# The Solow model and his 1956 paper

Reading through Robert Solow's 1956 paper, entitled "The Theory of Economic Growth", I was hoping to find his fundamental difference equation. I was wondering if the following equation is indeed that very equation, $$\dot{r}=s(r)F(r,1)-nr.$$ In my textbook, the fundamental difference equation is given by $$\dot{K}=f(K,L)-\delta K. \tag1$$ I am having a bit of trouble reading through Solow's paper (am a novice at economics), and was wondering if someone can locate the equation that is similar to $$(1)$$.

I will assume that you are talking about the Solow, R. M. (1956). A contribution to the theory of economic growth. The quarterly journal of economics, 70(1), 65-94. as to my best knowledge Solow did not published paper entitled the "Theory of Economic Growth" in 1956 and the first equation you use in your question is from the paper.

Moreover, I also think that you either made a mistake or your textbook assumed whole output is saved (which would be most unorthodox) because normally the equation (1) should be looking like this:

$$\dot{K} = s F(K,L) - \delta K$$

See for example the Romer's Advanced Macroeconomics or Blanchard's et al Macroeconomics an European Perspective (both authors use slightly different notation but the point is both textbooks show that change in capital depends on the fraction of saved output not the whole output).

In the 1956 paper the closest equivalent of equation (1) in your question would be equation (3) in Solow's paper:

$$\dot{K} = sF(K,L)$$

The reason why the depreciation is not explicitly there as a separate term is that Solow assumes that "Output is to be understood as net output after making good the depreciation of capital".

• @MB No it is not closer as it expresses the model per unit of effective labor, it would be closer if you would say that you are interested in $\dot{K/L} = sF(K/L,1) - \delta K/L$ - this is also valid specification of the model and also many textbooks include the one expressed per unit of effective labor but its not closer than the $\dot{K}= sF(K,L)$ to what you are asking for- in fact Solow equation 3 is equivalent he just assumes depreciation is already accounted for in $F(K,L)$. Also I am glad that you liked the answer in case you think it answered your Q consider accepting it.
– 1muflon1
Jun 16 '20 at 0:11
• @MB it does denote the proportion of saved output it is an inverse of marginal propensity to consume. $s=0.5$ would mean that half of what is produced is saved
– 1muflon1
Jun 16 '20 at 0:22
• @MB to my best knowledge none of the equations in Solow model is commonly referred to as fundamental difference equation but either the equation (3) or (6) can be considered fundamental. Indeed the only difference between them is that the second one expresses everything per unit of effective labor. If you seen that in textbook or paper they probably mean it fundamental with lowercase f, that is they just mean to say that it’s the critical part of the model from which result is derived not trying to give it a name. Also I think you mean differential equation not difference equation
– 1muflon1
Jun 16 '20 at 0:50
• @MB there is not much difference eq 2 says that rate at which stock of capital increases depends on proportion of saved output $Y$ (net of depreciation) and then equation 3 just show that output is equal to the production function F(K,L) (as Y=F(K,L) )so the production function is substituted in that’s it
– 1muflon1
Jun 16 '20 at 9:12
• @MB no it denotes the changes in capital $\dot{K}$ is the time derivative so it tells you how $K$ changes with respect to time. However, you can use it to determine what the future stock will be as the future stock of capital will be the old stock plus any change. By the way comments should be for clarification of answer but now we are moving to different questions you should post those as separate Q
– 1muflon1
Jun 16 '20 at 9:17