Question

Which of the following statements is TRUE? Select one: a. A Cobb-Douglas production function can have different returns to scale at different output levels. b. It is impossible to have increasing returns to scale for one output level, and decreasing returns to scale for a different output level. c. It is possible to have increasing returns to scale for one output level, and decreasing returns to scale for a different output level. d. None of the above.

Answer 1

Answer :d

d. None of the above.

If production or output is denoted by Q , and it is produced by two factors, labor (L) , and capital (K), then the production function can be written as,

Q = f(L,K)

A Cobb-Douglas production function is a special type of production function , in which the production function is of the following form,

Q = f(L,K) = A L K ,

where

Q = output

L= labor

K = capital

A >0 and constant . It shows the efficiency of the inputs or the technology of the production

>0 and constant

β> 0 and constant

+ β = 1

and  β indicate the degree of the labor and capital used in production , i.e. they are the relative shares used to produce output.

Now,if we increase both the inputs, L and K, by a particular amount , say '' , the Cobb-Douglas production function can be written as,

Q = f(L,K) = A (L) (K)

Or, Q = f(L,K) = ⁽⁺⁾ * A L K

Or, Q = ⁽⁺⁾ * Q

From the above equation, we see that if 'L' and 'K' increase by , the production or output will increase by ⁽⁺⁾ .

The returns to scale of a homogeneous production function can be measured by its degree of homogeneity.

Now, if

+ β = 1 , then Q = * Q , i.e.

output increases by the same proportion of the inputs.It is then called homogeneous of degree '1' , or constant returns to scale.

If, + β > 1 , then Q increases proportionately more than the increase in inputs. It is then called increasing returns scale.

If, + β < 1 , then Q increases proportionately less than the increase in inputs. It is then called decreasing returns scale.

Now, for Cobb-Douglas production function, + β = 1 , so when both the inputs are increased by a particular proportion, the output(Q) also increases by the same proportion or amount. So the Cobb-Douglas production function is homogeneous of degree 1, which shows a constant returns to scale .

The Cobb-Douglas production function can not have different returns to scale for different output levels. It always show a constant returns to scale for all the output levels.

The Cobb-Douglas production function always have constant returns to scale.

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