Mixing
Showing marginal product of capital is independent of the scale of production
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The image is pretty much self-explanatory. To add some context, I'm learning Solow-Swan Growth Theory and my professor said that the marginal product will not change if both capital and labor increase at the same scale.
I can intuitively understand that it makes sense but trying to apply a simple equation (the blue one, the definition of constant marginal product) is just not working.
It's either I'm not partial differentiating correctly or the whole theory is wrong.
I don't see anything wrong with what I've done but why are they not the same?
macroeconomics mathematical-economics
edited 44 mins ago
Ubiquitous♦
13.5k32363
asked 1 hour ago
zebralamy
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up vote
2
down vote
favorite
The image is pretty much self-explanatory. To add some context, I'm learning Solow-Swan Growth Theory and my professor said that the marginal product will not change if both capital and labor increase at the same scale.
I can intuitively understand that it makes sense but trying to apply a simple equation (the blue one, the definition of constant marginal product) is just not working.
It's either I'm not partial differentiating correctly or the whole theory is wrong.
I don't see anything wrong with what I've done but why are they not the same?
macroeconomics mathematical-economics
edited 44 mins ago
Ubiquitous♦
13.5k32363
asked 1 hour ago
zebralamy
132
New contributor
zebralamy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
The trick is to show the aggregate production function is homogenous of degree 1
– Pedro Cavalcante Oliveira
7 mins ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The image is pretty much self-explanatory. To add some context, I'm learning Solow-Swan Growth Theory and my professor said that the marginal product will not change if both capital and labor increase at the same scale.
I can intuitively understand that it makes sense but trying to apply a simple equation (the blue one, the definition of constant marginal product) is just not working.
It's either I'm not partial differentiating correctly or the whole theory is wrong.
I don't see anything wrong with what I've done but why are they not the same?
macroeconomics mathematical-economics
edited 44 mins ago
Ubiquitous♦
13.5k32363
asked 1 hour ago
zebralamy
132
New contributor
zebralamy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
The image is pretty much self-explanatory. To add some context, I'm learning Solow-Swan Growth Theory and my professor said that the marginal product will not change if both capital and labor increase at the same scale.
I can intuitively understand that it makes sense but trying to apply a simple equation (the blue one, the definition of constant marginal product) is just not working.
It's either I'm not partial differentiating correctly or the whole theory is wrong.
I don't see anything wrong with what I've done but why are they not the same?
macroeconomics mathematical-economics
macroeconomics mathematical-economics
edited 44 mins ago
Ubiquitous♦
13.5k32363
asked 1 hour ago
zebralamy
132
New contributor
zebralamy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 44 mins ago
Ubiquitous♦
13.5k32363
asked 1 hour ago
zebralamy
132
New contributor
zebralamy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 44 mins ago
Ubiquitous♦
13.5k32363
edited 44 mins ago
Ubiquitous♦
13.5k32363
edited 44 mins ago
Ubiquitous♦
13.5k32363
13.5k32363
asked 1 hour ago
zebralamy
132
New contributor
zebralamy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 1 hour ago
zebralamy
132
asked 1 hour ago
zebralamy
132
132
New contributor
zebralamy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
zebralamy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
zebralamy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
The trick is to show the aggregate production function is homogenous of degree 1
– Pedro Cavalcante Oliveira
7 mins ago
add a comment |Â
The trick is to show the aggregate production function is homogenous of degree 1
– Pedro Cavalcante Oliveira
7 mins ago
The trick is to show the aggregate production function is homogenous of degree 1
– Pedro Cavalcante Oliveira
7 mins ago
The trick is to show the aggregate production function is homogenous of degree 1
– Pedro Cavalcante Oliveira
7 mins ago
add a comment |Â
1 Answer
1
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oldest
votes
up vote
3
down vote
accepted
There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).
To compute the MPK, we must differentiate the production function with respect to the current level of capital: $partial F/partial K$.
But in your final line, you are not differentiating with respect to the current level of capital (which is $tildeK=lambda K$). You are instead differentiating with respect to $K$, which is a fraction $1/lambda$ of the current amount of capital.
If we compute the derivative with respect to $tildeKequiv lambda K$ instead of just $K$ then everything works as it should:
$$fracpartial F(lambda K,lambda L)partial lambda K=fracpartial (8(lambda K)^1/2(lambda L)^1/2)partial lambda K=4(lambda K)^-1/2(lambda L)^1/2=4fracsqrtLsqrtK.$$
This does not depend on $lambda$ so MPK is indeed independent of the scale of the economy. QED
answered 1 hour ago
Ubiquitous♦
13.5k32363
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).
To compute the MPK, we must differentiate the production function with respect to the current level of capital: $partial F/partial K$.
But in your final line, you are not differentiating with respect to the current level of capital (which is $tildeK=lambda K$). You are instead differentiating with respect to $K$, which is a fraction $1/lambda$ of the current amount of capital.
If we compute the derivative with respect to $tildeKequiv lambda K$ instead of just $K$ then everything works as it should:
$$fracpartial F(lambda K,lambda L)partial lambda K=fracpartial (8(lambda K)^1/2(lambda L)^1/2)partial lambda K=4(lambda K)^-1/2(lambda L)^1/2=4fracsqrtLsqrtK.$$
This does not depend on $lambda$ so MPK is indeed independent of the scale of the economy. QED
answered 1 hour ago
Ubiquitous♦
13.5k32363
add a comment |Â
up vote
3
down vote
accepted
There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).
To compute the MPK, we must differentiate the production function with respect to the current level of capital: $partial F/partial K$.
But in your final line, you are not differentiating with respect to the current level of capital (which is $tildeK=lambda K$). You are instead differentiating with respect to $K$, which is a fraction $1/lambda$ of the current amount of capital.
If we compute the derivative with respect to $tildeKequiv lambda K$ instead of just $K$ then everything works as it should:
$$fracpartial F(lambda K,lambda L)partial lambda K=fracpartial (8(lambda K)^1/2(lambda L)^1/2)partial lambda K=4(lambda K)^-1/2(lambda L)^1/2=4fracsqrtLsqrtK.$$
This does not depend on $lambda$ so MPK is indeed independent of the scale of the economy. QED
answered 1 hour ago
Ubiquitous♦
13.5k32363
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).
To compute the MPK, we must differentiate the production function with respect to the current level of capital: $partial F/partial K$.
But in your final line, you are not differentiating with respect to the current level of capital (which is $tildeK=lambda K$). You are instead differentiating with respect to $K$, which is a fraction $1/lambda$ of the current amount of capital.
If we compute the derivative with respect to $tildeKequiv lambda K$ instead of just $K$ then everything works as it should:
$$fracpartial F(lambda K,lambda L)partial lambda K=fracpartial (8(lambda K)^1/2(lambda L)^1/2)partial lambda K=4(lambda K)^-1/2(lambda L)^1/2=4fracsqrtLsqrtK.$$
This does not depend on $lambda$ so MPK is indeed independent of the scale of the economy. QED
answered 1 hour ago
Ubiquitous♦
13.5k32363
There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).
To compute the MPK, we must differentiate the production function with respect to the current level of capital: $partial F/partial K$.
But in your final line, you are not differentiating with respect to the current level of capital (which is $tildeK=lambda K$). You are instead differentiating with respect to $K$, which is a fraction $1/lambda$ of the current amount of capital.
If we compute the derivative with respect to $tildeKequiv lambda K$ instead of just $K$ then everything works as it should:
$$fracpartial F(lambda K,lambda L)partial lambda K=fracpartial (8(lambda K)^1/2(lambda L)^1/2)partial lambda K=4(lambda K)^-1/2(lambda L)^1/2=4fracsqrtLsqrtK.$$
This does not depend on $lambda$ so MPK is indeed independent of the scale of the economy. QED
answered 1 hour ago
Ubiquitous♦
13.5k32363
edited 1 hour ago
answered 1 hour ago
Ubiquitous♦
13.5k32363
answered 1 hour ago
Ubiquitous♦
13.5k32363
answered 1 hour ago
Ubiquitous♦
13.5k32363
13.5k32363
add a comment |Â
add a comment |Â
zebralamy is a new contributor. Be nice, and check out our Code of Conduct.
zebralamy is a new contributor. Be nice, and check out our Code of Conduct.
zebralamy is a new contributor. Be nice, and check out our Code of Conduct.
zebralamy is a new contributor. Be nice, and check out our Code of Conduct.
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The trick is to show the aggregate production function is homogenous of degree 1
– Pedro Cavalcante Oliveira
7 mins ago