consider-the-production-model-mathbfxc-mathbfxmathbfd-for-an-economy-with-two-sectors-where-cleftbeg

Question

Consider the production model $\mathbf{x}=C \mathbf{x}+\mathbf{d}$ for an economy with two sectors, where
$$C=\left[\begin{array}{cc}{.0} & {.5} \ {.6} & {.2}\end{array}ight], \quad \mathbf{d}=\left[\begin{array}{c}{50} \ {30}\end{array}ight]$$
Use an inverse matrix to determine the production level necessary to satisfy the final demand.

Michael Jacobsen · Accepted Answer

In this example, we are dealing with an economy that has the following consumption matrix C divide defined here as well as the demand vector d as here. What we like to do is to find out the production level which will call X given this matrix and this demand vector well, the first step is to four mount the model for this type of situation which is of the form X. The production level equals the intermediate demand, which is C Times X plus the final demand, which is deep Now if we rearrange this equation by subtracting, see extra bull sides, we have the equation I minus C times a vector X equals the demand vector d And then we know by our theorem in this section that this matrix is in vertebral. Since the column sums are both less than one. This then tells us that the solution will be X equals I minus C in verse times d. So let&#x27;s start off then in our solution by finding I minus c inverse here. So we&#x27;ll have I minus c is going to be first. I consist of just ones on the main diagonal, So we&#x27;re going to subtract one minus zero from the main Dagnall of sea to produce a one here and will have a negative 0.5 here, a negative 0.6. Then take one minus the point to, and we have a 0.8 altogether. But what we need here is its inverse. So let&#x27;s take the determinant of I minus C. It&#x27;s going to be the product of the main diagonal. So 0.8 subtract the product of the off diagonal, which is negative 0.6 times 0.5, resulting in a point 30 And we get all together that the determinant of this matrix is 300.5, which also tells us it is in vertical as we predicted. So let&#x27;s calculate its inverse Next I minus see in verse is going to be first take the main diagonal which is in purple and alternate it so point it goes here and 0.0.1 goes here. We&#x27;re going to then negates the off diagonal these entries here So we have a positive 0.6 a positive 0.5. Then divide by the determinant which is one over 10.5 Well, one over 10.5 results in two, so the inverse becomes 1.61 1.2 and there was an issue right here. This should have been a positive one or a point not 10.0.1, but just one itself multiplied by two and we get a two here. So this is our inverse matrix. This now tells us that the production level required to meet this demand vector will be X equals First, the inverse matrix of I minus C, which is 1.6 1.212 Multiply by the demand vector D, which says take 50 from the first sector, 30 from the second sector as the demand and multiply we see then that will require ah 110 units from the first sector of the economy and 120 units from the second sector and the solves our problem here.

Carlos Pinilla · Answer

all right. Suppose that we have, ah, production of us. Uh, well, you guys are revenue. These seek sex. The cost on the production of grease X, Q 966 plus steam. Thanks. So, Oh, you&#x27;d like to maximize the profit in this case, huh? Drop it. This would be just, uh, go even so, same revenue. Disgust. So Hassan&#x27;s pointless better, you know, a soul. Uh, p movie call to the Ram knew where I was. The cost. Which deaths of these functions is equal to seeks sex minus. Excuse me. I seek. Sex is squared. Plus 50. Next. So So you want to find emotional of the right food. So these people this year, so be prime. You&#x27;re able to seeks minus three x squared. He&#x27;s minus and minus. So that comes a plus. Plus six times still lost. 12. Cool. Thanks. Minus 15. 15. Just 15. Does he do doodle Flex? So is what, uh, what are we let out to my nose. Three extra square must. Well on. He&#x27;s C minus 15 6 is equal to mine. Was like so, uh, become becomes plugged or minus three from off that it&#x27;s not my house. Three access times X squared minus. Ah, you get Ah, uh, there was 12. Yeah. You stay here, huh? You should have seen no banks. It&#x27;s wrong to kill Flow X. So we found her. Three Rabia minus works. Glass three. So that, uh, we contractor dad us My three in here. The something needs to be minus four. Maria&#x27;s experience nothing. Times X minus something. The barrel has to be three. And there&#x27;s some muscle. We went as far as old minus one minus three. So, uh hey, look at our What is the tunnel lies with this sign. Brian Soapy Prime signed off on Brian Will be. Will be changing out one three. Right to it Is the sign here of years minus 41. Mine. Oh, sexy. Squares off. This is being reform claw minus negative. The receipt in velocity funding. Serene that this point. So many Mom. Uh, so these 0.3. The election. So you look good. What is the revenue three be able to seeks? Come three, That is 18. Nobody&#x27;s These are free cost because function at three. So he said cost three people too. Three Q minus six times squared plus 15 times three. So these are it is 45. My buddy said nine things. Six, she will do 54 minus 54. So this too added together they are minus nine. And in all these jamborees, 37 simply serve like his line. These people do, huh? Also eating, being will you compare? It is the same costumes they revenue. So them are small properties. Just tool Go. Even so, revenue is the same you see as the cost. So they probably this year that is in next month. What?

Haggai Liu · Answer

recall that the production matrix is given by X is equal to I minus a inverse times. Okay, okay, so let&#x27;s compute. I minus a I minus a iss. Well, we just subtract that the Agnos from one negate the rest with These are native point your point. Who this Is there a point to this is zero point. Okay, how do we invite this? Remember to calculate the inverse of a matrix. We need to take one over the determinant of that matrix, which are which we can calculate by applying the diagnose and, uh, and applying the advent, subtracting the results, putting the left the down down hill product first so that you maybe I&#x27;ll use different colors. So remember, the blue product comes 1st 0.2 times 7 43 Hey, now, plus now the green. But I, uh What do you want to? And Ideo too? That&#x27;s a, uh, that should be minus. And then we&#x27;re doing We need to multiply by certain matrix. What matrix would that be? Well, we we want to swap the diagonal entries. Hold on. We want to swap the diagonal entries, and we want to negate the rest. The off diagonal entries. Okay, so let&#x27;s work this out. Eso This is wine over. So the blue product is 0.6 their point. There are six. Fired me on the green product. It&#x27;s to a 0.4 So this is 0.2 Harder to okay, it&#x27;s a quiet three. No point to no point to to What is that? Give us that gives us well, remote fly everything here by a 50. So that&#x27;s 15. That&#x27;s 10 and 10. Okay, Now buy up X or just be 15 10. And And this matrix multiplied by our demand matrix, which is 23 All right. Okay, so let&#x27;s both plague Multiply this out a bit. Bus 30 plastic. 30 s the 60 and 50. Okay. And that gives us the production matrix. Thank you for watching.

Haggai Liu · Answer

So what we need to find is the production Matrix X, which is given by I might as a inverse times. Okay, So first we need to solve where I minus a so that that can be obtained by subtracting the diagonal from one give 79 here that 0.4 hadn&#x27;t negating the off diagonal. But when there was really here and zero point Okay, so now let&#x27;s find, I might say in verse Oh, sorry about that. Everyone is a book. So that&#x27;s one over the determined off of I minus a which we calculate by a multiply taking the product of the diagonal entries, subtracting from subtracting the product off the often ignore it tree. So it&#x27;s the blue product minus the green product so that this is their point Night, huh? Times zero point for minus. Native. Your 0.3 times native 0.7 on. We need what, five by a matrix. So what we need to do is swap the Diagne Oh, entries that require 40.9 here and then they gate the off diagonal entries. Okay, so then Okay, lets see. So that&#x27;s simplified is a little bit and its compute. What is the bottom? I&#x27;ll give you the 0.3579 ah, times this matrix. Hey, now what? What do we want to do now? We&#x27;re gonna Now we got a compute X, which is just that rain matrix. But we have to multiply, Pa Yeah, demand Major George&#x27;s 5 10 That&#x27;s sure. Yeah, by 10. So would you. One over. 0.3579 And then what I want to do is I want to go se the screen number times There&#x27;s green number. Okay? Plus the blue number times this blue number. So if I take the green product, that would be to on If I take the red product, that will be to appoint three. Yeah, that&#x27;s still career on red like that. So multiplied. Agree numbers together the best 0.35 on the blue number Ski if you nine. Yeah, and that&#x27;s simplified a little on three. 9.3 by thanks. So now let&#x27;s have to do with some division here. 2.3, divided by 0.3579 There. Yes, 6.43 This is approximately, of course. And, uh, then if I take a nine take a 9.35 divided by point 3579 I get around 26 0.12 Hey, that&#x27;s the final answer. Thank you for watching.

Haggai Liu · Answer

All right. So, uh, what we have to do here is ro reduce the augmented matrix I minus a are acquainted with. Okay? And this is easy enough to write down. But that&#x27;s Ah, 7.20 native 0.1. Okay. And, uh, this one native dio 11 don&#x27;t point. My maid Teoh point to on this one is 0.3. Yes, the extensible. All right, so yeah, let&#x27;s crippling roll reduction to this. So the So Now I want to turn this native 011 on the first column there to zero. So what I the row operation I want to do is I want I want to add half of ah ro one, then road who plus 1/2 row one. Okay, so what do I get if I do this? Okay, so of course, the first row doesn&#x27;t change and asked the road out. That doesn&#x27;t change. And then the night of their point to and you&#x27;re adding when half a row, one which is and is open, you find So this so his negative 0.25 All right. And ah, so it will also doesn&#x27;t change. Okay, so one here. Three here. Now, this six. I just added to have a 13 pumps. 6.5. Okay, so now what do we want to do? We would like to Well, let&#x27;s actually just proceed to backwards. Substitution. Okay, So since the Matrix lives up the coefficient matrix upward for Anglo lets it write down equations. But we have the third Where the last rogue, if that&#x27;s still 0.3 X three years equals three. So extra reiz equal to 10. Okay, so now the second road gives us the five axe to minus 0.25 Yeah. Yes, it&#x27;s a 0.5 x three is equal to 6.5. So that means 0.5 x two minus. So they went to five times a 10 gives us two point by and then 0.2 x two gives who pined me. 2.515 x two gives us six months. My plus 2.5 for just nine. So x two is x two is 18. Okay. And finally, the post equation. Give us still point to x one plus Well, excuse me. Minus minus 0.1 x three equal one Okay, that&#x27;s that. Looks good. Very good. And, ah, so but this is just so here. You just subtracting or what? Right? Cause actually is 10. So then 0.2 x one is equal to two people to to So acts one, It&#x27;s equal to 10. Okay, so finally we have our production, They tricks to be 10 18. And thank you for watching.

Michael Twiton · Answer

Let&#x27;s find where they intersect each other. So 35 minus Q squared equals three plus Q squared, that is, two Q Squared is equal to 30 to divide went to. To get 16 is equal to Q Square, so the equilibrium quantity his four the positive square root of 16. And that means that the equilibrium price I can plug in and either of them um, so it could be three plus 16 which is 19. No for a plot. We have one problem opening downwards and one opening upwards. This one started 35 going down. This one starts at three, and it goes up. And we know the interest section here is for on the Q access and 19 on the P exits. So we just need to find the area here, which is the integral from 0 to 4 off 19 minus. They supply curve three plus que square a que, and that is the interval between zero and four off. 16 minus Q. Squared. Thank you, which is 16 que minus Cube cube over three. You evaluated between Hugh Egos four and zero, but I don&#x27;t really need to evaluate it. Zero, because that will give a zero, so it&#x27;s really just 16 times four minus for Cube over three, and that is 42 2/3.

Problem

Repeat Exercise 5 with $C=\left[\begin{array}{cc}…

Problem 5 Medium Difficulty

Answer

$=\left[\begin{array}{l}{110} \\ {120}\end{array}\right]$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Anna Marie V.

Heather Z.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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$=\left[\begin{array}{l}{110} \\ {120}\end{array}\right]$

Linear Algebra and Its Applications

Grace H.

Anna Marie V.

Heather Z.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Anna Marie V.

Heather Z.

Michael J.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications