This is “The Solow Growth Model”, section 16.18 from the book Theory and Applications of Macroeconomics (v. 1.0).
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The analysis in Chapter 6 "Global Prosperity and Global Poverty" is (implicitly) based on a theory of economic growth known as the Solow growth model. Here we present two formal versions of the mathematics of the model. The first takes as its focus the capital accumulation equation and explains how the capital stock evolves in the economy. This version ignores the role of human capital and ignores the long-run growth path of the economy. The second follows the exposition of the chapter and is based around the derivation of the balanced growth path. They are, however, simply two different ways of approaching the same problem.
There are three components of this presentation of the model: technology, capital accumulation, and saving. The first component of the Solow growth model is the specification of technology and comes from the aggregate production function. We express output per worker (y) as a function of capital per worker (k) and technology (A). A mathematical expression of this relationship is
y = Af(k),where f(k) means that output per worker depends on capital per worker. As in our presentation of production functions, output increases with technology. We assume that f() has the properties that more capital leads to more output per capita at a diminishing rate. As an example, suppose
y = Ak^{1/3}.In this case the marginal product of capital is positive but diminishing.
The second component is capital accumulation. If we let k_{t} be the amount of capital per capita at the start of year t, then we know that
k_{t}_{+1} = k_{t}(1 − δ) + i_{t}.This expression shows how the capital stock changes over time. Here δ is the rate of physical depreciation so that between year t and year t +1, δk_{t} units of capital are lost from depreciation. But during year t, there is investment (i_{t}) that yields new capital in the following year.
The final component of the Solow growth model is saving. In a closed economy, saving is the same as investment. Thus we link i_{t} in the accumulation equation to saving. Assume that saving per capita (s_{t}) is given by
s_{t} = s × y_{t}.Here s is a constant between zero and one, so only a fraction of total output is saved.
Using the fact that savings equals investment, along with the per capita production function, we can relate investment to the level of capital:
i_{t} = sAf(k_{t}).We can then write the equation for the evolution of the capital stock as follows:
k_{t}_{+1} = k_{t}(1 − δ) + sAf(k_{t}).Once we have specified the function f(), we can follow the evolution of the capital stock over time. Generally, the path of the capital stock over time has two important properties:
To be more specific, the steady state level of capital solves the following equation:
k* = k*(1 − δ) + sAf(k*).At the steady state, the amount of capital lost by depreciation is exactly offset by saving. This means that at the steady state, net investment is exactly zero. The property of stability means that if the current capital stock is below k*, the economy will accumulate capital so that k_{t}_{+1} > k_{t}. And if the current capital stock is above k*, the economy will decumulate capital so that k_{t}_{+1} < k_{t}.
If two countries share the same technology (A) and the same production function [f(k)], then over time these two countries will eventually have the same stock of capital per worker. If there are differences in the technology or the production function, then there is no reason for the two countries to converge to the same level of capital stock per worker.
In this presentation, we explain the balanced-growth path of the economy and prove some of the claims made in the text. The model takes as given (exogenous) the investment rate; the depreciation rate; and the growth rates of the workforce, human capital, and technology. The endogenous variables are output and physical capital stock.
The notation for the presentation is given in Table 16.10 "Notation in the Solow Growth Model": We use the notation g_{x} to represent the growth rate of a variable x; that is, ${g}_{x}=\frac{\Delta x}{x}=\%\Delta x\text{.}$
There are two key ingredients to the model: the aggregate production function and the equation for capital accumulation.
Table 16.10 Notation in the Solow Growth Model
Variable | Symbol |
---|---|
Real gross domestic product | Y |
Capital stock | K |
Human capital | H |
Workforce | L |
Technology | A |
Investment rate | i |
Depreciation rate | δ |
The production function we use is the Cobb-Douglas production function:
Equation 16.1
Y = K^{a}(HL)^{1−}^{a}A.If we apply the rules of growth rates to Equation 16.1, we get the following expression:
Equation 16.2
g_{Y} = ag_{K} + (1 − a)(g_{L} + g_{H}) + g_{A}.The condition for balanced growth is that g_{Y} = g_{K}. When we impose this condition on our equation for the growth rate of output (Equation 16.2), we get
$${g}_{Y}^{BG}=a{g}_{Y}^{BG}+(1-a)({g}_{L}+{g}_{H})+{g}_{A}\text{,}$$where the superscript “BG” indicates that we are considering the values of variables when the economy is on a balanced growth path. This equation simplifies to
Equation 16.3
$${g}_{Y}^{BG}={g}_{L}+{g}_{H}+\left(\frac{1}{1-a}\right){g}_{A}\text{.}$$The growth in output on a balanced-growth path depends on the growth rates of the workforce, human capital, and technology.
Using this, we can rewrite Equation 16.2 as follows:
Equation 16.4
$${g}_{Y}=a{g}_{K}+(1-a){g}_{Y}^{BG}\text{.}$$The actual growth rate in output is an average of the balanced-growth rate of output and the growth rate of the capital stock.
The second piece of our model is the capital accumulation equation. The growth rate of the capital stock is given by
Equation 16.5
$${g}_{K}=\frac{I}{K}-\delta \text{.}$$Divide the numerator and denominator of the first term by Y, remembering that i = I/Y.
Equation 16.6
$${g}_{K}=\frac{i}{K\text{/}Y}-\delta \text{.}$$The growth rate of the capital stock depends positively on the investment rate and negatively on the depreciation rate. It also depends negatively on the current capital-output ratio.
Now rearrange Equation 16.6 to give the ratio of capital to gross domestic product (GDP), given the depreciation rate, the investment rate, and the growth rate of the capital stock:
$$\frac{K}{Y}=\frac{i}{\delta +{g}_{K}}\text{.}$$When the economy is on a balanced growth path, g_{K} = ${g}_{Y}^{BG}$, so
$${\left(\frac{K}{Y}\right)}^{BG}=\frac{i}{\delta +{g}_{Y}^{BG}}\text{.}$$We can also substitute in our balanced-growth expression for ${g}_{Y}^{BG}$ (Equation 16.3) to get an expression for the balanced-growth capital output ratio in terms of exogenous variables.
$${\left(\frac{K}{Y}\right)}^{BG}=\frac{i}{\delta +{g}_{L}+{g}_{H}+\frac{1}{1-a}{g}_{A}}\text{.}$$The proof that economies will converge to the balanced-growth ratio of capital to GDP is relatively straightforward. We want to show that if K/Y < ${\left(\frac{K}{Y}\right)}^{BG}\text{,}$ then capital grows faster than output. If capital is growing faster than output, g_{K} − g_{Y} > 0. First, go back to Equation 16.4:
$${g}_{Y}=a{g}_{K}+(1-a){g}_{Y}^{BG}\text{.}$$Subtract both sides from the growth rate of capital:
$${g}_{K}-{g}_{Y}={g}_{K}-a{g}_{K}-(1-a){g}_{Y}^{BG}=(1-a)\left({g}_{K}-{g}_{Y}^{BG}\right)\text{.}$$Now compare the general expression for ratio of capital to GDP with its balanced growth value:
$$K\text{/}Y=\frac{i}{\delta +{g}_{K}}\text{(generalexpression)}$$and
$${(K\text{/}Y)}^{BG}=\frac{i}{\delta +{g}_{Y}^{BG}}\text{(balancedgrowth)}\text{.}$$If K/Y < ${\left(\frac{K}{Y}\right)}^{BG}\text{,}$ then it must be the case that g_{K} > ${g}_{Y}^{BG}$, which implies (from the previous equation) that g_{K} > g_{Y}.
If we want to examine the growth in output per worker rather than total output, we take the per-worker production function (Equation 16.2) and apply the rules of growth rates to that equation.
$${g}_{Y\text{/}L}=\left(\frac{a}{1-a}\right){g}_{K\text{/}Y}+{g}_{H}+{g}_{A}\text{.}$$ (1 − a)g_{Y} = a[g_{K} − g_{Y}] + (1 − a)[g_{L} + g_{H}] + g_{A} = a[g_{K} − g_{Y}] + (1 − a)[g_{L} + g_{H}] + g_{A}.We then we divide by (1 − a) to get
$${g}_{Y}=\frac{a}{(1-a)}[{g}_{K}-{g}_{Y}]+{g}_{L}+{g}_{H}+\left(\frac{1}{1-a}\right){g}_{A}$$and subtract g_{L} from each side to obtain
$${g}_{Y}-{g}_{L}=\frac{a}{(1-a)}[{g}_{K}-{g}_{Y}]+{g}_{H}+\left(\frac{1}{1-a}\right){g}_{A}\text{.}$$Finally, we note that g_{Y} − g_{L} = g_{Y}_{/}_{L}:
$${g}_{Y\text{/}L}=\frac{a}{(1-a)}[{g}_{K}-{g}_{Y}]+{g}_{H}+\left(\frac{1}{1-a}\right){g}_{A}\text{.}$$With balanced growth, the first term is equal to zero, so
$${g}_{Y\text{/}L}^{BG}={g}_{H}+\left(\frac{1}{1-a}\right){g}_{A}\text{.}$$In this analysis, we made the assumption from the Solow model that the investment rate is constant. The essential arguments that we have made still apply if the investment rate is higher when the marginal product of capital is higher. The argument for convergence becomes stronger because a low value of K/Y implies a higher marginal product of capital and thus a higher investment rate. This increases the growth rate of capital and causes an economy to converge more quickly to its balanced-growth path.
Take the production function
Y = K^{a}(HL)^{1−}^{a}A.Now assume A is constant and $H={\left(\frac{B}{A}\right)}^{1\text{/}(1-a)}\times (K\text{/}L)\text{,}$ so
$$\begin{array}{ccc}\hfill Y& =& {K}^{a}{\left(L{\left(\frac{B}{A}\right)}^{1\text{/}(1-a)}(K\text{/}L)\right)}^{1-a}A\hfill \\ \hfill & =& {K}^{a}{\left({\left(\frac{B}{A}\right)}^{1/(1-a)}(K)\right)}^{1-a}A\hfill \\ \hfill & =& BK\text{.}\hfill \end{array}$$