Mixing
Decision over “max†production function:
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I've been presented with the following problem:
$$y=3(x_3)^frac13(maxx_1,8x_2)^frac13$$
And the objective is to both maximize profit and minimize cost. First of all, if the problems are dual, does that mean the result will be the same in variables such as demands?
STILL, MY BIGGEST ISSUE IS THIS: When you get rid of the max, this becomes a Cobb Douglas piece of cake. But I just don't understand how to do that. So far, all I've got is that this function allows corner solutions, so it's not solved just as a Leontieff function. How would you go about choosing each good? I'm also sure it has to do with input prices $w_1$ and $w_2$
production-function cost profit-maximization cost-functions
edited Sep 1 at 18:42
Herr K.
5,63421034
asked Sep 1 at 18:20
mudcake
211
add a comment |Â
up vote
4
down vote
favorite
I've been presented with the following problem:
$$y=3(x_3)^frac13(maxx_1,8x_2)^frac13$$
And the objective is to both maximize profit and minimize cost. First of all, if the problems are dual, does that mean the result will be the same in variables such as demands?
STILL, MY BIGGEST ISSUE IS THIS: When you get rid of the max, this becomes a Cobb Douglas piece of cake. But I just don't understand how to do that. So far, all I've got is that this function allows corner solutions, so it's not solved just as a Leontieff function. How would you go about choosing each good? I'm also sure it has to do with input prices $w_1$ and $w_2$
production-function cost profit-maximization cost-functions
edited Sep 1 at 18:42
Herr K.
5,63421034
asked Sep 1 at 18:20
mudcake
211
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I've been presented with the following problem:
$$y=3(x_3)^frac13(maxx_1,8x_2)^frac13$$
And the objective is to both maximize profit and minimize cost. First of all, if the problems are dual, does that mean the result will be the same in variables such as demands?
STILL, MY BIGGEST ISSUE IS THIS: When you get rid of the max, this becomes a Cobb Douglas piece of cake. But I just don't understand how to do that. So far, all I've got is that this function allows corner solutions, so it's not solved just as a Leontieff function. How would you go about choosing each good? I'm also sure it has to do with input prices $w_1$ and $w_2$
production-function cost profit-maximization cost-functions
edited Sep 1 at 18:42
Herr K.
5,63421034
asked Sep 1 at 18:20
mudcake
211
I've been presented with the following problem:
$$y=3(x_3)^frac13(maxx_1,8x_2)^frac13$$
And the objective is to both maximize profit and minimize cost. First of all, if the problems are dual, does that mean the result will be the same in variables such as demands?
STILL, MY BIGGEST ISSUE IS THIS: When you get rid of the max, this becomes a Cobb Douglas piece of cake. But I just don't understand how to do that. So far, all I've got is that this function allows corner solutions, so it's not solved just as a Leontieff function. How would you go about choosing each good? I'm also sure it has to do with input prices $w_1$ and $w_2$
production-function cost profit-maximization cost-functions
edited Sep 1 at 18:42
Herr K.
5,63421034
asked Sep 1 at 18:20
mudcake
211
edited Sep 1 at 18:42
Herr K.
5,63421034
edited Sep 1 at 18:42
Herr K.
5,63421034
edited Sep 1 at 18:42
Herr K.
5,63421034
5,63421034
asked Sep 1 at 18:20
mudcake
211
asked Sep 1 at 18:20
mudcake
211
asked Sep 1 at 18:20
mudcake
211
211
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
Hint
For profit maximization, either $x_1$ or $x_2$ (but not both) must be zero. If not, say $x_1^*>x_2^*>0$ at the optimum, then one could increase profit by lowering cost by reducing $x_2^*$ without affecting output and thus revenue.
Let $z=maxx_1,8x_2$. The profit function can be written as
beginequation
p[3(x_3)^1/3(z)^1/3]-w_3x_3-c_zz,tag1
endequation
where
beginequation
c_z=
begincases
w_1&textif x_1>8x_2\
w_2/8&textif x_1<8x_2\
minw_1,w_2/8&textif x_1=8x_2
endcases
endequation
Solve $(1)$ for the condition determining the optimal level of $z$, and compare the costs of achieving this level using either $x_1$ or $x_2$. Use the one that entails the lower cost.
answered Sep 1 at 19:22
Herr K.
5,63421034
I like this response so much, it is extremely clear, thank you! However, wouldn't it be the other way around? You'd choose z based on the comparison of w1 against w2/8, or am I in the wrong? My point is: you decide product after evaluating prices, I'm not sure how you'd choose z and thus the w to use if you don't know wether x1 or 8x2 is greater. I hope you can help me once more. Thank you!
– mudcake
Sep 1 at 22:49
@mudcake: You're welcome. And yes, ultimately, whether $x_1$ or $x_2$ gets chosen depends on the comparison between $w_1$ and $w_2/8$. Note that $z$ is just a placeholder in the maximization. Perhaps put in another way, you'd have compare $$pi_1=p[3(x_3)^1/3(x_1)^1/3]-w_3x_3-w_1x_1$$ with $$pi_2=p[3(x_3)^1/3(8x_2)^1/3]-w_3x_3-w_2x_2$$ and see which input leads to a higher profit. The above two profit maximization problems should be straightforward to solve and compare.
– Herr K.
Sep 2 at 1:36
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Hint
For profit maximization, either $x_1$ or $x_2$ (but not both) must be zero. If not, say $x_1^*>x_2^*>0$ at the optimum, then one could increase profit by lowering cost by reducing $x_2^*$ without affecting output and thus revenue.
Let $z=maxx_1,8x_2$. The profit function can be written as
beginequation
p[3(x_3)^1/3(z)^1/3]-w_3x_3-c_zz,tag1
endequation
where
beginequation
c_z=
begincases
w_1&textif x_1>8x_2\
w_2/8&textif x_1<8x_2\
minw_1,w_2/8&textif x_1=8x_2
endcases
endequation
Solve $(1)$ for the condition determining the optimal level of $z$, and compare the costs of achieving this level using either $x_1$ or $x_2$. Use the one that entails the lower cost.
answered Sep 1 at 19:22
Herr K.
5,63421034
I like this response so much, it is extremely clear, thank you! However, wouldn't it be the other way around? You'd choose z based on the comparison of w1 against w2/8, or am I in the wrong? My point is: you decide product after evaluating prices, I'm not sure how you'd choose z and thus the w to use if you don't know wether x1 or 8x2 is greater. I hope you can help me once more. Thank you!
– mudcake
Sep 1 at 22:49
@mudcake: You're welcome. And yes, ultimately, whether $x_1$ or $x_2$ gets chosen depends on the comparison between $w_1$ and $w_2/8$. Note that $z$ is just a placeholder in the maximization. Perhaps put in another way, you'd have compare $$pi_1=p[3(x_3)^1/3(x_1)^1/3]-w_3x_3-w_1x_1$$ with $$pi_2=p[3(x_3)^1/3(8x_2)^1/3]-w_3x_3-w_2x_2$$ and see which input leads to a higher profit. The above two profit maximization problems should be straightforward to solve and compare.
– Herr K.
Sep 2 at 1:36
add a comment |Â
up vote
4
down vote
Hint
For profit maximization, either $x_1$ or $x_2$ (but not both) must be zero. If not, say $x_1^*>x_2^*>0$ at the optimum, then one could increase profit by lowering cost by reducing $x_2^*$ without affecting output and thus revenue.
Let $z=maxx_1,8x_2$. The profit function can be written as
beginequation
p[3(x_3)^1/3(z)^1/3]-w_3x_3-c_zz,tag1
endequation
where
beginequation
c_z=
begincases
w_1&textif x_1>8x_2\
w_2/8&textif x_1<8x_2\
minw_1,w_2/8&textif x_1=8x_2
endcases
endequation
Solve $(1)$ for the condition determining the optimal level of $z$, and compare the costs of achieving this level using either $x_1$ or $x_2$. Use the one that entails the lower cost.
answered Sep 1 at 19:22
Herr K.
5,63421034
I like this response so much, it is extremely clear, thank you! However, wouldn't it be the other way around? You'd choose z based on the comparison of w1 against w2/8, or am I in the wrong? My point is: you decide product after evaluating prices, I'm not sure how you'd choose z and thus the w to use if you don't know wether x1 or 8x2 is greater. I hope you can help me once more. Thank you!
– mudcake
Sep 1 at 22:49
@mudcake: You're welcome. And yes, ultimately, whether $x_1$ or $x_2$ gets chosen depends on the comparison between $w_1$ and $w_2/8$. Note that $z$ is just a placeholder in the maximization. Perhaps put in another way, you'd have compare $$pi_1=p[3(x_3)^1/3(x_1)^1/3]-w_3x_3-w_1x_1$$ with $$pi_2=p[3(x_3)^1/3(8x_2)^1/3]-w_3x_3-w_2x_2$$ and see which input leads to a higher profit. The above two profit maximization problems should be straightforward to solve and compare.
– Herr K.
Sep 2 at 1:36
add a comment |Â
up vote
4
down vote
up vote
4
down vote
Hint
For profit maximization, either $x_1$ or $x_2$ (but not both) must be zero. If not, say $x_1^*>x_2^*>0$ at the optimum, then one could increase profit by lowering cost by reducing $x_2^*$ without affecting output and thus revenue.
Let $z=maxx_1,8x_2$. The profit function can be written as
beginequation
p[3(x_3)^1/3(z)^1/3]-w_3x_3-c_zz,tag1
endequation
where
beginequation
c_z=
begincases
w_1&textif x_1>8x_2\
w_2/8&textif x_1<8x_2\
minw_1,w_2/8&textif x_1=8x_2
endcases
endequation
Solve $(1)$ for the condition determining the optimal level of $z$, and compare the costs of achieving this level using either $x_1$ or $x_2$. Use the one that entails the lower cost.
answered Sep 1 at 19:22
Herr K.
5,63421034
Hint
For profit maximization, either $x_1$ or $x_2$ (but not both) must be zero. If not, say $x_1^*>x_2^*>0$ at the optimum, then one could increase profit by lowering cost by reducing $x_2^*$ without affecting output and thus revenue.
Let $z=maxx_1,8x_2$. The profit function can be written as
beginequation
p[3(x_3)^1/3(z)^1/3]-w_3x_3-c_zz,tag1
endequation
where
beginequation
c_z=
begincases
w_1&textif x_1>8x_2\
w_2/8&textif x_1<8x_2\
minw_1,w_2/8&textif x_1=8x_2
endcases
endequation
Solve $(1)$ for the condition determining the optimal level of $z$, and compare the costs of achieving this level using either $x_1$ or $x_2$. Use the one that entails the lower cost.
answered Sep 1 at 19:22
Herr K.
5,63421034
answered Sep 1 at 19:22
Herr K.
5,63421034
answered Sep 1 at 19:22
Herr K.
5,63421034
answered Sep 1 at 19:22
Herr K.
5,63421034
5,63421034
I like this response so much, it is extremely clear, thank you! However, wouldn't it be the other way around? You'd choose z based on the comparison of w1 against w2/8, or am I in the wrong? My point is: you decide product after evaluating prices, I'm not sure how you'd choose z and thus the w to use if you don't know wether x1 or 8x2 is greater. I hope you can help me once more. Thank you!
– mudcake
Sep 1 at 22:49
@mudcake: You're welcome. And yes, ultimately, whether $x_1$ or $x_2$ gets chosen depends on the comparison between $w_1$ and $w_2/8$. Note that $z$ is just a placeholder in the maximization. Perhaps put in another way, you'd have compare $$pi_1=p[3(x_3)^1/3(x_1)^1/3]-w_3x_3-w_1x_1$$ with $$pi_2=p[3(x_3)^1/3(8x_2)^1/3]-w_3x_3-w_2x_2$$ and see which input leads to a higher profit. The above two profit maximization problems should be straightforward to solve and compare.
– Herr K.
Sep 2 at 1:36
add a comment |Â
I like this response so much, it is extremely clear, thank you! However, wouldn't it be the other way around? You'd choose z based on the comparison of w1 against w2/8, or am I in the wrong? My point is: you decide product after evaluating prices, I'm not sure how you'd choose z and thus the w to use if you don't know wether x1 or 8x2 is greater. I hope you can help me once more. Thank you!
– mudcake
Sep 1 at 22:49
@mudcake: You're welcome. And yes, ultimately, whether $x_1$ or $x_2$ gets chosen depends on the comparison between $w_1$ and $w_2/8$. Note that $z$ is just a placeholder in the maximization. Perhaps put in another way, you'd have compare $$pi_1=p[3(x_3)^1/3(x_1)^1/3]-w_3x_3-w_1x_1$$ with $$pi_2=p[3(x_3)^1/3(8x_2)^1/3]-w_3x_3-w_2x_2$$ and see which input leads to a higher profit. The above two profit maximization problems should be straightforward to solve and compare.
– Herr K.
Sep 2 at 1:36
I like this response so much, it is extremely clear, thank you! However, wouldn't it be the other way around? You'd choose z based on the comparison of w1 against w2/8, or am I in the wrong? My point is: you decide product after evaluating prices, I'm not sure how you'd choose z and thus the w to use if you don't know wether x1 or 8x2 is greater. I hope you can help me once more. Thank you!
– mudcake
Sep 1 at 22:49
I like this response so much, it is extremely clear, thank you! However, wouldn't it be the other way around? You'd choose z based on the comparison of w1 against w2/8, or am I in the wrong? My point is: you decide product after evaluating prices, I'm not sure how you'd choose z and thus the w to use if you don't know wether x1 or 8x2 is greater. I hope you can help me once more. Thank you!
– mudcake
Sep 1 at 22:49
@mudcake: You're welcome. And yes, ultimately, whether $x_1$ or $x_2$ gets chosen depends on the comparison between $w_1$ and $w_2/8$. Note that $z$ is just a placeholder in the maximization. Perhaps put in another way, you'd have compare $$pi_1=p[3(x_3)^1/3(x_1)^1/3]-w_3x_3-w_1x_1$$ with $$pi_2=p[3(x_3)^1/3(8x_2)^1/3]-w_3x_3-w_2x_2$$ and see which input leads to a higher profit. The above two profit maximization problems should be straightforward to solve and compare.
– Herr K.
Sep 2 at 1:36
@mudcake: You're welcome. And yes, ultimately, whether $x_1$ or $x_2$ gets chosen depends on the comparison between $w_1$ and $w_2/8$. Note that $z$ is just a placeholder in the maximization. Perhaps put in another way, you'd have compare $$pi_1=p[3(x_3)^1/3(x_1)^1/3]-w_3x_3-w_1x_1$$ with $$pi_2=p[3(x_3)^1/3(8x_2)^1/3]-w_3x_3-w_2x_2$$ and see which input leads to a higher profit. The above two profit maximization problems should be straightforward to solve and compare.
– Herr K.
Sep 2 at 1:36
add a comment |Â
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