Problem Set 5: Foundations of Finance

Patience

Nikolai has an income of $€800$ this year, and he expects an income of $€1000$ next year. He can borrow money from his future self and save money for his future self at an interest rate of 25%. Suppose that consumption goods cost $€1$ each and that Nikolai has the utility function $u(c_1, c_2) = \ln c_1 + { 1 \over 3 }\ln c_2$.

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. 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).

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.

Bond Pricing: Inflation and Risk

Let’s think of a very simple bond pricing model. Assume value function for money is given by $v(c) = \sqrt{c}$. For simplicity, assume $\beta = 1$: that is, you don’t discount your future utility at all. You believe there is a probability $\delta$ that the issuer of the bond pays you back; therefore your von Neumann-Morgenstern (expected utility) function is \(u(c_1,c_2) = \sqrt{c_1} + \delta \sqrt{c_2}\) The bond has a nominal interest rate of $r$, and inflation is expected to be $\pi$. You have a current income of $m_1$, but you’re about to retire, and will receive no income in the future: that is, $m_2 = 0$. All of this means your budget line is given by \(c_1 + \left({1 + \pi \over 1 + r}\right) c_2 = m_1\)

Solve for your utility-maximizing value of $c_1$. What fraction of your income should you optimally save, as a function of $r$, $\pi$, and $\delta$? (Don’t worry about simplifying completely, but it will help you if each of your variables appears as few times as possible in the expression you find.)

Draw an indifference curve/budget line diagram that shows the effect of an increase in the inflation rate $\pi$. (That is, it should have a before and after, with arrows indicating what curve(s) shift and how the optimal bundle changes.) When inflation is higher, do you save more or less? How would you explain this intuitively?

Repeat part (b) to show the effect of a decrease in $\delta$ – that is, an increase in the risk of default.

Prediction Markets

You’re a political junkie, and you think you know more about the mood of California voters heading into the 2026 gubernatorial election than most people. You have $150 which you’re considering betting on whether or not Republican Steve Hilton advances to the general election. Right now, as I write this question on April 30, 2026, you can buy a “yes” contract on this market in Kalshi for 75¢, and a “no” contract for 25¢. In other words, for each “yes” contract you buy for 75¢, you will get €1 if Hilton makes it to the general election, but €0 if he doesn’t. On the other hand, for each “no” contract you buy for 25¢, you will get €1 if Hilton doesn’t make it to the general election, but €0 if he does.

Draw your budget line, assuming “good 1” is the money you have if Hilton makes it to the general election, and “good 2” is the money you have if he doesn’t. What is the equation of this line?

Suppose you believe that there’s only a 50% chance Hilton will advance to the general election. If your value function is $v(c)=\sqrt{c}$, this means your expected utility from consuming $c_1$ if Hilton advances and $c_2$ if he doesn’t is given by\(\mathbb{E}[v(c)] = 0.5\sqrt{c_1} + 0.5\sqrt{c_2}\)How many “no” contracts should you buy?

Relative and Absolute Risk Aversion (50Q only)

Problem credit to Stefano DellaVigna

Consider the exponential value function $v(c) = -e^{-\alpha c}$. Show using calculus that it is increasing and concave for all $c$ as long as $\alpha > 0$, that is, as long as the agent is risk-averse. Show that this function has constant absolute risk aversion coefficient given by $\alpha$.

Consider the power value function $v(c) = \frac{c^{1 - \gamma}}{1 - \gamma}$. Show using calculus that it is increasing and concave for all $c > 0$ as long as $\gamma > 0$, that is, as long as the agent is risk-averse. Show that this function has constant relative risk aversion coefficient given by $\gamma$.

Consider the log value function $v(c) = \ln (c)$. Show using calculus that it is increasing and concave for all $c > 0$. Show that this function has constant relative risk aversion coefficient equal to 1. (In fact, it is possible to show $\lim_{\gamma \rightarrow 1} \frac{c^{1 - \gamma}}{1 - \gamma} = \ln(c)$ — you are not required to prove this.)