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Consider an economy with three sectors, Chemicals \& Metals, Fuels \& Power, and Machinery. Chemicals sells 30$\%$ of its output to Fuels and 50 $\%$ to Machinery and retains the rest. Fuels sells 80$\%$ of its output to Chemicals and 10$\%$ to Machinery and retains the rest. Machinery sells 40$\%$ to Chemicals and 40$\%$ to Fuels and retains the rest.a. Construct the exchange table for this economy.b. Develop a system of equations that leads to prices at which each sector's income matches its expenses. Then write the augmented matrix that can be row reduced to find these prices.c. $[\mathbf{M}]$ Find a set of equilibrium prices when the price for the Machinery output is 100 units.

a. See solutionb. See Solutionc. The general solution is $p_{C}=1.417 p_{M}, p_{F}=.917 p_{M}$ , with $p_{M}$ free. If $p_{M}$ is assigned the value $100,$ then $p_{C}=141.7$ and $p_{F}=91.7 .$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 6

Applications of Linear Systems

Introduction to Matrices

Missouri State University

McMaster University

Harvey Mudd College

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to start with. Let's define some by rebels. Let's the prize off the art foods off the chemical sector pc the prize off all the art ports, all the forest sector, it Dia's and the prize of old outputs off the machinery sector. It feel off course. The PC is also the price off all the outputs off the chemical sector. Why C F and M represents chemical sector for a sect oh a machinery sector respectively. Now to generally table for their distribution like this. Be seen if and see if and way it's called represents. They explained itself if we take them and each rule who represent the income so that will see the decision off the chemical sector. But chemical sex sector distributes 30% off its outputs to the four sector Fugitive Saints office output No much never sick toe and we'll take you there many percent not to be trying to present late written us, though points u P c. Therefore sector. This is where it's 80% off its outputs to the chemical sector, 80% within hours 0.8 the if and 10% off each article to the Machin ery sector 10% is 0.1 the if retains their remaining which is going to be is 0.1. But this time percent yes. And finally for the imaginary sicked up it's distributes for the presents is old point for off its output to the chemical sector 14% don't bring fall off its outwards to the four sector and the retains There were many ones which will be ton of my saints written us no point to Okay In fact, they shouldn't be in How old is she became? They should end generating the equations four expanded job to the equestrian Come Yes, expenditure off the chemical sector is old 0.2 pc plazo points h He is last, you know going for bien No, me goes to the output The income is going to his opening to PC plus opening three p supplies open five pc As a mother of fun, we've staked it already that late the prize of all the art foods on the chemical sector pieces. So this will be the income of the chemical sector. Never simply filings equation were taken all times to the right and side that would give us his 0.8 BC Let me use it of in color 0.8 PC then minus is open to eight. He if minuses opened four b m it close to zero, they should consider it as equation one. Let's look at the second sector 246 It's expenditure. This column. We'll be his air 0.3 pc plus 0.1 Do you have? Plus is opening for fear 0.4 p. M. Across toe the income off the F simply five These lay stick all the times the left in sight we have He's old by three pc pc then pf comes too late Inside we have minus so points nine he if and then we have classes opening for b m He close to zero No va, no time here. And that should give us equation too. It's with the second equation generation that site equation expend expenditure for the machinery sector is going to open five Does a column on of the M to Britney US is opening five Que si class is all point to one? Yes, lies. Is that all fine to you, B m. You're close to the prize of all the outputs of the National Satar Rescue Cities Income PM Taking all the things to delay it inside. Simply fire this equation. He should off 0.5. Do you see glass? His old points one he if, and hoping to PM on is PM we give us negative you don't 0.8 PM It cost those. Oh, Indust gives us a third equation So now we have now we have three questions. Let us arrested three questions. We have a question on a question to you. Question three. Presenting basing mattresses, the coefficients or viable after verbal said PCP F P M That will be will go for the variable column. Volumetrics and the 0.8 joke When I get 0.8 negatives open four is open three and it was open. 90.4 is open. Five is opened one and negatives opened. Eight. We'll take East Omar tricks. This will be three by three matrix. So let's do that. Which arrived in my trace in the form a Patrick's a My trace P in course to my tricks Be now this cool day zero was has to be so let us right is out the coefficient for the first room. The festival we have 0.8 negatives opened eight and negatives open for 0.8 negative. 0.8, and then it leaves open for the second rule. We have his opening three, and as your soul points nine, is opening full. His whole 0.3. No. 0.9. He's opening full. And the third equation. The last room Open. Five. No punchline. Negatives. Open to eat. That's my foot, of course. Efficient. And for the variable ever I was Will be PC p of in pm and close to He's in. Okay, So these now, Ma Trix, we asked to present it in automated. My trick for no major market fall These are these Whitney his old planes. Thanks. Negative 0.8 nuggets use opening for is open three negatives open. I you're playing for in the last row. You opened five. Mrs. David. Weakness is your French wine. Anything for his opens eight. Now we used said the official line. I do wish just to separate it. Phone is there was so there's actually the automated metrics form to solve this problem toe. Find the global prices and in time myself, The prize called the Machin ery Saito. Well, we do is you can decide to represent these. All my tricks live different variable or which they have a year from here to year. Is these magics A So let's maintain is finally reduced. Role National phone off my tricks, eh? My usual model out. Dese rational form off my checks. E That should give you one. Izzo's negative. 1.4 167 and zero. Why any team? Zero point nine flying. Thanks. Seven They last for me. PM is free Does for Matics A so represented this well, this automated form So they wrote there would usually have snow from off the automatic metrics is better within this way one, you know And it is one point for why se seven this will is over. Yeah, one, I guess you zero point 91 se serving Miss Rizzo on these gifts. Yeah, until what becomes? I said this implies that this is one full of arable p. C. He's implying that PC one PC he's imposed toe. So let's take this to the other side. Let's these monies these is because today So all this Basically these equals two The stick into moving these to the rights and side. We have 1.4 16 six seven tm thing for the season NPF close to they cost toe equals the point 91 six serving. Okay, so these are equilibrium prizes. And then finally, if PM if we close to 100 unit What's with the price of the raced and these implies ducks PC. It will be close to five points for one C soon times 100 ready and that will give us one for one. Finds shakes. Sorry. Yes, and he This will be closed toe So points nine 167 say so intense. 100 that would give us 91 wines. She's seven UNIX. That is our answer. So finances these. These these and these

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