ECON7074 Chap.5 The Production Model and Productivity
The Production Model and Productivity
This chapter builds the static production model and uses it to ask how much of the gap between rich and poor countries capital can explain. It sets up a closed economy producing one good from capital and labour with a Cobb-Douglas technology, shows why the function has constant returns overall but diminishing returns to each input, and writes it per person as y = Ak^(1/3).
Firms maximise profit in competitive markets, so the real rental rate equals the marginal product of capital and the real wage equals the marginal product of labour. Solving the model gives output, factor prices and income shares of one third and two thirds with zero profit.
Development accounting then compares predicted and actual incomes, treats total factor productivity as a residual, and finds that productivity explains most of the gap, before weighing human capital, technology and institutions as explanations. Every calculation uses invented numbers chosen to make the method transparent.
What this chapter covers
- 01
Cobb-Douglas technology and constant returns
- 02
Diminishing returns to capital per person
- 03
Profit maximisation and marginal products
- 04
Equilibrium wages, rents and factor shares
- 05
Development accounting and the TFP residual
- 06
Human capital, technology and institutions
Worked example · free
Splitting a twelve-fold output gap between capital and the productivity residual
- +1With equal A, the predicted output ratio is (1/27)^(1/3) = 1/3, so capital per person explains a factor of 3.
- +1The actual ratio is 1/12, so the productivity ratio is (1/12) / (1/3) = 1/4: Country B's TFP is a quarter of the leader's.
- +1Check the split multiplies back: 3 times 4 = 12, the full income gap.
- +1In logs, ln 12 (about 2.48) equals ln 3 (about 1.10) plus ln 4 (about 1.39), so capital explains about 44% and the residual about 56%.
Key terms
- Constant returns to scale
- Property of a production function whereby scaling every input by the same factor scales output by exactly that factor.
- Marginal product of capital
- Extra output produced by one more unit of capital with labour held fixed; for Cobb-Douglas it equals one third of Y/K when a is 1/3.
- Real rental rate
- Price of renting a unit of capital measured in goods, which competitive firms push to equal the marginal product of capital.
- Development accounting
- Method that uses a production model to split cross-country income gaps into parts due to inputs and to productivity.
- Human capital
- Stock of skills and knowledge, built through education and training, that raises how much each worker can produce.
The Production Model and Productivity FAQ
Why does the production model use a Cobb-Douglas function?
Cobb-Douglas is simple to solve and matches a key fact: labour's share of income has been roughly two thirds and capital's one third over long periods. With exponents of one third and two thirds summing to one, the function has constant returns to scale, while each input separately shows diminishing returns.
How do firms decide how much capital and labour to use?
Each competitive firm chooses inputs to maximise profit, measured in units of output. It keeps adding capital while its marginal product exceeds the real rental rate and keeps hiring while the marginal product of labour exceeds the real wage. At the optimum, MPK equals r and MPL equals w.
Why are profits zero in the production model?
With constant returns and competitive markets, paying each input its marginal product uses up all output exactly. Capital receives one third of output as rent and labour two thirds as wages, so nothing is left as economic profit, and factor incomes add up to GDP as in the national accounts.
What is total factor productivity and why is it called a residual?
Total factor productivity is the parameter A that scales output for given capital and labour. It is not observed directly, so economists compute it as whatever part of output per person the model cannot attribute to capital per person. Because it is backed out as a leftover, it also absorbs any measurement error.
How much of the gap between rich and poor countries does capital explain?
Development accounting in the course finds that differences in capital per person account for about one third of the output gap between the richest and poorest countries, while productivity accounts for about two thirds. Rich countries have more capital, but they mainly differ in how efficiently they use capital and labour together.
Exam move
Practise this chapter by solving the model from scratch on blank paper. Write the production function, the profit problem and the two first-order conditions, then derive r = (1/3)Y/K and w = (2/3)Y/L and show that the payments add up to output. Rework the solution with your own invented values for A, K and L until the arithmetic is quick.
Then drill development accounting: take a capital ratio, cube-root it, compare with an output ratio, and back out productivity. Finish each session by writing two sentences explaining why productivity, not capital, carries most of the cross-country gap, naming one candidate explanation and its limit. Keep the one-third and two-thirds evidence on your summary sheet.
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