The model behind Long and Variable

The game runs a quarterly, discrete-time version of the dynamic AD–AS model in Cowen & Tabarrok, Modern Principles of Economics (4th ed.):

• ch. 13 (macro ch. 32), Business Fluctuations: Aggregate Demand and Supply: the AD curve from the dynamic quantity theory, the vertical LRAS curve at the Solow growth rate, and the upward-sloping SRAS curve indexed by expected inflation;
• ch. 15 (macro ch. 34), The Federal Reserve System and Open Market Operations: the federal funds rate as the operating target, open market operations, quantitative easing at the zero lower bound;
• ch. 16 (macro ch. 35), Monetary Policy: the “best case” of an AD shock, the negative-real-shock dilemma, lags and uncertainty, credibility and disinflation.

Everything is in percent per year. A period is one quarter; flows are annualized. The code is in js/model.js.

Equations

Solving AD and SRAS together each quarter:

g_t = (S_t − E_tπ − φ·(u* − u_t−1) − c_t + κ·g*_t) / (1 + κ), π_t = S_t − g_t.

So a one-point increase in spending growth raises growth by 1/(1+κ) = 0.67 points and inflation by κ/(1+κ) = 0.33 points on impact, which is C&T’s statement that “in the short run, an increase in spending will be split between increases in inflation and increases in real growth.”

Why these choices

• The interest rate enters through spending growth. C&T express monetary policy as a change in M̃ that shifts AD. Ch. 15 explains that in practice the Fed buys or sells bonds until the funds rate hits its target. The game turns that around: the player chooses the funds rate, and the real-rate gap moves S = M̃ + ṽ. When the real rate equals its neutral value r*, spending grows at expected inflation plus normal real growth, so the economy stays put.
• Real rates matter. Spending depends on i − E(π). If expectations drift up and the funds rate is left alone, policy loosens. Players have to move the funds rate more than one-for-one with inflation to stabilize it (the Taylor principle). Leaving the rate unchanged is therefore not neutral; it is the “Fed does too much” or “too little” case, depending on the shock.
• Self-correction (γ). Under a fixed money-growth rule, C&T’s model returns to the Solow rate on its own as expectations adjust. Under a fixed interest rate that mechanism is missing, so the γ term stands in for the slow natural recovery of spending when slack is large (pent-up demand for durables and investment). Without policy help, recovery is slow.
• Okun’s law with θ. C&T describe both AD shocks and real shocks as causing recessions. Unemployment therefore responds to growth relative to its normal pace ḡ. When growth changes because of a real shock, half of the change (θ = 0.5) is in output per worker rather than in jobs. This lets a negative real shock raise both inflation and unemployment (the ch. 16 dilemma) and a productivity boom lower inflation without an unemployment collapse.
• Labor-market pressure in SRAS (φ). Without the φ term, SRAS responds only to growth relative to the Solow rate. Expected inflation and unemployment are then running totals of the same growth gaps: ΔE(π) = λκ(g − g*) and Δu = −(β/4)(g − ḡ), so with anchoring off E(π) − 2 = −1.2·(u − u*). Bringing unemployment back to its natural rate after a boom takes a spell of below-Solow growth, which undoes exactly the rise in expectations, so inflation always drifts back to its starting rate. The φ term makes inflation depend on the level of slack as well. Over an episode, E(π)_end − E(π)_start = −1.2·(u_end − u_start) + λφ·(point-quarters of unemployment below its natural rate). Once unemployment is back at its natural rate the first term is zero and the second remains, so a lasting stimulus can leave the economy on LRAS at a higher inflation rate, as in C&T’s long-run analysis. This is the standard expectations-augmented Phillips curve, and C&T’s own account of SRAS shifting up as “workers demand even higher wages to catch up.”

Parameters

Small random disturbances hit AD (an AR(1) with persistence 0.5, s.d. 0.5), LRAS (s.d. 0.25) and SRAS (s.d. 0.2) every quarter. Instructors can change the natural rate, the Solow rate, the term length and the QE option from the Instructor panel or the URL.

Policy lags

The Policy lags slider in the Instructor panel (URL parameter lags=0…4) sets how long monetary policy takes to work and how predictable that timing is, in the spirit of Friedman’s “long and variable lags” and C&T’s warning that the Fed’s control is “subject to uncertain lags.”

Length. The real-rate gap passes through k first-order smoothing stages, each with speed ω. With one stage the effect starts at once and decays geometrically; with two or three stages the response is hump-shaped and builds over several quarters. The mean lag is k(1 − ω)/ω quarters.

Variability. Each quarter the speed is ω_t = ω·exp(σε_t − σ²/2), with ε_t ~ N(0,1), clipped to [0.05, 0.95]. The same rate move can therefore bite quickly in one game and slowly in another. The lag draws come from their own random stream, so changing the slider does not change the shocks a given seed produces, and the end-of-game benchmarks face the same lag draws as the player.

(200 seeds per shock; “AD shocks” are fear, boom, fiscal expansion and fiscal tightening.) As lags lengthen, a rule that reacts to last quarter’s data does worse, and the gap between active policy and doing nothing narrows. That is the classic argument for caution, or for rules, when lags are long and variable. At the default setting the lag is the original single-stage lag.

Credibility

Credibility (0–100%) measures how firmly the public expects 2% inflation. It enters only through the expectations equation, as the weight a = 0.12 × cred that pulls expected inflation toward 2% each quarter:

• At 100%, about 12% of any gap between expected inflation and 2% closes each quarter on its own (a half-life of about five quarters), so SRAS drifts back toward 2% without the Fed having to create as much slack.
• At 0%, expectations are purely adaptive: they follow realized inflation and nothing pulls them back to target.

It rises 3 points a quarter while inflation is within 1 point of 2%, and falls 3 points for each point of miss beyond that band (undershooting counts the same as overshooting). It is quick to lose and slow to rebuild: from zero to 100% takes 34 on-target quarters.

The Starting credibility slider in the Instructor panel (URL parameter cred=0…100, default 100) sets credibility at the start of the term; it is held at that level through the pre-game quarters. Starting expectations are 2% at every setting, so a low setting gives a Fed whose 2% target is not yet believed, not an economy that has already inherited high inflation. The opening headline tells players when markets doubt the Fed.

Starting credibility at 0% does not switch anchoring off for the whole game: credibility rebuilds whenever inflation is within a point of 2%. To make expectations purely adaptive throughout, untick Anchor expectations to 2% in the Instructor panel (URL parameter anchor=0). The credibility meter then reads “Off.”

(200 seeds per shock; “AD shocks” are fear, boom, fiscal expansion and fiscal tightening.) Without credibility every shock leaves a longer trail in expectations. Passive policy suffers most, because a fixed funds rate lets the real rate fall as expectations rise. Supply shocks such as oil become much harder, which is C&T’s point that credible policy makes disinflation cheaper.

A lasting stimulus: moving up LRAS

C&T’s long-run experiment is a permanent rise in money growth: AD shifts right once, expectations catch up, and the economy returns to the Solow growth rate at a higher inflation rate. Two things matter for reproducing it in the game.

1. Policy. Holding the funds rate low is not a one-time rise in money growth. As expected inflation rises the real rate keeps falling, so AD keeps shifting right and inflation accelerates without limit (Wicksell’s cumulative process; Friedman’s point that the Fed cannot peg the interest rate). The game’s version of a one-time increase is to cut for a while and then raise the funds rate one-for-one with expected inflation, keeping the real rate at its neutral 2% (the “Real interest rate” tile shows it).
2. Settings. Switch anchoring off; otherwise expectations are pulled back to 2%, which is the point of a credible inflation target.

With anchoring off, the funds rate at 0% for six quarters and then a neutral real rate (no random shocks):

With the default φ, about 31 point-quarters of unemployment below its natural rate leave expectations λφ × 31 ≈ 2.8 points higher, and the economy settles on LRAS with inflation near 5%. Use ?scenario=none&quarters=24&anchor=0 for a classroom demonstration.

Shocks

One major shock hits each game, in a random quarter between 2 and 5. Students are not told which. The categories follow C&T’s tables of factors that shift AD and LRAS.

Scoring

Each quarter’s miss is (π − 2)² + (u − u*)². The distance from goals is the square root of the average miss over the term, in percentage points.

At the end of the game the same shocks are replayed under two benchmark policies so students can compare:

• Never change the rate: the funds rate stays where it started.
• Taylor rule: i = r* + π̄ + 0.5(π̄ − 2) + 0.5·gap, where π̄ is inflation over the past four quarters and the output gap is −2(u − u*) from Okun’s law.

Calibration check

Median distance and share of games reappointed, 200 random seeds per shock (npm test checks the main patterns):

Patterns worth pointing out in class:

• AD shocks are the “best case” (ch. 16): a timely response gets both goals back, and the Taylor rule does far better than doing nothing.
• Real shocks (oil, trade war) are the dilemma: no policy keeps both goals close. The Taylor rule usually scrapes by with a C, and doing nothing fails more often than not.
• The financial crisis drives the right rate below zero. The Taylor rule hits the zero lower bound; a player who cuts early and adds QE can do better than the rule, but rarely gets an A.
• Two shocks (strong dollar, productivity boom) can often be ridden out without any change, as in the original San Francisco Fed game. Not every shock needs a response.