The Solow model: production, saving and growth
The Solow model explains output per worker, capital accumulation and growth with a few relationships applied one year at a time. On this site you will run the model yourself: inspect a single year’s accounts, follow an economy through time, change its saving, depreciation or productivity, and explain what follows.
Every chart, equation and number on a page comes from the same calculation. When you select a year, the equation shows that year’s numbers, the diagram marks that year’s capital stock, and the time path highlights the same year.
Save a prediction, explore the model, then choose “I’m ready to explain what I found” to open the After question. Your answers stay in this tab and reset when you reload the page.
EC306 Intermediate Macroeconomics. Uses the course’s continuing example: A = 2, L = 27 workers, K0 = 27, s = 0.20, δ = 0.10.
Production: capital per worker and productivity
What can make output per worker higher?
Before anything moves through time, we need to know what an economy can produce with what it has. The production function y = A k1/3 says how much output each worker produces in a year given the capital per worker k and the productivity parameter A (total factor productivity, or TFP).
Two different changes can raise output per worker. More capital per worker moves the economy along the curve. Higher A shifts the whole curve up. Try both below and watch which one the readout reports.
Algebra: from Y = A K1/3 L2/3 to y = A k1/3
Divide both sides by L: Y/L = A K1/3 L2/3 / L = A K1/3 L−1/3 = A (K/L)1/3. Dividing powers with the same base subtracts exponents (2/3 − 1 = −1/3), and powers with the same exponent combine into a power of the ratio. Writing y = Y/L and k = K/L gives y = A k1/3. Nothing about the economy changed; we only measured both quantities per worker.
The power 1/3 is a cube root: 271/3 = 3 because 3 × 3 × 3 = 27, and 81/3 = 2. So at A = 2, multiplying capital per worker by 8 (from 1 to 8) only doubles output per worker (from 2 to 4).
The marginal product of capital is the slope of the curve: MPK = (A/3) k−2/3. It is positive and falls as k rises. That is what diminishing returns means.
One year of accumulation
When does an economy’s capital stock grow?
Now the economy acquires its capital. Each year follows the same order: the capital inherited at the start of the year produces output; a share s of that output is invested and the rest consumed; a fraction δ of the starting capital wears out. What survives, plus what was invested, is the capital stock at the start of next year.
This rule is the law of motion for capital: it takes this year’s stock and produces next year’s. Answer the question below first, then use the year ledger to check.
Repeating the rule: transitions and the steady state
If the same rule applies every year, where does the economy end up?
A simulation is nothing more than the year ledger applied again and again: each year’s kt+1 becomes the next year’s starting stock. The left chart has capital on the horizontal axis and shows the two flows that decide the change. The right chart has time on the horizontal axis and shows the path that results. Step through a few years before you play a long run.
Find a capital stock that reproduces itself
A steady state is a capital stock that the law of motion maps to itself: start the year with it, apply the rule, and you end the year with the same stock. Enter a value of k and apply one year of the baseline rule to test it.
Running a model experiment
If saving, depreciation or productivity changes, what happens immediately, during the transition, and eventually?
An experiment starts from a known situation, changes one thing, and holds everything else fixed. Here the economy starts at its baseline steady state, k = 8, and a parameter changes permanently at the start of year 5. The dashed path is the same economy with no change, so you can compare.
Notice the difference from Section 3: there, a different starting capital stock gave a different path to the same steady state. Here a parameter changes, which moves the steady state itself. The capital stock still has to get there through the law of motion.
A higher level is not permanently faster growth
Does a richer steady state imply permanently faster growth?
The level of output per worker is how much each worker produces in a year. Its growth rate is the change relative to the previous year, gt = (yt − yt−1) / yt−1. All four paths below start at k = 8, y = 4 in year 0 and change one assumption at a time. Predict first.
The Golden Rule: output is not consumption
Which steady state supports the most consumption per worker?
People consume goods; capital is useful because it helps produce them. A larger capital stock produces more, but it also needs more replacement investment every year just to stay the same size. At a steady state, investment equals depreciation, so the consumption that can be sustained is c* = f(k*) − δ k*: output minus replacement.
Getting there: the transition cost
The Golden Rule compares steady states. An actual economy has to move from where it is. During the transition, actual consumption is ct = (1 − s) yt, not f(k) − δk, because investment is also building or running down capital.
Explore the transition. Start at the s = 0.20 steady state and set the new saving rate to 1/3. Compare consumption in year 0 with its later level.
Productivity, ideas and AI
What would AI have to change to raise living standards, and what would it have to change to raise growth persistently?
The model gives three separate channels. Each one predicts a differently shaped output path, so “AI raises growth” is not yet a claim the model can evaluate. You have to say which channel operates.
More practice: classify AI scenarios
More physical capital
Servers, data centers and related equipment raise k. The economy moves along an unchanged production curve. This costs investment and faces diminishing returns.
A one-time rise in productivity
A tool lets the same workers and equipment produce more, once. A rises to a new level and stays there. Output jumps, capital follows, and growth returns to zero.
Faster production of ideas
If AI made useful, adopted improvements arrive every year, A would keep rising. Only this channel produces continuing growth in output per worker in the model.
What one study measures
The course reading, Brynjolfsson, Li and Raymond, Generative AI at Work (working paper, November 2024 version), studies the staggered introduction of an AI assistant among 5,172 customer-support agents. It reports an average 15 percent increase in issues resolved per hour, with larger gains for less experienced and lower-skilled agents.
Self-check: what does that finding establish, and what does it not?
It measures labor productivity on one task, in one firm’s setting: issues resolved per hour. That is evidence for the second channel (a gain in performing an existing task).
It does not measure economy-wide TFP, aggregate output, or continuing growth. An aggregate claim would also need evidence on how widely similar gains apply, what equipment, training and other inputs they require, whether output quality changed, and whether productivity keeps improving after the initial adoption. A permanent growth claim needs evidence for the third channel specifically.
The model paths on this page are illustrative calculations, not estimates from the study.
Diagnose an unknown change
Can you work backward from the consequences to the cause?
An economy is at its baseline steady state. At the start of year 5, one thing happens, and you see only its consequences. Use the year-5 numbers first: which relationships could have produced an immediate change in those quantities, and which could not? Then use the later years to decide.
Explain it as an economist would
Write a short explanation of the case above. This site cannot grade written answers, so compare yours with the checklist and the model explanation. Your text stays in this browser tab.
A template for any new scenario
What should an explanation using the model contain?
Optional · Click to learn an important extension
Build a model with a growing workforce
Can you adapt the one-year ledger when the number of workers changes?
So far, the workforce has stayed fixed. Here you will change that assumption yourself. Begin with numbers, then build a general rule. The experiments in sections 1–8 still use a fixed workforce.
Try each step before opening a hint. “Show explanation” becomes available after you save an attempt; it explains the approach without filling in the answers. Your work stays in this tab until you reload the page. Written answers are not automatically graded.