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Suppose a plant produces engines each week using assembly machines and workers.
The number of engines produced each week is
Each assembly machine rents for dollars per week and each worker costs dollars per week.
Part one of the question is, how much does it cost to produce engines, if the plant is cost-minimizing?
I am assuming that this is a constrained maximization problem where we need to maximize subject to
I'm trying to solve this with the Lagrangian
that is, with
I think I worked out from the first two conditions that and so
Then from the last condition
I get and
So the overall cost function is
Have I done this correctly, or is there another way to do this?
I am also asked what are the average and marginal costs for producing engines.
For the average cost I presume you just divide the total cost by , while for the marginal cost you take the derivative of the total cost with respect to
In this case the average and marginal costs are the same. Does this indicate constant returns to scale?
118.208.40.147 ( talk) 02:28, 14 July 2011 (UTC)
I have read the Square root of a matrix article.
I have a real square positive-definite but possibly non-symmetric matrix M, and I seek a (preferably real) matrix S such that transpose(S)*S=M (the * indicates matrix product). I don't have to be able to construct S, it suffices to know that it exists (does it?).
Cholesky decomposition will not do, as the matrix M is not guaranteed to be symmetric.
A square root based on diagonalization won't help either, as that would give S*S=M and not transpose(S)*S=M
The Square root of a matrix article writes further
"In linear algebra and operator theory, given a bounded positive semidefinite operator (a non-negative operator) T on a complex Hilbert space, B is a square root of T if T = B* B, where B* denotes the Hermitian adjoint of B. citation needed According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator T½ such that T½ is itself positive and (T½)2 = T. The operator T½ is the unique non-negative square root of T. citation needed"
This seems very close to what I'm looking for, but, alas, "citation needed". I know next to nothing about continuous functional calculus and that article isn't very helpful.
213.49.89.115 ( talk) 17:47, 14 July 2011 (UTC)
Mathematics desk | ||
---|---|---|
< July 13 | << Jun | July | Aug >> | July 15 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Suppose a plant produces engines each week using assembly machines and workers.
The number of engines produced each week is
Each assembly machine rents for dollars per week and each worker costs dollars per week.
Part one of the question is, how much does it cost to produce engines, if the plant is cost-minimizing?
I am assuming that this is a constrained maximization problem where we need to maximize subject to
I'm trying to solve this with the Lagrangian
that is, with
I think I worked out from the first two conditions that and so
Then from the last condition
I get and
So the overall cost function is
Have I done this correctly, or is there another way to do this?
I am also asked what are the average and marginal costs for producing engines.
For the average cost I presume you just divide the total cost by , while for the marginal cost you take the derivative of the total cost with respect to
In this case the average and marginal costs are the same. Does this indicate constant returns to scale?
118.208.40.147 ( talk) 02:28, 14 July 2011 (UTC)
I have read the Square root of a matrix article.
I have a real square positive-definite but possibly non-symmetric matrix M, and I seek a (preferably real) matrix S such that transpose(S)*S=M (the * indicates matrix product). I don't have to be able to construct S, it suffices to know that it exists (does it?).
Cholesky decomposition will not do, as the matrix M is not guaranteed to be symmetric.
A square root based on diagonalization won't help either, as that would give S*S=M and not transpose(S)*S=M
The Square root of a matrix article writes further
"In linear algebra and operator theory, given a bounded positive semidefinite operator (a non-negative operator) T on a complex Hilbert space, B is a square root of T if T = B* B, where B* denotes the Hermitian adjoint of B. citation needed According to the spectral theorem, the continuous functional calculus can be applied to obtain an operator T½ such that T½ is itself positive and (T½)2 = T. The operator T½ is the unique non-negative square root of T. citation needed"
This seems very close to what I'm looking for, but, alas, "citation needed". I know next to nothing about continuous functional calculus and that article isn't very helpful.
213.49.89.115 ( talk) 17:47, 14 July 2011 (UTC)