Harrod–Domar model

Let Y represent output, which equals income, and let K equal the capital stock. S is total saving, s is the savings rate, and I is investment. δ stands for the rate of depreciation of the capital stock. The Harrod–Domar model makes the following a priori assumptions:

${\displaystyle \ Y=f(K)}$	1: Output is a function of capital stock only (labor is irrelevant).
${\displaystyle \ {\frac {dY}{dK}}=c\Rightarrow {\frac {dY}{dK}}={\frac {Y}{K}}}$	2: The marginal product of capital is constant; the production function exhibits constant returns to scale. This implies capital's marginal and average products are equal.
${\displaystyle \ f(0)=0}$	3: Capital is necessary for output.
${\displaystyle \ sY=S=I}$	4: The product of the savings rate and output equals saving, which equals investment
${\displaystyle \ \Delta \ K=I-\delta \ K}$	5: The change in the capital stock equals investment less the depreciation of the capital stock

Derivation of output growth rate:

${\displaystyle {\begin{aligned}&c={\frac {dY}{dK}}={\frac {Y(t+1)-Y(t)}{K(t)+sY(t)-\delta \ K(t)-K(t)}}\\[8pt]&c={\frac {Y(t+1)-Y(t)}{sY(t)-\delta \ {\frac {dK}{dY}}Y(t)}}\\[8pt]&c(sY(t)-\delta \ {\frac {dK}{dY}}Y(t))=Y(t+1)-Y(t)\\[8pt]&cY(t)\left(s-\delta \ {\frac {dK}{dY}}\right)=Y(t+1)-Y(t)\\[8pt]&cs-c\delta \ {\frac {dK}{dY}}={\frac {Y(t+1)-Y(t)}{Y(t)}}\\[8pt]&s{\frac {dY}{dK}}-\delta \ {\frac {dY}{dK}}{\frac {dK}{dY}}={\frac {Y(t+1)-Y(t)}{Y(t)}}\\[8pt]&sc-\delta \ ={\frac {\Delta Y}{Y}}\end{aligned}}}$

A derivation with calculus is as follows, using dot notation (for example, ${\displaystyle \ {\dot {Y}}}$) for the derivative of a variable with respect to time.

First, assumptions (1)–(3) imply that output and capital are linearly related (for readers with an economics background, this proportionality implies a capital-elasticity of output equal to unity). These assumptions thus generate equal growth rates between the two variables. That is,

${\displaystyle \ Y=cK\Rightarrow log(Y)=log(c)+log(K)}$

Since the marginal product of capital, c, is a constant, we have

${\displaystyle \ {\frac {d\log(Y)}{dt}}={\frac {d\log(K)}{dt}}\Rightarrow {\frac {\dot {Y}}{Y}}={\frac {\dot {K}}{K}}}$

Next, with assumptions (4) and (5), we can find capital's growth rate as,

${\displaystyle \ {\frac {\dot {K}}{K}}={\frac {I}{K}}-\delta \ =s{\frac {Y}{K}}-\delta \ }$

${\displaystyle \ \Rightarrow {\frac {\dot {Y}}{Y}}=sc-\delta \ }$

In summation, the savings rate times the marginal product of capital minus the depreciation rate equals the output growth rate. Increasing the savings rate, increasing the marginal product of capital, or decreasing the depreciation rate will increase the growth rate of output; these are the means to achieve growth in the Harrod–Domar model.