Answer:
my hands hurt bcz of this
Explanation:
We have the production function as Y=AL
t
a
K
t
3
.
Where Y is the output, L
t
is the labor, A is the total factor productivity, K
t
is the physical capital, and α is the capital's share in output.
To find ∂L
i
∂Y
, we take the partial derivative of Y with respect to L
i
∂L
i
∂Y
=αY/L
i
This shows that the marginal productivity of labor is equal to α times the output per worker.
To find ∂K
t
∂Y
, we take the partial derivative of Y with respect to K
t
∂K
t
∂Y
=3(1−α)Y/K
t
This shows that the marginal productivity of capital is equal to 3(1-α) times the output per unit of capital.
If β=1-α, then we have
Y=AL
t
a
K
t
3(1−β)
Substituting β=1-α, we get
Y=AL
t
a
K
t
3α
Now,
∂K
t
∂Y
=3Y/K
t
Thus, the marginal productivity of capital is now equal to 3 times the output per unit of capital.