3 Cobb-Douglas Production Function The Cobb-Douglas production function, in its stochastic form, may be expressed as Yi

[answerhappygod]

3 Cobb-Douglas Production Function
The Cobb-Douglas production function, in its stochastic form, may
be expressed as
Yi = β0Xβ22i Xβ33i eui (3)where Yi is output, X2i is labor input,
X3i is capital input, ui is error term, and e is the base of
natural
logarithm. From equation (2), it is clear that the relationship
between output and the two inputs is non-linear.
However, if we log-transform this model, we obtain:
ln Yi = ln β0 + β2 ln X2i + β3 ln X3i + ui
= β1 + β2 ln X2i + β3 ln X3i + ui (4)
where β1 is dened as β1 = ln β0. Thus the model (3) is linear in
the parameters β1, β2, and β3 and is
therefore a linear regression model. Assume that the classical
assumptions are satised.
2(a) What is the interpretation of β2 and β3?
The sum (β2 + β3) gives information about the returns to scale,
that is, the response of output to a proportionate
change in the inputs. If this sum is 1, then there are constant
returns to scale (CRTS), that is, doubling
the inputs will double the output, tripling the inputs will triple
the output. If the sum is less than 1, there are
decreasing returns to scale (DRTS), that is, doubling the inputs
will less than double the output. Finally,
if the sum is greater than 1, there are increasing returns to scale
(IRTS), that is doubling the inputs will
more than double the output.
(b) We want to test whether there are constant returns to scale or
not. Specify a null hypothesis.
(c) Write down the restricted model under H0 you specied in
(b).
(d) Write down the unrestricted model.
(e) Suppose that R2
R (R2 from the restricted model) is 0.977 and R2
U (R2 from the unrestricted model) is 0.9951.
What is the test statistic? If you don't think you can calculate
the test statistic using the information above,
state the reason clearly