Why is the Cobb-Douglas production function so popular?

Question

As relatively novice quantitative analyst/ Cost analyst, Ive been asked to estimate the level of a given organizations productivity more than once, and then forecast for the next couple of periods. The place I work at is a relatively small non-profit (around 30 people) dedicated to food bank donation distribution and volunteer solicitation, so Im not sure if firm size has something to do with this.

Most of the time I've been asked for specific units and not percentage changes or elasticities, so I'm forced to present one of two production functions.

1. $$f(x_1,...,x_n)=\Sigma_{i=1}^n\beta_i x_i$$
2. $$f(x_1,...,x_n)=\gamma \min(x_1,...,x_n)$$

Yet when I read economic literature I see the cobb douglas (or some variation of it like the stone-gerry) being used all the time.

I know it has the property of mathematically showing diminishing returns to scale for a single factor of production, however Im having difficulty seeing it in my line of work. Is it a production function exclusive to manufacturing of real goods?

Answer 1

The reason why Cobb Douglas production functions are so popular stem from the fact that the following assumptions are satisfied while remaining statistically rigorous¹:

Recall the Cobb- Douglas production function form:

$$F(K,AL)=K^{\alpha}(AL)^{1-\alpha}$$

where $0<\alpha <1$ (i.e. the share of output that goes to capital)

$1)$ Positive marginal products:

$${\partial{F(K,AL)}\over{\partial K}}>0 \space , \space\space{\partial{F(K,AL)}\over{\partial (AL)}}>0$$

$2)$ Diminishing Marginal Products (as you already mentioned)

$${\partial^2{F(K,AL)}\over{\partial K^2}}<0 \space , \space\space{\partial^2{F(K,AL)}\over{\partial (AL)^2}}<0$$

$3)$ Constant Returns to Scale (this is how most production processes work)

$$F(\lambda K, \lambda AL)= \lambda F(K, AL)$$

for any $\lambda \ge 0$ (E.g. "Double input $\Rightarrow$ Double output")

Hope this helps!!

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_{¹ Source: https://en.wikipedia.org/wiki/Cobb%E2%80%93Douglas_production_function}

Answer 2

As you hint at in your question, the real underlying reason (in my opinion) behind the C-D production function's popularity is mathematical convenience. The fact that the sum $\alpha + \beta$ is a nice, intuitive representation of "returns to scale" is very convenient.

I think of its use as similar to the use of exponential or power utilities in mathematical finance. Unrealistic? Perhaps, but oh so friendlier to work with.

Answer 3

The Cobb-Douglas production function $^1$

$$Y=F(K,L)=AK^{\alpha}L^{1-\alpha}\; 0<\alpha<1, A>0 \qquad (1)$$

as pointed out in the other answers, is very popular because of several formal properties, that make it convenient both from an economic and a mathematical point of view.

To recall some of them:

1. homogeneity of degree one (constant returns to scale);
2. decreasing marginal product of both factors;
3. map of the isoquants strictly convex with respect to the origin;
4. elasticity of substitution between factors constant and equal to $1$.

The last property implies that in perfect competition (with the prices of capital and labor equal to the value of their marginal productivity), the share of capital and labor are constant and equal to, respectively, $\alpha$ and $1-\alpha$.

The Cobb-Douglas has been used in many theorical models, among them models of economic growth.

Therefore, I want to add here an example, coming from economic growth theory, that illustrates the mathematical convenience of the Cobb-Douglas (and also the beauty of the growth model in that case): the use of the Cobb-Douglas in the Solow growth model, that is, how the model dramatically simplifies and becomes mathematically treatable using a Cobb-Douglas.

With a generic form of the production function the Solow model cannot be solved analytically, but only qualitatively through a graph.

Let’s recall the fundamental equation of growth of the Solow model$^2$:

$$\dot k= sf(k)-(n+d)k \qquad (2),$$ where the variables have the usual meaning: $k=K/L$, $s$= saving rate, $n$=rate of growth of population, $d$= rate of depreciation of the capital, $f(k)$ is the intensive production function with the usual properties.

Equation $(2)$ is the differential equation governing the dynamic of capital accumulation in the model, which in turn determines the dynamics of the other endogenous variables.

This equation can be solved qualitatively, through the usual graph of the Solow model:

Through this graph it is possible to analyze many important features of the model, but we haven't any mathematically explicit, analytical, solution of the fundamental equation $(2)$.

If, instead of using a generic production function, we use a Cobb-Douglas, the model becomes extremely easy to solve from a mathematical point of view, as the fundamental equation reduces to a well- known type of differential equation, the Bernoulli equation, which is one of the (few) types of differential equations that we know how to solve.

Consider again the Solow model, in which now we use a Cobb-Douglas function

$$Y=K^{\alpha}L^{1-\alpha}\quad (1')$$

where we have set, for simplicity, $A=1$. As it is homogeneous of degree $1$, we can write its intensive form, which is :

$$f(k)= k^{\alpha}.$$

Substituting this last function in the place of the generic function $f(k)$, and setting for simplicity $d=0$, equation $(2)$ becomes

$$\dot k= sk^{\alpha}-nk \qquad (2').$$

This type of differential equation is called Bernoulli equation and a general method for solving equations of this type exists.$^3$

The solution of equation $(2')$, together with an initial condition $k(0)=k_0$, is

$$k(t)= [(k_0 ^{1-\alpha} -s/n )e^{-n(1-\alpha)t}+s/n]^{1/(1-\alpha)}\qquad (3)$$

Equation $(3)$ describes explicitly the evolution of the capital-labor ratio $k$ in time, and shows that, as time $t$ goes to infinity, $k$ converges to the steady state value $(s/n)^{1/(1-\alpha)}$.$\;^4$

Therefore, also the stability of the steady state equilibrium is established: if the production function is a Cobb-Douglas, not only the existence of the steady state equilibrium, but also its stability can be analytically proved.

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$^1$ ^{This is the form introduced by Cobb and Douglas in their paper of 1928: Cobb, C.W. and Douglas, P.H. (1928) A Theory of Production, American Economic Review, 18, 139-165.}

$^2$ ^{Here we have set $A=1$ for simplicity.}

$^3$ ^{The general form of a Bernoulli equation is $y'+p(t)y+q(t) y^{\alpha}=0$, where $p(t)$ and $q(t)$ are known continuous functions on an open interval. A Bernoulli equation can be reduced, by means of a substitution, to a linear non homogeneous equation. See also, for this substitution in the Cobb-Douglas case and a discussion of the Solow model with a Cobb-Douglas, Gandolfo G., Economic Dynamics, Springer, 2009}

$^4$ ^{The value of the steady state $k$, $k^*$, can be derived from equation $(2')$ setting $\dot k=0$.}

Answer 4

The Cobb-Douglas production function is so popular, only because it is one of the very few functions for which you can compute explicitly input demand (and output supply) functions. The Cobb-Douglas production function is usually used at the bachelor level (lectures, exams and exercises) because we can solve the system of first order conditions. However, this functional form is very restrictive, and its validity is empirically rejected. So that at master levels, we restate microeconomic theory using unrestricted forms for the production function (see e.g. the Mas Colell et al. textbook), and use either the implicite function theorem or duality theory for deriving comparative static results.

Answer 5

Okay I think I made a lot of implicit assumptions because I was confused by your functions in the last answer, which made my answer very confusing itself. So I try to be a bit more explicit this time. I will only consider your first function.

Now if this is a production function, then the $x_i$ are inputs, and you only have one output. Now your production function has the following properties: $\frac{df}{dx_i}=\beta_i$ and $\frac{df}{dx_jdx_i}=0$. This implies that your inputs are completely independent from each other. You can reach the same output with only $x_1$ or with only $x_2$. This makes no sense if your inputs are Capital and Labor (at least not until we have fully automized production where you don't need a human anywhere in the production process). Because if either of them are 0 in the real world, you would get no output.

This consideration lead me to believe that your $x_1$ model methods of production, which implies that they already contain mixes of Capital and Labor. In that case you would simply optimize over these different methods of production, and pick the best one. The one with the highest $\beta_i$ to cost ratio. Because at a company level, marginal costs would most likely be constant.

This is a reasonable model, from a companies perspective. You can only choose between those production methods, so the fact that you mashed together Capital and Labor does not concern you. You optimize the function and pick the best method of production given fixed costs. The (presumed) simplification is, that you don't consider how expansion or production changes the marginal cost within a production method.

(If the expansion would be on a economic scale, then you would increase the wage of workers by employing more of them, which would at some point result in you picking a more capital heavy production method)

An economists approaches this from a different angle: They are interested in the Capital/Labor ratio and not in the specific method of production. They want a function that takes Capital and Labor as input, selects the best method of production given that input, and returns an output. Their assumption is, that on this scale there are so many different methods of production, that you can brush over them and basically get a continuous function in Capital and Labor.

They want a model which has the property $\frac{df}{dx_jdx_i}>0$ which the Cobb-Douglas function provides.

You are basically comparing particle simulation with fluid simulation. The equations to model a single water molecule will be different from modelling a stream of water. And it might look like one doesn't have anything to do with the other.

The other possibility I thought of, was that this function is actually a cost function of producing the outputs $(x_1, ...,x_n)$ but then you have a fixed marginal cost of $x_i$ by definition. Which again, is an assumption that is reasonable to make on the micro level but not on the macro level.