AdvancedMacroeconomics · Chapter 1About 9 min

AI lifts productivity in a single quarter,and a newcomer builds a world in miniature

The note

Your first Monday at the Central Bank of Halcyon. Fourth floor. Twelve desks, an old coffee machine, a whiteboard nobody has fully erased in years.

1. IV
   Ilse VargaRead this. Productivity jumped last quarter. The statistics office says it’s AI tools.
2. You
   YouFirms using AI for scheduling, design, customer service?
3. IV
   Ilse VargaExactly. The Governor wants two answers by Friday.
4. IV
   Ilse VargaWill it last? And what will people do with it?
5. IV
   Ilse VargaBring a model, not an opinion.

Predict first

AI raises productivity by 1% this quarter. What do households do with the extra output?

A world in miniature

A model at the Bank means a small economy that is complete:

• everyone makes choices for a reason,
• every market clears,
• random shocks push it around over time.

That is a dynamic stochastic general equilibrium (DSGE) model.

Start small. One household lives forever, owns the capital and works a fixed amount. One firm produces:

Y_t = Z_t K_t^{\alpha}

Z_t is productivity — what AI just raised. What isn’t consumed is invested. Capital wears out at rate \delta:

K_{t+1} = (1-\delta) K_t + Y_t - C_t

The household maximises the discounted sum of \ln C_t. That gives the Euler equation: saving one more unit today must pay back tomorrow.

\frac{1}{C_t} = \beta \, \mathbb{E}_t \left[ \frac{1}{C_{t+1}} \left( \alpha Z_{t+1} K_{t+1}^{\alpha-1} + 1 - \delta \right) \right]

Productivity is persistent, with surprises:

\ln Z_{t+1} = \rho \ln Z_t + \varepsilon_{t+1}

\rho is the Governor’s first question. Near 0: the AI boost is a one-off. Near 1: it lasts for years.

What happens after the shock

Step through what the model does after a 1% productivity jump (\rho = 0.9).

1. A 1% productivity jump arrives in quarter 0.
2. Output rises 1% at once — capital and labour are fixed this quarter.
3. Investment jumps about 3.5%. The windfall goes into capital.
4. Consumption rises only a little — about 0.2%.
5. Everything returns to normal. How fast depends on \rho.

To get these paths, find the steady state (where the economy rests with Z = 1) and approximate the model by linear equations around it. Solved, they give rules for capital and consumption. The bench applies them.

Model bench — an AI productivity shock

Challenge. Find a persistence at which consumption rises at least half as much as output on impact.

Quarterly model with log utility and a fixed labour supply; α = 0.33 and δ = 0.025 are held fixed. Lines show percent deviations from the steady state.

Output on impact never moves with \rho. What moves is how the extra output is used. The shorter the gain, the more is invested. That is consumption smoothing.

An AI’s answer

Spot the error

Question to an AI assistant In a real business cycle model, how do households respond to a productivity gain from AI?

Answer (written for this exercise)

¹ A productivity gain raises output on impact. ² Households smooth consumption rather than spending a windfall at once. ³ So most of a temporary gain is saved, which in this model means invested. ⁴ The more persistent the gain, the larger the share that is invested on impact.

Which sentence is wrong?

Sentence 1 Sentence 2 Sentence 3 Sentence 4

Friday

1. You
   YouOutput rises one for one with productivity, on impact.
2. You
   YouIf the AI boost is short-lived, most of it is invested. If it lasts, people spend more right away.
3. G
   The GovernorSo the question is whether AI is a one-off burst or a lasting change.
4. You
   YouYes. That decides the split.
5. G
   The GovernorOne more thing. In your little world, prices adjust instantly. On our high street they don’t. What then?

Your prediction

You did not make a prediction at the start of the chapter. Here is what it asked:

AI raises productivity by 1% this quarter. What do households do with the extra output?

Output rises one for one on impact. How the gain is split depends on how long it is expected to last.

Check yourself

With \rho = 0.9, which rises more in percent on impact after a positive productivity shock?

Consumption Investment Both the same

As \rho rises towards 1, the impact response of consumption:

Rises Falls Doesn’t change

\beta = 0.99, \alpha = 0.33, \delta = 0.025. What is the steady-state K/Y (capital over quarterly output)? (Two significant digits.)

Your answer

Take it with you

This chapter on two pages

The ideas, the formulas, a worked example, four traps — and six questions to try on your own, with answers.

Download the sheet (PDF) 2 pages · 140 KB

Take it to your AI

Want to go further? Paste this into the AI you use. It asks the AI to quiz you, not to hand you answers.

I am learning the real business cycle model. Suppose the adoption of AI raises an economy's productivity by 1% this quarter, and households have log utility, a fixed labour supply and a capital stock that depreciates at 2.5% a quarter.
Please act as my tutor. Do not give me the answers. Ask me one question at a time to help me work out:
1. why output rises one for one with productivity on impact,
2. why most of a temporary gain is invested rather than consumed, and
3. how the answer changes if the gain is expected to be permanent.
Wait for my answer after each question. At the end, tell me what I got right and correct anything I got wrong.

Key points

• A DSGE model is a small, complete economy: people optimise, markets clear, shocks arrive at random.
• The steady state is where the model rests without shocks. Here K/Y = \alpha / (1/\beta - 1 + \delta).
• Log-linearising around the steady state gives rules that can be solved.
• An impulse response shows each variable’s path after one shock.
• A short-lived gain is mostly invested. A long-lived one is consumed more.

The Governor’s last question stays on your desk all weekend: what if shops don’t reprice overnight? To be continued — Chapter 2: sticky prices and the New Keynesian model.