his endowment point, the horizontal intercept, and the vertical intercept.
Like many footballers, Roy is a “live for today” sort of gent. His preferences over present consumption ($c_1$) and future consumption $(c_2)$ may be represented by a Cobb-Douglas utility function $u(c_1,c_2) = \ln c_1 + \beta \ln c_2$. For what values of $\beta$ will Roy choose to neither borrow at 20% interest nor save at 5% interest, but consume at his endowment?
Suppose Roy’s discount factor is actually $\beta = 19/21$. How much would he optimally borrow or save, or would he do neither? Add his optimal bundle to your graph from part (a).
What’s your advice?
Consider a two-period model of retirement savings for a white-collar worker named Pat. Period 1 is Pat’s adult life, when they’re earning money. Period 2 is retirement, when they live off their savings plus social security income from the government.
Suppose Pat’s income in period 1 is $m_1$, measured in current dollars. Their social security income in period 2 is $m_2$ and is inflation-adjusted, meaning that it’s measured in units of period-2 consumption. They can save present income at an interest rate of $r$, and the inflation rate is $\pi$. Therefore, as we showed in class (and is derived in the book), their budget constraint is given by \(c_2 = m_2 + \frac{1 + r}{1 + \pi}(m_1 - c_1)\) Pat’s utility function is given by $u(c_1,c_2) = \ln c_1 + \beta \ln c_2$.
For the purposes of this problem, assume Pat’s social security income is small relative to their earnings, so they will certainly save and not borrow.
Solve for Pat’s optimal amount of savings, $s = m_1 - c_1$, as a function of $r$, $\beta$, and $\pi$ (as well as $m_1$ and $m_2$).
In three separate budget line/indifference curve diagrams, illustrate the effect of an increase in $r$, $\pi$, and $\beta$ on Pat’s optimal savings and consumption choice.
Suppose you get a job after Stanford as a financial advisor, and Pat is one of your clients. Thinking about your answers to (b), come up with a scenario – a life event or a change in the economy – that would cause you to advise them to save more money than they previously were planning on saving. Write an email explaining the situation to them, without using any technical terms – Pat hasn’t studied economics.
Analyzing a Lottery
Suppose there are two states of the world (state 1, 2), that occur with probability $\pi$ and $(1 - \pi)$, respectively. Your indirect utility from money within any state of the world is given by $u(c) = c^{1 \over 4}$.
You currently have €1296, but face a one-third chance of losing €1215, leaving you with just €81. Let state 2 be the “good” state in which you have $c_2 = 1296$, and state 1 be the “bad” state in which you have $c_1 = 81$; there is probability $\pi = \frac{1}{3}$ of state 1 occurring.
What is the expected (consumption) value of this lottery, $\mathbb E[c]$?
What is your expected utility from this lottery, $\mathbb E[v(c)]$? (It might be helpful to note that $3^4 = 81$ and $6^4 = 1296$…)
Compare your answer from (b) to the utility of your answer from (a) – that is, the utility you would receive if you consumed your expected value of the lottery for sure. What does this say about whether you are risk-averse, risk-neutral, or risk-loving?
What is your certainty equivalent ($CE$) for this lottery?
What is your risk premium ($RP$)?
Draw two graphs like the ones on this page. Label the coordinates of all the points on each graph.
Relative and Absolute Risk Aversion (50Q only)
Problem credit to Stefano DellaVigna