# Solow Growth Model

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Categories: CategoryModel CategoryMacro CategoryGrowth

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Created: 2008-07-23

Suggested Citation: See license page.

Contents

## Introduction

Solow is the best-known neoclassical growth model and is based on the assumption of exogenous technological progress. Under the specifications of the model, policy (notably changing the savings rate) has no effect on long-run growth, rather countries converge to a 'steady state' growth rate as determined by exogenous parameters. (Cite Solow).

## Model

There is some aggregate production function $$ Y=F(K,L) $$ which has the following properties:

- Positive but diminishing returns to factor inputs:
$$ \frac{\partialF(K,L)}{\partialK} = F_K > 0 $$ and $$ \frac{\partial^2F(K,L)}{\partialK^2} = F_{KK} < 0 $$

$$ \frac{\partialF(K,L)}{\partialL} = F_L > 0 $$ and $$ \frac{\partial^2F(K,L)}{\partialL^2} = F_{LL} < 0 $$

Constant returns to scale: $$ F(\lambdaK,\lambdaL) = \lambdaF(K,L) \forall \lambda>0 $$

- Satisfies the INADA conditions:
- $$ \lim_{K \rightarrow 0} F_K = \infty $$ and $$ \lim_{K \rightarrow \infty} F_K = 0 $$
- $$ \lim_{L \rightarrow 0} F_L = \infty $$ and $$ \lim_{L \rightarrow \infty} F_L = 0 $$

This is a competitive economy, such that inputs are paid their marginal product ($$ r = F_K $$ and $$ w = F_L $$) and factor payments exhaust all output ($$ wL+rK = Y $$).

Labour-augmenting technical progress $$ A $$ can also be included in the aggregate production function, which now becomes $$ Y=F(K,AL) $$ where $$ AL $$ represents units of 'effective labour'. The 3 properties described above still hold for this new production function. It can be written in per worker terms as $$ y = f(k) $$ or in terms of effective labour units as $$ \tilde{y} = f(\tilde{k}) $$ . Lower-case letters denote a per-capita variable ($$ x = \frac{X}{L} $$) whereas a tilde indicates that a variable is per-effective-worker ($$ \tilde{x} = \frac{X}{AL} $$) .

$$ A_t = A_0e^{gt} $$ and $$ L_t = L_0e^{nt} $$ i.e. $$ A $$ grows at a constant rate $$ g $$ and $$ L $$ grows at a constant rate $$ n $$ .

The capital stock K grows at the rate of net investment: gross investment $$ I $$ minus depreciation $$ \deltaK $$ where $$ \delta $$ is a constant. The economy is assumed closed (in terms of the National Income Identity $$ Y = C+I+G+NX $$ , $$ G=NX=0 $$), so since households save at a constant rate $$ s $$ where $$ 0 \le s \le 1 $$ then $$ I = S = sY $$ . This gives the fundamental Solow equation of $$ \dot{K} = sY - \deltaK $$ , which can also be written in terms of effective labour units as $$ \dot{\tilde{k}} = s\tilde{y} - (n+g+\delta)\tilde{k} $$ .

## Solving

In the **steady state** equilibrium, all variables follow a **balanced growth path** such that $$ \dot{\tilde{k}} = \dot{\tilde{y}} = 0 $$ . Using the fundamental equation as stated above, this occurs where total saving equals depreciation, or $$ s\tilde{y} = (n+g+\delta)\tilde{k} $$ . The *existence* of an equilibrium is guaranteed if the INADA conditions are satisfied, and this equilibrium is *unique* if the production function is continuous and concave.

Per-capita variables grow at rate $$ g $$ and aggregate variables grow at rate $$ (n+g) $$ :

- $$ \frac{\dot{k}}{k} = \frac{dlogk}{dt} = \frac{dlogA}{dt}+\frac{dlog\tilde{k}}{dt} = g + 0 = g $$
- $$ \frac{\dot{K}}{K} = \frac{dlogK}{dt} = \frac{dlogL}{dt}+\frac{dlogk}{dt} = n + g $$

The parameters $$ s $$ and $$ \delta $$ affect the *levels* of $$ \tilde{k*} $$ and $$ \tilde{y*} $$ but not their *growth rates* , whereas $$ n $$ and $$ g $$ affect both levels and growth rates . Exactly how depends on the functional form of $$ Y = F(K,L) $$ .

### Transitional Dynamics and Convergence

The Solow model predicts that an economy will converge monotonically to its steady state level of capital $$ \tilde{k*} $$ . During the transition phase from the initial level of capital to the steady state level there is growth (either positive or negative) in capital per effective worker such that $$ \frac{\tilde{\dot{k}}}{\tilde{k}} \ne 0 $$ :

Too little initial capital means $$ \tilde{k} < \tilde{k}* $$ thus $$ s\tilde{y} > (n+g+\delta)\tilde{k} $$ so $$ \frac{\tilde{\dot{k}}}{\tilde{k}} > 0 $$

Too much initial capital means $$ \tilde{k} > \tilde{k}* $$ thus $$ s\tilde{y} < (n+g+\delta)\tilde{k} $$ so $$ \frac{\tilde{\dot{k}}}{\tilde{k}} < 0 $$

However, this growth is only short-term. Once the steady state level of capital has been achieved, the economy reverts to a balanced growth path and $$ \frac{\tilde{\dot{k}}}{\tilde{k}} = 0 $$ .

The growth rate $$ \frac{\tilde{\dot{k}}}{\tilde{k}} = \frac{s\tilde{y}}{\tilde{k}} - (n+g+\delta) = \frac{sf(\tilde{k})}{\tilde{k}} - (n+g+\delta) $$ is decreasing in $$ \tilde{k} $$ (from the concavity of $$ f(\tilde{k}) $$) . This implies that economies with lower levels of $$ \tilde{k} $$ relative to their steady state will grow faster, and leads to two theories of convergence:

**Absolute convergence**- poor countries grow faster since those who are further below $$ \tilde{k*} $$ experience faster growth. However, this is assuming that all countries have the same steady state.**Conditional convergence**- countries converge to their own steady state $$ \tilde{k_i*} $$ as determined by the parameters $$ (s,n,g,\delta) $$ , therefore countries with similar steady states should have similar growth rates (assuming similar initial capital).

### Golden Rule

The **golden rule** gives the savings rate which maximises consumption in the steady state, $$ s_{GR} $$ . Since it is assumed that $$ \tilde{y} = \tilde{c} + \tilde{i} $$ , $$ s_{GR} $$ is found by $$ \max_s \tilde{c}* = (1-s)f(\tilde{k*}) = f(\tilde{k*}) - (n+g+\delta)\tilde{k*} $$ which gives the FOC $$ \frac{\partialf(\tilde{k_{GR}*})}{\partial\tilde{k_{GR}*}} = (n+g+\delta) $$ , and from this $$ s_{GR} $$ can be determined.

## Example

Let the aggregate production function be Cobb-Douglas such that $$ Y = K^{\alpha}(AL)^{(1-\alpha)} $$ where $$ \alpha $$ is the factor share of capital and $$ \alpha<1$$ (i.e. diminishing returns to capital). Now $$ y = k^{\alpha}A^{1-\alpha} $$ and $$ \tilde{y} = \tilde{k}^{\alpha} $$ . The fundamental equation for capital accumulation is as before, meaning that along a balanced growth path $$ \frac{\tilde{\dot{k}}}{\tilde{k}} = \frac{\tilde{\dot{y}}}{\tilde{y}} = 0 $$ . Converting this to per-capita terms, the growth rate of per capita output is given by $$ \frac{\dot{y}}{y} = \alpha\frac{\dot{k}}{k}+(1-\alpha)\frac{\dot{A}}{A} = \alphag+(1-\alphag) = g $$ .

The steady-state *levels* of output and capital per effective worker are:

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

which means that steady-state output and capital per capita are:

- $$ k* = A\tilde{k*} = A\left(\frac{s}{(n+g+\delta)} \right)^{\frac{1}{(1-\alpha)} $$
- $$ y* = A\tilde{y*} = A\left(\frac{s}{(n+g+\delta)} \right)^{\frac{\alpha}{(1-\alpha)} $$

As noted above, the savings rate $$ s $$ and population growth $$ n $$ have no effect on the long-run growth rate, only on the level of output per capita (although changes to these parameters may have transitional effects as they determine an economy's steady-state and thus affect short-run growth). Technology $$ A $$ is exogenous to the model and has both level and growth effects - countries with higher technological growth rates and/or higher levels of technology grow faster in per-capita terms.

## Applications and Extensions

Human capital is included as a factor of production within a Solow framework in Human_Capital_Growth_Models .