# CRS assumption in Solow model

Can you explain why does Solow growth model assume "constant returns to scale"?

From an economic point of view, the assumption of Constant Returns to Scale can have several reasons, and they are not specific of the Solow model.

I can quote what Solow himself says about Constant Return to Scale assumption in his classical paper A contribution to the Theory of Economic Growth$$^1$$:

About production all we will say at the moment is that it shows constant returns to scale. Hence the production function is homogeneous of first degree. This amounts to assuming that there is no scarce non augmentable resource like land. Constant returns to scale seems the natural assumption to make in a theory of growth. (p. 67)

But in Solow model the assumption of Constant Returns to Scale has a great importance from a formal point of view, as it allows a formulation of the model that is mathematically easy to deal with, at least graphically.

In short, it allows to transform the model into an ordinary differential equation that is possible to solve graphically and allows the usual analysis of steady state and comparative dynamics analysis.

And which can also be analytically solved if we assume particular forms of the production function, in particular a Cobb-Douglas.$$^2$$

The reason why Constant Returns to Scale allow this simplification is that they consent to write the so-called intensive production function, and they transform the model into a form that is formulated in terms of per capita variables.

Consider a generic production function that has the usual properties

$$Y=F(K, L)\;\; (1)$$

where the variables have the usual meaning: $$Y$$ is output, $$L$$ is labor, $$K$$ is capital. This is a function of two variables.

Constant Returns to Scale mean that if you multiply each production factor, that is $$K$$ and $$L$$, by a positive constant $$\lambda$$ the output also is multiplied by $$\lambda$$, that is:

$$\lambda Y= F(\lambda K, \lambda L)\;\; (2)$$.

If we take a particular value of $$\lambda$$, $$1/L$$, we can rewrite $$(2)$$ as $$\frac{1}{L}Y = F(\frac{1}{L}K, \frac{1}{L}L)\;\; (3)$$

and defining the per capita variables $$y\equiv Y/L$$ and $$k\equiv K/L$$ we can rewrite $$(3)$$ as

$$y=F(k,1)$$

or, deleting the $$1$$ in the brackets

$$y=f(k)\;\;(4)^3$$.

This last equation is the so-called intensive production function, which links the output per capita to the capital per capita, irrespective of the absolute levels of $$K$$ and $$L$$.

All that allows us to write the model in terms of the per capita variables only. Thus, we have a dramatic simplification, because the production function now encompasses only one independent variable $$k$$, not two variables, $$Y$$ and $$K$$.

As a consequence, the model can be reduced to an ordinary differential equation, the so called fundamental equation of growth, that is a differential equation which involves only one unknown function, $$k(t)$$ (that is $$k$$ as a function of time):$$^4$$

$$\dot k(t)= sf(k(t))- (n+d)k(t)\;\;(5)$$

where $$s$$ is the saving rate, $$d$$ is depreciation, and $$n$$ is the rate of growh of population.

Setting $$\dot k=0$$ in equation $$(5)$$ we have the equation that gives us the value of $$k$$ for which $$k$$ is constant, which we denote $$k^*$$.

$$\dot k(t)= sf(k(t))- (n+d)k(t)=0\;\;(5')$$

Solow then shows how to solve qualitatively this ordinary differential equation $$(5')$$, with the usual nice graph below. Notice that a two dimension graph is possible because we have now only one indipendent variable, $$k$$.

The solution of the differential equation $$(5')$$ is $$k^*$$, which we know as the steady state value of $$k$$ of the model.

$$^1$$ Solow, Robert M., ‘A Contribution to the Theory of Economic Growth’, The Quarterly Journal of Economics, Vol. 70, No. 1 (Feb., 1956), pp. 65-9.

$$^2$$ Of course, the formulation of a model mathematically easy to handle is not a sterile, end in itself, formal exercise. The aim of Solow is to show that with more general assumptions, one can go beyond the conclusions of the Harrod-Domar model, which he criticizes in the introduction of his paper. His focus is to show that the conclusions of the Harrod-Domar model, that in the long run the economic system is at best on a 'knife-edge’ equilibrium, or is condemned to instability, depend crucially on too restrictive assumptions about the production function. Instead, Solow shows that with a more general production function (the so-called neoclassical production function with constant return to scale) the economic system can reach a steady state long run equilibrium, which under some assumptions is stable: “The basic conclusion of this analysis is that[…]there may not be in fact […] any knife-edge. The system can adjust to any given rate of growth of the labor force, and eventually approach a state of steady proportional expansion.“ (Solow, cit. p.73)

$$^3$$ The lowercase $$f$$ here, instead of the capital $$F$$, denotes that now we have a different function, a function of one variable, k, instead of a function $$F$$ of two variables, k and another variables fixed at 1.

$$^4$$ In case we had two unknown functions, we have a partial differential equation, that would be much more complicated to solve, if not impossible.

One way to conceptualize constant returns to scale is by envisioning multiple plants employing the same technology, where it is feasible to initiate as many plants as desired to produce the desired output. This assumption is consistent with a competitive setup. Additionally, in the Solow model without technological growth, constant returns to scale (CRS) ensures that the rate of growth of aggregate output equals the population growth rate in the steady state.