# Solow Model

## Summary

Capital based model that can explain a lot of, although not all the differences between countries.

## Assumptions

1. Capital has five key characteristics: 1. It is productive 2. It is produced 3. It is limited (rival) 4. It can earn a return 5. It wears out (depreciates)
2. Production function Y=F(K,L) where we assume constant returns to scale (i.e. F(zK,zL)=zF(K,L) & thus y=f(k) and we assume diminishing marginal product.
3. Cobb Douglas Production function has an additional assumption where where α is capital's share of income where F(K,L)=AK^αL^(1-α) implies y=Ak^α. In a competitive economy factors of production will be paid their marginal products (i.e. wage=MPL, rent=MPK) and α=MPK*K/Y and 1-α=MPL*L/Y
4. Constant quantity of labour, L (or at least exogenously determined population & growth rate).
5. Constant production function i.e. no change in technology A
6. Capital accumulation is modelled as Δk=i-d where i=γy and d=𝛿k means  Δk=γf(k)-𝛿k

## Model/Theory

Steady state can be solved for as γf(k)=𝛿k

• Kss increases with an increase in γ, or a decrease in 𝛿

ss=A(kss)^α

=A^(1/1-α)( γ/𝛿)^(α/(1-α))

Savings=government savings + private savings

## Predictions

Convergence Towards the Steady State (Solow as a Theory of Relative Growth Rates)

• If two countries have the same rate of investment but different levels of income, the country with lower income will have higher growth.

• If two countries have the same level of income but different rates of investment, then the country with a higher rate of investment will have higher growth.

• A country that raises its level of investment will experience an increase in its rate of income growth.

## Evidence

• Solow Model as a Theory of Income Differences is not very good (Figure 3.7).
• Solow Model as a Theory of Relative Growth Rates - further away from steady state the fastest a country will grow.

## Evaluation

• Before the 19th century land was a more important factor of production than capital.
• What matters are differences in the rate of investment, which depends upon the savings rate. Critics of the Solow Model argue the savings rate is endogenous and that people save more when they are rich rather than they are rich because they save more i.e. Solow doesn't explain why there are differences in investment rates.
• 'Can't afford to save' argument doesn't work for countries just above subsistence. Although perhaps a broken savings line could potentially lead to multiple steady state equilibria.
• Model doesn't explain long-term growth - as long-term steady state has no growth and shifts in savings rate are only temporary and finite.
• If the model includes population growth than you can get capital dilution where   Δk=γf(k)-𝛿k-nk where again we can solve for yss=A(kss)^α. However the model, with only a 34% predicted difference, is insufficient to explain the large differences in gdp/capita that we see in the data.