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1 vote
(Challenge): Let L

t

,K
t

be variables at time t. Find
∂L
i


∂Y

and
∂K
t


∂Y

for Y=AL
t
a

K
t
3

. What if β=1−α ?

1 Answer

6 votes

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

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.

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