The Solow Growth Model that I talked about in my previous post, Is India Operating Below Its Golden Rule Level of Capital? A Quick Numerical Exercise, like most economic models, has a mathematical base. Robert Solow used the Cobb-Douglas production function to build this model i.e. he assumed that the production function of an economy is of the form:
Q = F (K, L) =
A K α L β (this is the Cobb-Douglas production function)
Where Y = output, K = capital input, L = labour input and A = total factor productivity.
Solow also assumed that α + β = 1, with α and β both being constants with values lying between o and 1.
I will talk about the Cobb-Douglas function in detail in this post since it is important to understand the mathematical properties of this function clearly if one hopes to understand and apply the Solow model successfully to real world economies.
In my next post, I will discuss how these properties lend themselves to the Solow model.
The Desirable Properties of the Cobb-Douglas Production Function
In its most standard form, the Cobb-Douglas (CD) production function is expressed as we’ve shown above. Though Solow assumed that α + β must be = 1 for his growth model, this is not true for the standard Cobb-Douglas function. α + β can be < = or > 1.
The CD function has some desirable, mathematically appealing qualities that make it easy to use and a great choice for depicting production functions. Let’s see what these are.
1. The Cobb-Douglas production function exhibits Diminishing Returns to Labour and Capital; Marginal Products of both labour and capital are positive.
Production functions of real-world economies always tend to exhibit Diminishing Marginal Returns to Labour and Capital i.e. if you keep the amount of labour and level of technology constant, and keep adding additional units of Capital, the Marginal Product (MP) of each successive unit of capital will be lower than the previous one. Same for labour.
The Cobb-Douglas function exhibits diminishing marginal returns to both K and L. Let’s demonstrate.
Marginal Product of Capital (MPK) = ΔQ/ ΔK = A α K α-1 L β = A α L β/ K 1-α
Note: We know that α lies between 0 and 1. This means that α - 1 is negative. Remember (basic exponential math rule), a x
= 1/ a -x. This implies K α-1 = 1/ K 1-α
where 1 - α is positive.
This tells us that MPK for the Cobb-Douglas production function decreases with an increase in K (units of capital). Thus, it exhibits diminishing marginal returns to capital. The same can be demonstrated for labour. This property allows the CD function to realistically depict real life economies.
Also, as in real-world economies, the marginal products of both labour and capital are positive (as long as L, K are positive) for the CD production function.
2. The Cobb-Douglas function meets all the Inada conditions.
The Inada conditions are assumptions about the shape/properties of a production function that guarantee the stability of the economic growth path in the Solow model. Read more here (Wikipedia link).
What this essentially means is that the Inada conditions guarantee that a unique, stable steady-state exists in the Solow model (where s f (k*) = (γ + n + g) k*). Read my post Is India Operating Below Its Golden Rule Level of Capital? A Quick Numerical Exercise for more on the steady-state.
Remember, the Inada conditions are purely technical, mathematical assumptions. They cannot be observed empirically. Their purpose is to provide us an ideal, theoretical, mathematically robust production function, a model based on which can be used to guide policy analysis for real-world economies.
We’ve already discussed the assumptions regarding positive marginal products and diminishing marginal returns (these are part of the Inada conditions). In addition, the Cobb-Douglas function also satisfies the following conditions.
MPL and MPK approach infinity as the amount of L and K (respectively) approach zero, while they approach zero as L and K (respectively) approach infinity.
3. α and β represent the Output Elasticities of Capital (K) and Labour (L) in the Cobb-Douglas function; these elasticities are constant.
The two properties mentioned above make the Cobb-Douglas function eligible for representing production functions that can depict stable, growth paths for economies. However, the CD function also has other qualities that make it mathematically appealing and relatively easy to work with. The first of these deals with the output elasticities of labour and capital.
The output elasticity of Capital (K) is the % change in output (Q) in response to a 1% change in K (labour and technology are held constant).
Output Elasticity of Capital = (ΔQ/Q) / (ΔK/K)
For the CD
function, Q = A K α L β
ΔQ/ ΔK = A
α K α-1 L β (I’ve used a basic differentiation rule here)
This
implies:
ΔQ/ ΔK = A
α K α L β/ K = α (A K α L β/ K)
= α Q/K
So we get, ΔQ/ ΔK = α Q/K
Which implies that,
(ΔQ/Q) / (ΔK/K) = α
Hence proved, that in the Cobb-Douglas function, the output elasticity of capital is = the constant α.
Similarly, it can be demonstrated that the Output Elasticity of Labour is = the constant β for the standard Cobb-Douglas function.
The mathematical appeal of this property is obvious. If α is = 0.45, just by looking at the Cobb Douglas function, you’ll know that if you increase Capital by 1%, output will increase by 0.45%, and that this relationship will hold no matter where you are on the production curve.
4. If α + β = 1, the Cobb-Douglas function exhibits Constant Returns to Scale; if α + β < 1, it exhibits Decreasing Returns to Scale; if α + β > 1, Increasing Returns to Scale.
Lets see what happens when we increase both L and K by a factor “z” in the Cobb-Douglas production function.
A (zK) α (zL) β
= A z α+ β K α L β = z α+ β (A K α L β) = z α+ β Q
So if we increase both L and K by a factor “z”, output (Q) increases by a factor
z α+ β. If α + β = 1, then the output increases by the same factor “z” and the production function exhibits constant returns to scale. If α + β < 1, then the output increases by less than “z” and the production function exhibits decreasing returns to scale. If α + β > 1, the function has increasing returns to scale.
This quality of the Cobb-Douglas function is also rather attractive. Just by summing α and β, you’ll know the factor by which output will increase if you increase both inputs (K and L) by say 10%.
5. Elasticity of Substitution of the Cobb-Douglas production function is = 1.
The Elasticity of Substitution (σ) of a production function measures the ease with which one factor input (labour or capital) can be substituted for the other. More specifically, it measures the % change in K/L in response to a 1 % change in the Marginal Rate of Technical Substitution (MRTS).
To understand MRTS, one must be familiar with isoquants. An Isoquant is a curve that shows all combinations of L and K that produce the same level of output. In Figure 1 below, curve Q=Q0 depicts an isoquant where all the combinations of L and K (on the curve) produce an output = Q0.
The slope of an isoquant at any given point gives us the Marginal Rate of Technical Substitution of one factor for the other (at that point). In figure 1, the slope of the isoquant at point “A” gives us the MRTS of labour for capital i.e. the amount of K that has to be given up for an additional unit of L, so that the output remains constant at Q0.
Figure 1: Isoquants and Marginal Rate of Technical Substitution
MRTSLK = ΔK/ΔL = MPL/MPK
The MRTSLK (MRTS of labour for capital) is also = MPL/MPK. This is because when we add an additional unit of L, output will increase by MPL. This increase in output has to be = the decrease in output due the reduction in capital, given that along an isoquant, output is constant.
ΔL x MPL = ΔK x MPK
⇒ ΔK/ΔL = MPL/MPK
Let’s now get back to the Elasticity of Substitution.
σKL = Δ(K/L)/(K/L) / ΔMRTSLK/MRTSLK
In a practical sense, σ can be thought of as an index that measures the rate at which diminishing marginal returns set in as one factor is substituted for the other. When σ is low, a change in MRTS, results in a small change in K/L. When σ is high, a change in MRTS, leads to larger change in K/L.
Using some simple calculus rules, it can be shown that the elasticity of substitution of the Cobb-Douglas production function =1.
Now that we are familiar with the appealing mathematical properties of the Cobb-Douglas function, we can move on to why it lends itself nicely to the Solow growth model.
Read my next post for that discussion.
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