Intermediate Macroeconomics · Chapter 1

An AI programme

Your first week at the Ministry of Finance of Halcyon. A new poster in the corridor: Invest in AI, grow for good.

The MinisterWe raise investment from 20% to 30% of output. Mostly data centres and AI systems.
The MinisterWill the island grow faster — for good?
Your head of unitAnswer it with the Solow model. One page.
YouCan I ask one thing first? Is AI more machines, or a better way of producing?
Your head of unitThat’s the whole question. Show it on the chart.

Predict first

The island invests a bigger share of output in AI equipment. In the long run, what happens to the growth of output per worker?

Two curves

The model keeps only what the question needs:

Output per worker: $y = k^{\alpha}$ , where $k$ is capital per worker.
Each year a share $s$ of output is invested.
Capital wears out at rate $\delta$ . New workers arrive at rate $n$ .

k_{t+1} = \frac{s\,y_t + (1 - \delta)\,k_t}{1 + n}

Step through the diagram.

Saving per worker. It flattens: each new server adds less.
Break-even line. The investment needed to keep capital per worker constant.
Where they cross: the steady state $k^*$ . Capital stops growing.
More AI investment lifts the saving curve. A new, higher $k^*$ — then it stops again.
Better technology keeps lifting the curves. That is what keeps growth going.

At the crossing:

k^* = \left( \frac{s}{n + \delta} \right)^{\frac{1}{1-\alpha}}

Data centres and servers are capital. What keeps output per worker growing is technology — doing more with the same inputs — if it keeps improving. On the bench, it grows at rate $g_A$ .

The bench

The island starts at the steady state of a 20% saving rate, with no technological progress. In year 0, your settings take over.

Model bench — AI as capital, AI as technology

Challenge. Keep output per worker growing by about 1% a year or more in year 50.

Annual model with α = 0.33 and labour-augmenting technology; before year 0 the saving rate was 0.20 and technology did not grow.

Saving rate 0.30, no technological progress:

Growth: about 1% in year 1, under 0.1% by year 50.
Output per worker levels off about 22% higher.
Consumption per worker drops at once, from 80 to about 71. It takes about 25 years to get back above 80.

Now set the saving rate back to 0.20 and technological progress to 1%. Output per worker keeps growing about 1% a year. After a century it is about two and a half times today’s.

Push the saving rate up. Steady-state consumption rises until $s = 0.33$ , then falls. That is the golden rule: with $y = k^{\alpha}$ , it is $s = \alpha$ .

An AI’s answer

Spot the error

Question to an AI assistant Will a big programme of investment in AI equipment raise a country’s long-run growth rate?

Answer (written for this exercise)

¹ In the Solow model, output per worker depends on capital per worker and on technology. ² Investing more in AI equipment raises capital per worker. ³ Because of diminishing returns, each extra unit of capital adds less output than the one before. ⁴ So a one-time increase in AI investment raises the growth rate of output per worker permanently.

Which sentence is wrong? Sentence 1 Sentence 2 Sentence 3 Sentence 4

The answer on one page

YouInvesting 30% raises output per worker by about a fifth. Not the long-run growth rate.
The MinisterSo the poster is wrong?
YouHalf wrong. Growth lasts only if AI keeps making us more productive — technology, not just machines.
Your head of unitAnd the catch?
YouConsumption per worker falls first. It takes about a generation to recover.

Your prediction

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

The island invests a bigger share of output in AI equipment. In the long run, what happens to the growth of output per worker?

More AI equipment = more capital: a higher level, growth that fades. Lasting growth needs AI to keep improving technology.

Check yourself

Without technological progress, what is the long-run growth rate of output per worker? It rises with the saving rate Zero It equals the population growth rate n

With

y = k^{\alpha}

, which saving rate maximises steady-state consumption per worker?

s = \alpha

s = 1

s = n + \delta

s = 0.3

n = 0.01

\delta = 0.05

\alpha = 0.33

. What is

k^*

? (One decimal.)

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 · 130 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 Solow growth model. A government plans to raise investment from 20% to 30% of output, mostly in AI equipment, and says this will make the economy grow faster for good.
Please act as my tutor. Do not give me the answers. Ask me one question at a time to help me work out:
1. what the Solow model predicts for the level and the growth rate of output per worker,
2. why diminishing returns to capital matter for that answer, and
3. under what conditions AI could raise long-run growth.
Wait for my answer after each question. At the end, tell me what I got right and correct anything I got wrong.

Key points

Output per worker depends on capital per worker and on technology. Capital has diminishing returns.
Capital per worker settles where saving just covers wear and tear and new workers: the steady state.
Investing more — in AI or anything else — raises the level of output per worker, not its long-run growth.
Only continuing technological progress keeps output per worker growing.
Steady-state consumption is highest at the golden rule: $s = \alpha$ when $y = k^{\alpha}$ .

The Minister accepts the chart, then asks a harder question: why do households save what they save? To be continued — Chapter 2: consumption and saving.