Recent content on Ben Davies
Anirudh and I have a new paper . It explores the role that conceptual knowledge plays in human learning and decision-making. A motivating example Suppose a farmer wants to learn which fertilizers are most effective. He views them as black boxes : he knows that they help plants grow, but doesn’t know why . He doesn’t know how fertilizers’ effects relate and can’t extrapolate…
I make pancakes from scratch most mornings. Here’s my method: Blend 45 grams of rolled oats. Mix the blended oats with a tablespoon of cornstarch, half a teaspoon of baking powder, 40 grams of vanilla protein powder, a teaspoon of cinnamon, half a teaspoon of nutmeg, and a pinch of salt. In a separate bowl, combine two whisked eggs, 120 grams of greek yoghurt, a splash of milk, a finely…
Today I’ll speak at an event run by the Stanford Economics Association . They asked me to talk about (micro)economic theory and doing research on it. You can read my slides here and my notes below. They draw from my posts on economic models and how we judge them. What do theorists do? If you read an economic theory paper, you will probably see a lot of math. Modern theorists are…
In one of my favorite papers (summarized here ), Carnehl and Schneider (2025) study the choice between two types of research: research that “deepens” our knowledge by filling gaps between answered questions; research that “expands” our knowledge by answering questions beyond the frontier. Gans (2025) considers how this choice is impacted by artificial intelligence. He…
On this week’s episode of the ReThinking podcast, Adam Grant describes how people sabotage themselves: “Let’s say you’re going into a difficult math test and you’re afraid of finding out that you’re not smart enough to do it. What do you do? You don’t start studying until the night before, or you don’t study at all. And then, when you fail, you can…
Suppose I borrow \(B_0\) at an annual interest rate of \(r\) . I repay the loan by making constant payments of \(P\) at the end of each year for \(N\) years. So, after \(n\in\{0,1,\ldots,N\}\) years, the remaining balance on my loan equals $$B_n\equiv\frac{P}{r}\left(1-\frac{1}{(1+r)^{N-n}}\right).$$ Although I make the same payment \(P\) each year, the share $$\begin{align*} S_n &\equiv…
This post summarizes the past two years of my PhD program . 1 I describe my research and teaching experience, and offer some advice to other students. Research I worked at Motu before starting my PhD. My job was to write research papers. I had two supervisors— Dave Maré and Adam Jaffe —who guided me through the research process. I left Motu thinking research was easy and that I knew…
Statistical models have a formal definition: they comprise (i) a sample space and (ii) a probability distribution (or set of distributions) over that space. Economic models, in contrast, do not have a formal definition. The closest I’ve found is by Spiegler (2024, page 152) : “Most economic models have two built-in parts: a description of the model’s primitives and the solution…
Armchair Expert is a podcast hosted by Dax Shepard and Monica Padman . They interview celebrities, scientists, and other public figures. They also publish some subsidiary “shows” via the podcast’s main feed. The table below counts episodes by show: Show Episodes Interviews 655 Armchair Anonymous 98 Flightless Bird 94 Synced 52 Armchaired & Dangerous 21 Race to 35 12 We Are…
Let \(A\) be the \(n\times n\) matrix with \({ij}^\text{th}\) entry \(A_{ij}=\min\{i,j\}\) . From a previous post , we know \(A\) has a tridiagonal inverse \(A^{-1}\) with \({ij}^\text{th}\) entry 1 $$\left[A^{-1}\right]_{ij}=\begin{cases} 2 & \text{if}\ i=j<n \\ 1 & \text{if}\ i=j=n \\ -1 & \text{if}\ \lvert i-j\rvert=1 \\ 0 & \text{otherwise}. \end{cases}$$ For example, if \(n=4\) then…
Suppose I want to retire at time \(T>0\) . I make constant payments to a savings account that earns continuously compounded interest \(r>0\) . I want my retirement fund to be worth \(V>0\) today (time \(0\) ). How much bigger do my payments have to be if I delay them? Let \(X_d\) be the payments I have to make if I start saving at time \(d\in[0,T]\) . These payments form an annuity with value…
I have a new paper on Bayesian learning. It extends my model of paying for precision to a setting where the unknown state changes over time. This makes the agent keep buying new information as his existing information becomes out of date. I show how his demand for information depends on whether he is myopic or forward-looking, and on the Gaussian process defining how the state evolves. The paper…
Suppose I want to learn the value of a parameter \(\theta\in\mathbb{R}\) . My prior is that \(\theta\) is normally distributed with variance \(\sigma_0^2\) . I observe \(n\ge1\) signals $$\DeclareMathOperator{\Cor}{Cor} \DeclareMathOperator{\E}{E} \DeclareMathOperator{\Var}{Var} \newcommand{\R}{\mathbb{R}} \renewcommand{\epsilon}{\varepsilon} s_i=\theta+\epsilon_i$$ of \(\theta\) . The errors…
A (standard) Wiener process is a continuous-time stochastic process \(\{W(t)\}_{t\ge0}\) with initial value \(W(0)=0\) and instantaneous increments $$\newcommand{\der}{\mathrm{d}} \der W(t)\sim N(0,\der t).$$ We can simulate such a process as follows. First, create a sequence of times \(t\) at which to store the value of \(W(t)\) : t_max = 100 dt = 1e-2 t = seq ( 0 , t_max , by = dt ) Increasing…
Let \(0<x_1<x_2<\ldots<x_n\) and let \(A\) be the symmetric \(n\times n\) matrix with \({ij}^\text{th}\) entry \(A_{ij}=\min\{x_i,x_j\}\) . 1 This matrix has linearly independent columns and so is invertible. Its inverse \(A^{-1}\) is symmetric, tridiagonal , and has \({ij}^\text{th}\) entry $$[A^{-1}]_{ij}=\begin{cases} \frac{1}{x_1}+\frac{1}{x_2-x_1} & \text{if}\ i=j=1 \\…
Let \(X=(X_1,X_2)\) be a random vector in \(\mathbb{R}^2\) . Two realizations \(x\) and \(x'\) of \(X\) form a concordant pair if \((x_2'-x_2)\) and \((x_1'-x_1)\) have the same sign. What’s the probability of sampling a concordant pair when \(X\) is bivariate normal? For example, suppose \(X_1\) and \(X_2\) have zero means, unit variances, and a correlation of \(\rho\) . The scatter plots…
This post is about waiting for information before taking an action. It uses a simple model to explain when and why waiting is valuable. It formalizes some ideas discussed in my posts on climate change and pandemic policy. Suppose it costs \(c>0\) to take an action that pays \(b>c\) if it is beneficial ( \(\omega=1\) ) and zero otherwise ( \(\omega=0\) ). I take the action if its expected net…
This post describes a continuous-time model of Bayesian learning about a binary state. It complements the discrete-time models discussed in previous posts (see, e.g., here or here ). I present the model , discuss its learning dynamics , and derive these dynamics analytically. The model has been used to study decision times ( Fudenberg et al., 2018 ), experimentation ( Bolton and Harris, 1999 ;…
Suppose my payoff \(u(a,\mu)\equiv-(a-\mu)^2\) from taking an action \(a\in\mathbb{R}\) depends on an unknown state \(\mu\in\mathbb{R}\) . 1 I can learn about \(\mu\) by collecting data \(X=\{x_1,x_2,\ldots,x_n\}\) , where the observations \(x_i\) are iid normally distributed with mean \(\mu\) and variance \(\sigma^2\) : 2 $$x_i\mid \mu\sim N(\mu,\sigma^2).$$ I use these data, my prior belief…
Suppose I receive a noisy signal \(s\in\{0,1\}\) about an unknown state \(\omega\in\{0,1\}\) . The signal has false positive rate $$\renewcommand{\epsilon}{\varepsilon} \Pr(s=1\mid\omega=0)=\alpha$$ and false negative rate $$\Pr(s=0\mid\omega=1)=\beta$$ with \(\alpha,\beta\in[0,0.5]\) . 1 I use these rates, my prior belief \(p=\Pr(\omega=1)\) , and Bayes’ rule to form a posterior belief…
Imagine two portfolios of S&P 500 companies. One portfolio weights all companies equally; the other weights companies by their market capitalization (hereafter “value”). Which portfolio is the better investment? One way to answer this question is to look at historical data. For example, the Center for Research in Security Prices (CRSP) provides monthly returns on each portfolio between…
In this week’s episode of EconTalk , Tyler Cowen asks: “Is there any actual mathematical model of this process of how the world is supposed to end? … If you look, say, at COVID or climate change fears, in both cases, there are many models you can look at. … I’m not saying you have to like those models. But the point is: there’s something you look at and then…
Here’s a summary of my website’s traffic since the start of 2023: Notice the spike on April 9, when Tyler Cowen linked to my post of Marginal Revolution metadata . That post is now my second most-viewed ever (just behind my post on applying to economics PhD programs ). Where in the world did those views come from? Here’s a summary: Most visitors came from the US. This makes…
Suppose I take out a loan. It gains interest at rate \(r\) , compounded continuously. I repay the loan by making constant, continuous payments until time \(T\) . How does the repaid share of my loan vary over time? And how does it depend on \(r\) and \(T\) ? Let \(P_0\) be the initial value of my loan: the “principal.” Then my continuous payments \(C\) must satisfy $$\begin{align} P_0…
Today I released the R package MRposts . It contains data on Marginal Revolution blog posts: their authors , titles , publication times , categories , and comment counts . I describe these data below. They cover all 34,189 posts published between August 2003 and March 2023. Authors Marginal Revolution is run by Tyler Cowen and Alex Tabarrok . They wrote 86% and 13% of the posts in MRposts. The…
Today marks five years since my first blog post . This post is my 100th. It summarizes the words I’ve used and traffic I’ve received . Words used My first 99 posts contained more than 56 thousand words: I wrote 11 posts in March and April 2020, when the pandemic forced me to “work” from home. I’ve written 56 posts—about once every 16 days—since starting my…
Economists often use fixed effects to correct for selection bias . Intuitively, these effects “partial out” the reasons why our data include some observations but not others. But this intuition relies on the selection criteria being linear functions of the dependent variable. For example, suppose I have panel data on 100 individuals \(i\) at ten dates \(t\) . These data include pairs…
People talk for many reasons. One is to learn: to collect information that helps us make better choices. Another is to persuade : to convince others to make choices we think are best. Meng (2021) shows how wanting to learn and persuade can lead to homophily . He presents a model in which people choose conversation partners before taking actions. Everyone wants these actions to match an unknown…
This week’s Riddler Classic asks us to fend off an alien invasion: The astronomers of Planet Xiddler are back in action! Unfortunately, this time they have used their telescopes to spot an armada of hostile alien warships on a direct course for Xiddler. The armada will be arriving in exactly 100 days. (Recall that, like Earth, there are 24 hours in a Xiddler day.) Fortunately,…
We often use others’ opinions to guide our choices. For example, we use movie and Yelp reviews to decide what to watch and where to eat. But opinions can be hard to interpret because they depend on objective facts (e.g., movie/food quality) and subjective perspectives (e.g., reviewers’ tastes). So, when seeking opinions, we face a trade-off between “well-informed” sources…
stravadata is an R package I use to organize and analyze my Strava activity data. This post offers some example analyses: Computing annual totals Making activity heat maps Counting efforts Making training calendars Tracking personal records My examples use data on my running activities from the last five years: library ( dplyr ) library ( lubridate ) library ( stravadata ) runs = activities %>%…
IndianVillages is a new R package containing data on social networks in rural India. I derived these data from Banerjee et al.‘s (2013) surveys of households across 75 Karnatakan villages. This post describes the derived data and the networks they define. I also show that the networks are assortatively mixed with respect to caste . Data description IndianVillages provides two tables. The…
My previous post discussed how I should pace myself in a running race. I allowed the cost of running fast to vary during the race, and I showed how the costs I faced determined my optimal speeds and finish time. I assumed that I knew the costs in advance—for example, that I knew which parts of the race had the steepest hills and the strongest headwinds. But sometimes I don’t know the…
Suppose I’m running a race. I have a fixed amount of energy to “spend” on running fast. But the energy cost of running fast varies during the race (e.g., it’s high on hills and low on flats). How should I pace myself to minimize my race time? This post discusses my optimal pacing problem. I describe it mathematically, derive its solution in simple and general settings, and…
Suppose I have data \((a_i,b_i)_{i=1}^n\) on two random variables \(A\) and \(B\) . I store my data as vectors a and b , and compute their correlation using the cor function in R: cor ( a , b ) ## [1] 0.4326075 Now suppose I append a mirrored version of my data by defining the vectors alpha = c ( a , b ) beta = c ( b , a ) so that alpha is a concatenation of the \(a_i\) and \(b_i\) values, and…
People tend to be less popular than their friends. This paradox , first observed by Feld (1991) , is due to popular people appearing on many friend lists. For example, consider the social network among members of a karate club studied by Zachary (1977) : The network contains \(n=34\) nodes with mean degree $$\DeclareMathOperator{\Corr}{Corr} \DeclareMathOperator{\Cov}{Cov}…
This post shows how binary random variables can be defined by their mean, variance, and skewness. I use this fact to explain why variance does not (always) measure “riskiness.” Suppose I’m defining a random variable \(X\) . It takes value \(H\) or \(L<H\) , with \(\Pr(X=H)=p\) . I want \(X\) to have mean \(\mu\) , variance \(\sigma^2\) , and skewness coefficient…
A crop farmer wonders if he should use a new fertilizer. He asks his peers what fertilizer they use and what are their annual yields. He notices that some have different soil. “That’s annoying,” the farmer thinks. “If we all had the same soil, then I could estimate the benefit of using the new fertilizer by comparing the mean yields among farmers who do and don’t use…
One of the requirements for my PhD program is to write a “second-year paper.” You can read mine here . It discusses how career concerns impact the type of advice that experts provide. I consider two types of advice: “Simple” advice of the form “take this action;” “Complex” advice of the form “take this action under these conditions.”…
Dollar cost averaging (DCA) is a way to split a lump sum investment into many smaller investments. It involves regular purchases of a fixed value (rather than quantity ) of shares. This leads to buying more shares when their price is low and fewer when their price is high. DCA is less risky than investing the lump sum because: it reduces the chance of buying lots of shares before their price rises…
Yesterday Science published a study on social networks and job mobility. It suggests that there’s a causal, “inverted U-shaped” relationship between the number of mutual friends you share with someone and the probability that befriending them leads you to change jobs. The authors call for a theory to explain this relationship. In fact it has a simple explanation: homophily .…
This post summarizes the first two years of my PhD program . 1 I discuss my first - and second -year coursework, and my quality of life as a grad student. First-year courses I spent the first year taking the “core” micro, macro, and econometrics courses. Most of their content was familiar from undergrad. Some of my classmates waived out of courses they’d taken before. I…
In a previous post , I showed that if the truth doesn’t matter then I’m better off being an ideologue with ideological friends. I discussed the trade-off between (i) experiencing reality and (ii) experiencing what my friends experience. Truth-seeking made sense only when the benefit of (i) exceeded the cost of forgoing (ii). This post discusses another cost of truth-seeking: having to…
I published my first blog post in March 2018. Since then I’ve spent countless hours planning, drafting, and editing other posts. Academics might think those hours were wasted: “Why write blog posts when you could write research papers? Blogging won’t get you citations or tenure!” But I disagree with that criticism. Blogging complements my research rather than substitutes…
Last month I visited New Zealand for the first time since moving to the USA . Lots of people asked what living in the USA is like. Here’s what I told them: It’s always sunny in Palo Alto I live in Palo Alto, California . It’s near San Francisco and part of Silicon Valley. Palo Alto is officially a “city,” but it feels suburban: the streets are clean, there are trees…
Research is a cumulative process. New discoveries build on previous discoveries: researchers “stand on the shoulders of giants.” Carnehl and Schneider (2022) embed this idea in a model of how knowledge evolves. In their model, knowledge is the set of questions with known answers and research is the process of finding answers. The model has three main features: Existing knowledge…
People learn socially: they get information from their friends. Research on social learning takes as given that people want to learn the truth. 1 This assumption motivates worries about online misinformation: if your friends see something wrong and share it with you, then you might believe it and be wrong too. But people share for more reasons than learning. Sometimes we share to feel connected:…
I have a new paper on gender sorting in economic research teams. Here’s the abstract: I compare the co-authorship patterns of male and female economists, using historical data on National Bureau of Economic Research working papers. Men tended to work in smaller teams than women, but co-authored more papers and so had more co-authors overall. Both men and women had more same-gender co-authors…
My previous post showed that NBER research programs with higher female representation tend to have fewer papers published in the “Top Five” economics journals. A reader suggested comparing Top Five publication rates among men and women within each program. This comparison reveals whether men and women publish at different rates despite writing about similar topics. Here’s the…
The latest version of nberwp (1.2.0) contains information on where NBER working papers are published: Outlet Papers Share (%) Top Five journals 3,832 12.7 Other journals 14,792 49.2 Book/chapters 3,096 10.3 Unpublished 8,363 27.8 About 62% of working papers are published or forthcoming in peer-reviewed journals. One in five of these papers are in the “Top Five:” the American Economic…