Economics at Zero Marginal Cost: Concept and Properties of Cobb-Douglas Production Functions

Sunday, July 18, 2021

Concept and Properties of Cobb-Douglas Production Functions

Cobb-Douglas Production Function (1928)

Introduction:

The Cobb–Douglas production function is a particular functional form of the function, widely used to represent the technological relationship between the amounts of two or more inputs, particularly physical capital and labour and the amount of output produced by those inputs. The Cobb-Douglas production function takes its name from Prof Douglas, who inferred its properties from empirical observation and his colleague Cobb, a mathematician who suggested the mathematical form. Though the function has been suggested earlier by Knut Wicksteed, Douglas made the function popular with his vast amount of empirical support. The function they used to model production was of the form:
$Q=f(L,K)=AL^{\alpha }K^{\beta }$
Where:
Q = total production (the monetary value of all goods produced in a year)
L = labour input (the total number of person-hours worked in a year)
K = capital input (the monetary worth of all machinery, equipment, and buildings)
A = total factor productivity
α and β are the output elasticities of labour and capital, respectively. These values are constants determined by available technology.
Further,
If a + b = 1, returns to scale are constant.
If a + b > 1, returns to scale are increasing.
If a + b < 1, returns to scale are decreasing

Properties of Cobb-Douglas Production Function:

1. In the Cobb-Douglas production function, the average product of labour and capital depends on the ratio of inputs in the production.
2. In the Cobb-Douglas production function, the marginal product of labour and capital depends on the ratio of inputs in the production.
3. The multiplicative form of the production function, $Q=AL^{\alpha }K^{\beta }$ can be converted into its log-linear form as $log\ Q=log\ A\ +\ \alpha\ log\ L\ +\ \beta\ log\ K$ .
4. The exponents of Cobb-Douglas production function α and β measure output elasticities of inputs, labour, L and capital, K respectively. Output elasticity measures the responsiveness of output to a change in either labour or capital used in production, ceteris paribus.
5. The elasticity of substitution (s) between two factors, labour and capital, in Cobb-Douglas production is equal to unity or 1.
6. The Cobb-Douglas production function is a homogeneous production function, and the returns to scale are measured by the exponent α + β.
7. When the sum of the exponents α + β in the two factor Cobb-Douglas production function $Q=AL^{\alpha }K^{\beta }$ is equal to one, it would show constant returns to scale.
8. The Cobb-Douglas production function in its general form $Q=AL^{\alpha }K^{\beta }$ indicates that there will be zero production at zero cost.
9. The exponents α and β of the Cobb-Douglas production function represents the relative distributive share of inputs L and K.

Mathematical Proof of the Properties:

1. In the Cobb-Douglas production function, the average product of labour and capital depends on the ratio of inputs in the production.
The Cobb-Douglas production function is given by:
$Q=AL^{\alpha }K^{\beta }$
Proof:
$\\\\Average\ product\ of\ labour,\ AP_{L}=\frac{Q}{L}=\frac{AL^{\alpha }K^{\beta }}{L}\\$
$\\\ AP_{L}=\frac{Q}{L}\\\\ AP_{L}=\frac{AL^{\alpha }K^{\beta }}{L}\\\\ AP_{L}=A \frac{K^{\beta}}{L^{1}L^{-\alpha}}\\\\ AP_{L}=A \frac{K^{\beta}}{L^{1-\alpha}}\ (\because \alpha +\beta =1, \ \therefore \beta=1-\alpha)\\\\ AP_{L}=A \frac{K^{\beta}}{L^{\beta}}\\\\ AP_{L}=A \left ( {\frac{L}{K}} \right )^{\beta}\\$

$\\\\Average\ product\ of\ labour,\ AP_{K}=\frac{Q}{K}=\frac{AL^{\alpha }K^{\beta }}{K}\\$
$\\\ AP_{K}=\frac{Q}{K}\\ AP_{K}=\frac{AL^{\alpha }K^{\beta }}{K}\\\\ AP_{K}=A \frac{L^{\alpha}}{K^{1}K^{-\beta}}\\\\ AP_{K}=A \frac{L^{\alpha}}{K^{1-\beta}}\ (\because \alpha +\beta =1, \ \therefore \alpha =1-\beta)\\\\ AP_{K}=A \frac{L^{\alpha}}{K^{\alpha}}\\\\ AP_{K}=A \left ( {\frac{L}{K}} \right )^{\alpha}\\$

2. In the Cobb-Douglas production function, the marginal product of labour and capital depends on the ratio of inputs in the production.
The Cobb-Douglas production function is given by:
$Q=AL^{\alpha }K^{\beta }$
Proof:
Marginal Product of Labour $MP_{L}$ :
$Marginal\ product\ of\ labour,\ MP_{L}=\frac{\partial Q}{\partial L}$
$\\MP_{L}=\frac{\partial Q}{\partial L}\\ MP_{L}=\frac{\partial}{\partial L}(AL^{\alpha }K^{\beta })\\ MP_{L}=AK^{\beta }\frac{\partial}{\partial L}L^{\alpha }\\ MP_{L}=AK^{\beta }.\alpha L^{\alpha -1}\\ MP_{L}=A\alpha\frac{K^{\beta }}{L^{-(\alpha -1)}}\\ MP_{L}=A\alpha\frac{K^{\beta }}{L^{- \alpha +1 }}\\ MP_{L}=A\alpha\frac{K^{\beta }}{L^{1-\alpha}}\ (\because \alpha +\beta =1, \ \therefore \beta=1-\alpha)\\\\ MP_{L}=A\alpha\frac{K^{\beta }}{L^{\beta}}\\ MP_{L}=A\alpha\left ( \frac{K}{L} \right )^{\beta}\\$
Marginal Product of Capital, $\\MP_{K}$ :
$Marginal\ product\ of\ labour,\ MP_{K}=\frac{\partial Q}{\partial K}$
$\\MP_{K}=\frac{\partial Q}{\partial K}\\ MP_{K}=\frac{\partial}{\partial K}(AL^{\alpha }K^{\beta })\\ MP_{K}=AL^{\alpha }\frac{\partial}{\partial K}K^{\beta }\\ MP_{K}=AL^{\alpha }.\beta K^{\beta-1 }\\ MP_{K}=A\beta\frac{L^{\alpha }}{K^{-(\beta-1)}}\\ MP_{K}=A\beta\frac{L^{\alpha }}{K^{- \beta +1 }}\\ MP_{K}=A\beta\frac{L^{\alpha }}{K^{1-\beta }}\ (\because \alpha +\beta =1, \ \therefore \alpha=1-\beta)\\\\ MP_{K}=A\beta\frac{L^{\alpha }}{K^{\alpha}}\\ MP_{K}=A\beta\left ( \frac{K}{L} \right )^{\alpha}\\$

3. The exponents of Cobb-Douglas production function α and β measure output elasticities of inputs, capital, K and labour, L respectively. Output elasticity measures the responsiveness of output to a change in levels of either labour or capital used in production, ceteris paribus.
Proof:
The Cobb-Douglas production function is given by:
$Q=AL^{\alpha }K^{\beta }$
Output elasticity of labour,
$\\\varepsilon _{L}=\frac{\frac{\partial Q}{ Q}}{\frac{\partial L}{ L}}\\\\ \varepsilon _{L}={\frac{\partial Q}{ \partial L}}.{\frac{L}{ Q}}\\$
By differentiating the production function, $Q=AL^{\alpha }K^{\beta }$ , with respect to L, we get

$\\\frac{\partial Q}{\partial L}=\frac{\partial }{\partial L}\left ( AL^{\alpha }K^{\beta } \right )\\\\ \frac{\partial Q}{\partial L}=AK^{\beta } \frac{\partial }{\partial L} L^{\alpha} \\\\ \frac{\partial Q}{\partial L}=AK^{\beta } \alpha L^{\alpha -1} \\$

Now,
$\\\varepsilon _{L}={\frac{\partial Q}{ \partial L}}.{\frac{L}{ Q}}\\$
Substituting ${\frac{\partial Q}{ \partial L}}$
$\\\varepsilon _{L}=(AK^{\beta } \alpha L^{\alpha -1}).{\frac{L}{ Q}}\\\\ \varepsilon _{L}=\alpha (AK^{\beta } L^{\alpha -1}L^{1}).{\frac{1}{ Q}}\\ \varepsilon _{L}=\alpha (AK^{\beta } L^{\alpha -1+1}).{\frac{1}{ Q}}\\ \varepsilon _{L}=\alpha (AL^{\alpha}K^{\beta } ).{\frac{1}{ Q}}\\ \varepsilon _{L}=\alpha .Q.{\frac{1}{ Q}}\ (\because Q=AL^{\alpha }K^{\beta })\\ \therefore \varepsilon _{L}=\alpha$
Output elasticity of capital,
$\\\varepsilon _{K}=\frac{\frac{\partial Q}{ Q}}{\frac{\partial K}{ K}}\\ \\\varepsilon _{K}={\frac{\partial Q}{\partial K}}.{\frac{K}{ Q}}\\\\$
By differentiating the production function, $Q=AL^{\alpha }K^{\beta }$ , with respect to K, we get

$\\\frac{\partial Q}{\partial K}=\frac{\partial }{\partial K}\left ( AL^{\alpha }K^{\beta } \right )\\\\ \frac{\partial Q}{\partial K}=A L^{\alpha} \frac{\partial }{\partial K} K^{\beta } \\\\ \frac{\partial Q}{\partial K}=AL^{\alpha} \beta K^{\beta -1} \\$
Now,
$\varepsilon _{K}={\frac{\partial Q}{ \partial K}}.{\frac{K}{ Q}}\\$
Substituting $\frac{\partial Q}{ \partial K}$
$\\\varepsilon _{K}=\left ( AL^{\alpha} \beta K^{\beta -1} \right ).{\frac{K}{ Q}}\\ \varepsilon _{K}=\left ( AL^{\alpha} \beta K^{\beta -1}K^{1} \right ).{\frac{1}{ Q}}\\ \varepsilon _{K}=\beta \left ( AL^{\alpha} K^{\beta -1+1} \right ).{\frac{1}{ Q}}\\ \varepsilon _{K}=\beta \left ( AL^{\alpha} K^{\beta} \right ).{\frac{1}{ Q}}\\ \varepsilon _{K}=\beta . Q.{\frac{1}{ Q}}\ \left ( \because Q=AL^{\alpha} K^{\beta} \right )\\ \therefore \varepsilon _{K}=\beta$
Therefore, α and β measure output elasticities of inputs, labour, L and capital, K respectively.
4. The multiplicative form of the production function $Q=AL^{\alpha }K^{\beta }$ can be converted into its log-linear form as $log\ Q=log\ A\ +\ \alpha\ log\ L\ +\ \beta\ log\ K$ .
The Cobb-Douglas production function is given by:
$Q=AL^{\alpha }K^{\beta }$
Taking log on both sides:
$\\log\ Q=log \left (AL^{\alpha}K^{\beta} \right )\\ \\log\ Q=log\ A+log\ L^{\alpha}+log\ K^{\beta}\\ \\log\ Q=log\ A +\alpha log L+\beta log K\\$

5. The elasticity of substitution (s) between two factors, labour and capital, in Cobb-Douglas production is equal to unity (1).
Proof:
The Cobb-Douglas production function is given by:
$Q=AL^{\alpha }K^{\beta }$

Elasticity of substitution (s),
$\\\sigma =\frac{\frac{\mathrm{d} \frac{K}{L}}{ \frac{K}{L}}}{\frac{\mathrm{d}MRTS_{LK}}{ MRTS_{LK}}}\\\\ \sigma ={\frac{\mathrm{d} \frac{K}{L}}{ \frac{K}{L}}}.{\frac{MRTS_{LK}}{ \mathrm{d}MRTS_{LK}}}\\ \sigma ={\frac{\mathrm{d} \frac{K}{L}}{ \mathrm{d}MRTS_{LK}}}.{\frac{MRTS_{LK}}{ \frac{K}{L}}}\\$
Now,
$MRTS_{LK}= \frac{MP_{L}}{MP_{K}}\\$

$\\MP_{L}=\frac{\partial Q}{\partial L}\\ MP_{L}=\frac{\partial }{\partial L}\left (AL^{\alpha}K^{\beta} \right )\\ MP_{L}=\alpha AL^{\alpha-1}K^{\beta}\\ MP_{L}=\alpha AL^{\alpha}L^{-1}K^{\beta}\\ MP_{L}=\alpha \frac{AL^{\alpha}K^{\beta}}{L}\\ MP_{L}=\alpha \frac{Q}{L}\\\\$

$\\MP_{K}=\frac{\partial Q}{\partial K}\\ MP_{K}=\frac{\partial }{\partial K}\left (AL^{\alpha}K^{\beta} \right )\\ MP_{K}=\beta AL^{\alpha}K^{\beta-1}\\ MP_{K}=\beta AL^{\alpha}K^{\beta}K^{-1}\\ MP_{K}=\beta \frac{AL^{\alpha}K^{\beta}}{K}\\ MP_{K}=\beta \frac{Q}{K}\\\\$
Then,
$\\MRTS_{LK}=\frac{MP_{L}}{MP_{K}}\\ \\MRTS_{LK}=\frac{\alpha \frac{Q}{L}}{\beta \frac{Q}{K}}=\frac{\frac{\alpha }{L}}{ \frac{\beta}{K}}=\frac{\alpha }{L}.\frac{K}{\beta }=\frac{\alpha }{\beta}.\frac{K}{L}$
Also,
$\\d\left (MRTS_{LK} \right )=d\left ( \frac{\alpha }{\beta}.\frac{K}{L} \right )\\ \\d\left (MRTS_{LK} \right )=\frac{\alpha }{\beta} d\left ( \frac{K}{L} \right )\\ \\\frac{\beta }{\alpha }\ d\left (MRTS_{LK} \right )= d\left ( \frac{K}{L} \right )\\ \\\frac{\beta }{\alpha }\ =\frac{d\left ( \frac{K}{L} \right )}{d\left (MRTS_{LK} \right )} \\$
Next,
$\\\sigma ={\frac{\mathrm{d} \frac{K}{L}}{ \mathrm{d}MRTS_{LK}}}.{\frac{MRTS_{LK}}{ \frac{K}{L}}}\\\\ \sigma =\frac{\beta }{\alpha }.{\frac{\frac{\alpha }{\beta}.\frac{K}{L}}{ \frac{K}{L}}}\\\\ \sigma =\frac{\beta }{\alpha }.\frac{\alpha }{\beta}\\\\ \sigma =1\\\\$
Therefore, the elasticity of substitution (s) between two factors, labour and capital, in Cobb-Douglas production is equal to unity (1).

6. The Cobb-Douglas production function is a homogeneous production function and the returns to scale is measured by the exponent α+β.
Proof:
The Cobb-Douglas production function is given by:
$Q=AL^{\alpha }K^{\beta }$
Multiplying L and K by a constant λ, we have
$\\Q=f(L,K)=AL^{\alpha}K^{\beta}\\ \\Q=f(\lambda L,\lambda K)=A(\lambda L)^{\alpha}(\lambda K)^{\beta}\\ \\Q=f(\lambda L,\lambda K)=A \lambda^{\alpha}L^{\alpha}\lambda^{\beta}K^{\beta}\\ \\Q=f(\lambda L,\lambda K)=\lambda^{\alpha}\lambda^{\beta}.A L^{\alpha}K^{\beta}\\ \\Q=f(\lambda L,\lambda K)=\lambda^{\alpha+\beta}.Q\ \left ( \because Q=AL^{\alpha}K^{\beta} \right )\\$
Now, the returns to scale are measured by the exponent α+β:
If a + b = 1, returns to scale are constant.
If a + b > 1, returns to scale are increasing.
If a + b < 1, returns to scale are decreasing.

Additional Notes:
In 1928, Charles Cobb and Paul Douglas published a study in which they modelled the growth of the American economy during the period 1899 - 1922. They considered a simplified view of the economy in which production output is determined by the amount of labour involved and the amount of capital invested. While there are many other factors affecting economic performance, their model proved to be remarkably accurate.
Output elasticity measures the responsiveness of output to a change in levels of either labour or capital used in production, ceteris paribus. For example, if α = 0.15, a 1% increase in labour would lead to approximately a 0.15% increase in output.
Returns to scale refer to a technical property of production that examines changes in output subsequent to a proportional change in all inputs (where all inputs increase by a constant factor). If output increases by that same proportional change then there are constant returns to scale (CRTS). If output increases by less than that proportional change, there are decreasing returns to scale (DRS). If output increases by more than that proportion, there are increasing returns to scale (IRS).
A function is said to be homogenous of degree ‘k’, if multiplication of each of its independent variables by a constant λ will alter the value of the function by the proportion, .i.e., or, i.e., . Note that,, may be negative.
Reading Lists:
1) Samuelson, P. A., & Nordhuas, W.D (1992), Economics (14th edition). McGraw Hill International edition, U.S.
2) Samuelson, P. A., & Nordhuas, W.D (2013), Microeconomics (19th edition). McGraw Hill Education (India) Pvt. Ltd.
3) Koutsoyiannis, A. (1990), Modern Microeconomics (2nd edition). Macmillan, London.
4) Henderson, J. M. and R. E. Quandt (1980), Microeconomic Theory: A Mathematical Approach (3rd edition). McGraw Hill, New Delhi.
5) Dwivedi, D. N. (2016), Microeconomics: Theory and Applications (3rd edition). Vikas Publication House Pvt. Ltd. Noida (UP), India.
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