Clifton Economics · Week 1 — Economic Growth: the Facts and the Machinery

The Question This Unit Is Built Around

Framing the topic

A worker in the United States produces, in a year, something in the region of thirty times what a worker in the poorest countries produces. That is not a gap in effort, and it is not mostly a gap in hours. It is a gap in what an hour of work is able to make.

Put beside it the other half of the picture. The rich countries have been growing at roughly two per cent a year, per worker, for well over a century — not spectacularly, but with remarkable steadiness. A handful of countries, mostly in East and Southeast Asia, have grown at several times that rate for decades on end. And a depressingly long list, concentrated in tropical Sub-Saharan Africa, has barely moved at all in thirty years.

Why two per cent is not a small number. At 2% a year, output per worker doubles roughly every 35 years and multiplies by seven over a century. At 6% — the kind of rate the growth miracles sustained — it doubles every 12 years. A country growing at 6% while its neighbour grows at 2% will be four times richer per worker in 35 years, and sixteen times richer in 70 — starting from the same place. Nothing else in economics compounds like this, which is why Robert Lucas wrote that once you start thinking about growth it is hard to think about anything else.

So there are really two questions, and they are not the same question. Why is any country rich at all? And why are some so much richer than others? A theory that answers one and not the other has not finished.

What this week does

Week 1 builds no model. It does something less exciting and more necessary: it sets out the facts a model will have to match, and assembles the machinery — a production function, a set of assumptions about it, a story about how factors get paid, and the algebra of growth rates — that Week 2 will put to work. Everything here is a component. Next week they get bolted together into the Solow model.

Know what needs explaining

The twelve stylised facts, and which of them are about a single country over time versus differences across countries.

Work with the production function

Cobb-Douglas, constant returns, diminishing marginal products, the Inada conditions — and why each assumption is there.

Price the factors

Derive MPK and MPL, show they are what capital and labour get paid, and that paying them exhausts output exactly.

Handle growth rates

Dots, logs, and the three rules that turn a messy product into a sum of growth rates.

Twelve Facts a Growth Model Has to Match

What needs explaining

In 1957 Nicholas Kaldor noticed that the post-war developed economies, for all their differences, shared a short list of regularities. They have held up well enough that they still function as the entry exam for any growth model: produce these, or you are not describing the world.

Figure 1 — the stylised facts, and the two jobs they set

The first six are Kaldor's, and they describe one rich country watched over time. Output per worker grows, continuously. Its growth rate is roughly constant. The capital-output ratio has no trend. Capital per worker grows. The rate of return on capital is near-constant. And the shares of national income going to labour and to capital have no trend.

Read together, those six describe something quite specific: an economy where capital and output grow at the same rate, so their ratio holds still, while the return on capital and the division of the spoils stay put. Economists call that a balanced growth path, and producing one is the first thing a model must do.

The remaining six are about the world rather than one country in it: the enormous rich-poor gap, the huge variation in growth rates, the miracles, the disasters — and then two that are easy to skip over and should not be. The rate of return on capital does not vary much across countries. Nor does the capital-output ratio.

Those last two are the awkward ones. If poor countries were poor simply because they have little capital, then capital there would be scarce, and scarce capital earns a high return. Investment should pour in until the returns equalise. Facts 11 and 12 say the returns already are roughly equal — which means a pure capital-shortage story cannot be the whole explanation, and something about productive efficiency has to be doing work. Hold that thought: it is where Week 2 ends up.

What a Production Function Is, and Is Not

The core object

Everything rests on one object: a function that turns inputs into output.

Y = F(K, L) = BK^αL^1−α

Y is aggregate output, K the capital stock, L the labour force, and B an index of how efficiently the two are being combined. The parameter α sits strictly between 0 and 1.

You will also meet it written Y = K^α(AL)^1−α, with the efficiency term attached to labour. The two are the same function: set B = A^1−α and they coincide exactly. The choice is about which later result you want to fall out neatly, not about economics.

What B is called

Productive efficiency, productivity, total factor productivity (TFP), or the level of technology. Four names, one idea: given K and L, how much Y comes out.

What B actually contains

Technology, but also management quality, institutions, the rule of law, how well capital is allocated to its best use. Anything that makes the same inputs yield more. It is measured as a residual, which is both its strength and its weakness.

None of this exists. There is no such thing as "the capital stock" — there are lathes and lorries and laptops and office blocks, which are not commensurable. There is no homogeneous "labour" either. The aggregate production function is a deliberate fiction, justified by whether its predictions survive contact with data, not by its realism. Be clear-eyed about that: it buys enormous analytical power at the cost of assuming away a great deal.

Two more pieces complete the setup. Capital wears out at rate δ, so with no investment at all K̇ = −δK. And the labour force grows at rate n, which may be positive, zero or negative — much of Europe and East Asia is now in the third case, and the model handles it without complaint.

Constant Returns, and the Trick They Buy

Assumption 1

A production function has constant returns to scale if scaling every input by the same factor scales output by that factor too:

F(λK, λL) = λF(K, L) = λY for every λ > 0

Check it on Y = 9K^1/3L^2/3. Triple both inputs:

9(3K)^1/3(3L)^2/3 = 3^1/33^2/3 × 9K^1/3L^2/3 = 3Y

It works because the exponents sum to one. That is the whole of why Cobb-Douglas is written with α and 1−α rather than two free parameters.

Figure 2 — scaling both inputs together

The economic content is a replication argument. If you build a second factory identical to the first, staff it identically, and the two do not interfere with each other, you should get twice the output. Constant returns says there is nothing special about being big — no economies of scale at the level of the whole economy, and no congestion either.

The trick

Set λ = 1/L. Then

F(K/L, 1) = Y/L → y = Bk^α

with y ≡ Y/L output per worker and k ≡ K/L capital per worker. A function of two variables has become a function of one.

This is the single most useful consequence of constant returns, and the whole of Week 2 depends on it. The Solow model is a story about one number, k, moving over time. Without constant returns there is no per-worker representation, no single state variable, and no diagram to draw. With our numbers: y = 9k^1/3, so a country with k = 64 produces y = 9 × 4 = 36 per worker, and one with k = 216 produces 54.

Diminishing Returns, and the Conditions at the Edges

Assumptions 2 and 3

The second assumption is that each factor has a positive but diminishing marginal product. More capital always helps; it just helps less and less.

Differentiate Y = 9K^1/3L^2/3 with respect to K:

MPK = ∂Y/∂K = 3K^−2/3L^2/3 = 3k^−2/3 > 0

∂MPK/∂K = −2K^−5/3L^2/3 < 0

Positive, and falling. The same two lines for labour give MPL = 6K^1/3L^−1/3 > 0 with ∂MPL/∂L < 0.

Figure 3 — output per worker, and the marginal product as its slope

The figure is worth sitting with, because the whole of Week 2 is a consequence of its shape. Moving from k = 64 to k = 512 means eight times as much capital per worker. Output per worker goes from 36 to 72 — it merely doubles. And the marginal product falls from 0.1875 to 0.0469, a quarter of what it was.

So a country can get richer by accumulating capital, but each additional machine buys less than the one before. That is the mechanism that will eventually stop growth dead in the Solow model, and it is why the model needs something other than capital to explain growth that does not stop.

The Inada conditions

The third assumption pins down the behaviour at the two extremes:

lim_k→0 MPK = ∞ and lim_k→∞ MPK = 0

Capital is unbelievably productive when there is almost none of it, and worth almost nothing when there is a vast amount. Both hold for 3k^−2/3 by inspection.

These look like technical housekeeping. They are doing real work. The first limit guarantees that an economy starting with almost no capital finds it worth accumulating — so a steady state is reached from below rather than the economy collapsing. The second guarantees that accumulation eventually stops being worth it — so the economy does not grow for ever on capital alone. Between them they force the sy curve and the break-even line in Week 2 to cross exactly once, at a positive level of k. Without the Inada conditions that crossing is not guaranteed, and the entire Solow diagram falls apart.

What Capital and Labour Actually Get Paid

Competitive markets

Assume all markets are perfectly competitive and factor markets are frictionless — firms take prices as given, and K and L can be adjusted instantly at no cost. A firm then chooses both inputs to maximise

π = pF(K, L) − (r + δ)K − wL

The cost of using a unit of capital for a period is r + δ: the return r its owner could have earned elsewhere, plus the δ of it that wears out in the process. Setting the two derivatives to zero and normalising p = 1:

MPK = r + δ and MPL = w

Supply is inelastic: the quantity of capital and labour in existence at a moment does not depend on what they are paid. So a vertical supply line meets a downward-sloping marginal-product curve, and the intersection sets the price.

Figure 4 — equilibrium in the two factor markets

Work it through with numbers. Take L = 2,500 workers and K = 160,000, so k = 64 and output is Y = 90,000. Then MPK = 0.1875 and MPL = 24.00, and the factor bill comes to

0.1875 × 160,000 + 24.00 × 2,500 = 30,000 + 60,000 = 90,000 = Y

Paying both factors their marginal products uses up output exactly. Nothing left over, nothing short. That is not a coincidence of these numbers — it is Euler's theorem, and it holds for any constant-returns function.

And it fixes the factor shares at α and 1−α. Capital's share is MPK·K/Y = 30,000/90,000 = 1/3 = α. Labour's is 2/3 = 1−α. Since α is a parameter, the shares are constant — whatever happens to K, L or B. That is Kaldor's sixth fact, and Cobb-Douglas delivers it automatically. Which is less impressive than it sounds: the fact is why economists picked this functional form. Labour's share in the real world has in fact drifted down since about 1980, which is one of the live reasons to be suspicious of the whole setup.

The Last Machine and the Average Machine

A distinction that recurs

Two different ratios get called "the return on capital", and keeping them apart matters.

Marginal product, MPK

∂Y/∂K = αBk^α−1. What one more unit of capital adds. This is what capital gets paid.

Average product, APK

Y/K = Bk^α−1. What the typical unit produces. Its reciprocal is the capital-output ratio.

Compare the two expressions and the relationship is immediate:

MPK = α × APK

At α = 1/3, the average product is always exactly three times the marginal product — at every level of capital, in every country, at every date.

Figure 5 — the two curves, and the fixed ratio between them

Why must the average exceed the marginal? Because the marginal product is falling. Every unit of capital installed before this one went in when capital was scarcer, and therefore when it was more productive. The average drags all those earlier, better units along with it, so it sits above the last one. The gap is wide precisely because diminishing returns are strong.

What the capital-output ratio tells you. K/Y is 1/APK, so a constant capital-output ratio means a constant average product, and since MPK = α·APK with α fixed, it means a constant return on capital too. Kaldor facts 3 and 5 are therefore not two independent observations — under Cobb-Douglas with competitive markets, either one implies the other. Spotting that saves you work: a question asking what happens to MPK when K/Y is constant has already been answered.

Growth Rates, Dots and Logs

The working notation

Macroeconomics works in continuous time, so variables are functions of t and the notation has to keep up. A dot means a derivative with respect to time:

Ẏ ≡ dY(t)/dt

Ẏ is an absolute rate of change — so many pounds of extra output per year. Usually the more interesting quantity is the proportional rate:

g_Y = Ẏ/Y

And then the single most useful identity in the course:

g_Y = d(ln Y)/dt

which is just the chain rule: d(ln Y)/dt = (d ln Y/dY)(dY/dt) = Ẏ/Y.

Three rules, all from that one identity

Take logs first, then differentiate. Logs turn products into sums, quotients into differences and powers into multiples, and differentiating then turns each term into a growth rate:

Products

Z = XY → g_Z = g_X + g_Y

Ratios

Z = X/Y → g_Z = g_X − g_Y

And for powers, Z = X^α gives ln Z = α ln X, hence g_Z = αg_X.

Applied to the production function itself, those three rules do in one line what would otherwise be a page of quotient rule:

Y = BK^αL^1−α → g_Y = g_B + αg_K + (1−α)g_L

Try it. Suppose productivity grows at 1.5%, the capital stock at 5% and the labour force at 2%, with α = 1/3. Then

g_Y = 1.5 + (1/3)(5) + (2/3)(2) = 1.5 + 1.667 + 1.333 = 4.5%

and output per worker grows at g_y = g_Y − g_L = 2.5%. Check that against the other route: g_y = g_B + αg_k = 1.5 + (1/3)(3) = 2.5%. Same answer, as it must be.

This decomposition is growth accounting, and it is the most-used tool in the field. Measure g_Y, g_K and g_L in the data, use α = capital's income share, and whatever is left over is g_B — the Solow residual. It is called a residual because it is not measured; it is what is unexplained after capital and labour have been accounted for. In most countries it comes out larger than the capital and labour contributions combined, which is a slightly embarrassing way of saying that the thing doing most of the work is the thing we understand least.

Why Logs Are Worth the Trouble

Reading growth off a chart

Anything growing at a constant proportional rate follows P(t) = P(0)e^nt. Take logs and the exponential becomes a straight line:

ln P(t) = ln P(0) + nt

Intercept ln P(0), gradient n. The growth rate stops being something you infer from a curve's bend and becomes something you read off as a slope.

Figure 6 — the same population, levels and logs

Suppose a country's population is 52.6 million in 2025 and is projected to reach 71.8 million by 2055. The formula inverts directly:

n = [ln(71.8) − ln(52.6)] / 30 = 0.31116 / 30 = 1.037% a year

At that rate the population doubles in ln2/n = 66.8 years. The familiar shortcut — divide 70 by the percentage growth rate — gives 67.5, overstating it by about eight months. The rule of 70 is an approximation because ln2 = 0.693, not 0.70.

The real payoff is comparison. On a log scale, two economies growing at the same rate plot as parallel lines whatever their levels, and a country that is catching up shows as a converging one. On a levels chart the rich country's curve rises so steeply that everything else is squashed flat against the axis and you cannot see anything at all. Essentially every growth chart you will meet in this unit has a log scale, and now you know why.

Where Output Goes, and What Gets Saved

The accounting

One piece remains. From national accounts,

Y = C + I + G + (X − M)

For now, assume no government and a closed economy — G = 0 and X = M = 0. Both are strong assumptions and both get relaxed later in the unit. What is left is

Y = C + I → S ≡ Y − C = I

In a closed economy with no government, saving and investment are the same thing, by definition rather than by any behavioural assumption. Output not consumed is output added to the capital stock.

Then the one genuinely behavioural assumption in the whole model: households save a constant fraction s of income.

S = sY = I and so C = (1 − s)Y

This is the weakest link in the model, and it is worth knowing that now. A constant saving rate is not a decision anybody makes — nobody wakes up and resolves to save 24% of their income regardless of interest rates, their age, or what they expect next year. Real saving responds to all of those. The assumption is here because it is simple enough to let the rest of the model be solved on a single diagram, and because the predictions turn out to be surprisingly robust to replacing it with something better. Third-year growth units replace it with optimising households and rederive most of this week from choice rather than assumption.

What is now on the table

Week 1 has assembled six components, and that is all it has done:

A production function

y = Bk^α, with constant returns, diminishing marginal products and the Inada conditions.

Competitive factor markets

MPK = r+δ, MPL = w, shares fixed at α and 1−α.

Two laws of motion

Capital depreciates at δ; the labour force grows at n.

A saving rule

S = sY = I, so investment is a fixed fraction of output.

Next week these get combined into a single differential equation in k, and that equation turns out to have exactly one resting point. Everything the Solow model has to say about why countries are rich or poor follows from where that point sits.