consider-an-economy-with-three-sectors-chemicals-metals-fuels-power-and-machinery-chemicals-sells-30

Question

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.

James Nyumutsu · Accepted Answer

to start with. Let&#x27;s define some by rebels. Let&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s distributes for the presents is old point for off its output to the chemical sector 14% don&#x27;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&#x27;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&#x27;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&#x27;s look at the second sector 246 It&#x27;s expenditure. This column. We&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;ll take East Omar tricks. This will be three by three matrix. So let&#x27;s do that. Which arrived in my trace in the form a Patrick&#x27;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&#x27;s opening full. And the third equation. The last room Open. Five. No punchline. Negatives. Open to eat. That&#x27;s my foot, of course. Efficient. And for the variable ever I was Will be PC p of in pm and close to He&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;s implying that PC one PC he&#x27;s imposed toe. So let&#x27;s take this to the other side. Let&#x27;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&#x27;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&#x27;s seven UNIX. That is our answer. So finances these. These these and these

Runpeng Li · Answer

okay to solve this problem. When you do first we need Thio set up the table. Here we have the four sectors A for agriculture, E for energy and for manufacturing and Tibor Transportation. And this column represents the her just fine purchased bye and a e m t. All right, so by our assumption sector A sales 10% to e and 25% to m. So that is 21 T and 25% to M and retains the rest that it is 65% and signal t cells. By our reception, he sails 30% to a 10% two and 35% to m in 25% to t. Now I am sales 30% to a 15% to eat, 15% thio, 40% to t and transportation sales, 20% to t to A and excuse me and 10% teat. 30% too and 40% to teach. All right, so now we can write down the equations. That is 0.65 times price of a plus 0 23 times price of you plus 0.3 times price love plus 0.2 times price of tea, and that is price off. And second question will be 0.1 times price up, Kate. Plus one times price of heat US 0.15 times price of that. Because when one times price up T equals two. So this whole, this whole summation equals price of and the 30 question here would be 0.25 price. Okay, plus 0.35. Price of he plus 0.15 price up, then plus or three. Price of teeth equals price of heat. The last equation here will be zero times price of a plus 0.25 times price of you us point four times myself. Van lives 0.4. Sorry, I&#x27;ll rewrite this. So this is 0.4. Okay, point four ties. Price of thine should be equal to price of tea. Um, right here and it&#x27;s a table. Be price off them. Okay. Now we can write out the The Matrix by using this coefficient top of these equations. And remember, this is we have four sectors in this case, so the matrix will be a war by for matrix. So in this case, um, 0.61 0.65 minus one because we put the way. We need to put the price of a from the right hand side two left inside and several 20.61 minus one will be negative. Point 35 and the rest, oh, remains to say 13 0.3 12 And similarly, we have second role, which is one connective point. NY 0.15 0.1. The third rule 0.25 1 35 and negative 0.85 and 0.3. Fourth row will be zero 0.25 and point for a negative 0.6. All right, So the rest will be We need Thio used the gardening illumination Do turn these meetings to a redo station one. Now, he&#x27;s quite hard to do this by hand because we have, you know, a lot of decimals here. And one thing I would recommend to do is to used the Wolfram Alva or Houston men that or numb pie. Just use the programming. Just use the program. Mean to, um to help you, in this case, out just right down the solution Here, uh, the next page. Just write down the final reduce station form here. So Daddy&#x27;s 10 zero negative two point 0 28 Second role will be 01 zero negative to a point. Life 31 third rule will be 001 and negative one point 1 68 And the last rule would be all zeros. So our vector here were created from for the quitting Bram prices. That is, we have Ah, remember, we have. This is the price of Hey and price of price off E and price of, um, and price of t. So price of tea will be repairable here and price of a will be too. Time will be 2.0 28 times myself, T and price of e will be. Is there a point 5 31 times Price of D and price of M will be 1.1 68 times price of two and last country. It&#x27;s just price empty because price of d, he&#x27;s, uh, pre bearable here. Now we take out scaler. Price up, T. Here&#x27;s our victor. Two point 0.8, zero point five, 31 1.1 68 and one. Okay. Now, to find a set of equipment prices, we just take, um let&#x27;s say we take price up t to be equal. Two 1000 now. So price off a determined by price of tea would be two 0 to 8. 2028? Yeah. All right. So price of price off e, determined by price of tea will be 5 31 and price off AM will be 11. 68. And here&#x27;s our final solution.

Mrinal Rana · Answer

So if X units are sold, the cost per month is gonna be the fixed costs, which is $400 plus the costs per unit, which is $45 per unit. So have plus 45 x. So that&#x27;s the total cost on the revenue. There&#x27;s gonna be ISS $69 per unit. So that&#x27;s 69 x on and to, ah, break even. I&#x27;ll just equate the costs on the total revenue. I have to find the number of units, so I have 8400 plus 45 x equals 69 x, and I have 24 X equals 8400. Therefore, X is just 8400 divided by 24. That&#x27;s equal to 350 units said these men units must be sold to break even. Onda Part B asks the revenue required to break even. So that&#x27;s just 69 times through 50 not sequel toe. 24,001. 50 dollars

Sarah Lewites · Answer

So this is a multi part problem, and we&#x27;re thinking about the supply and demand and how they both the pedal price. So we&#x27;re giving these two equations, and they both really, these variables p and G In this first equation, this is about the demand, and demand is based on price, so D stands for demand, and that&#x27;ll change based on the price going up or down in the second equation. Deace standing for supply and P is the price, so suppliers will only make a certain amount if they have a certain price of their working with and D. It means two things, but it&#x27;s given by the same variable because it actually will mean the same thing. When we get to part B of this cause, we&#x27;ll be looking for an equilibrium point. So first, though, we have Teoh Saul for when supply is going to be greater than zero. So we have to find what&#x27;s the minimum price at which manufacturers are going to start supplying oil. If it the price isn&#x27;t high enough, there&#x27;s not going to be any supply because they won&#x27;t be making enough money. So we need to take the second equation And when you defined the price P that gets de higher than zero. So to do that, we&#x27;re going to set D equals zero and solve, and then we&#x27;ll round up to the next to nearest sent. So we&#x27;ll set d equal to zero. So doing a piece squared, minus it here minus zero. So we don&#x27;t you try it, it equals 12. And so we have a quadratic that we gotta solve. So eight p squared minus a p minus 12 equals zero. We subtract 12 so we can factor if it can factor. Or we can have our A, B and C ready to go into the court attic formula. This does not factor. So we&#x27;re gonna use the quadratic formula X equals eight cause a Rennes, the square root of negative eight screw, which is 64 minus four times eight times negative. 12. That&#x27;s all over 16. And so we&#x27;ll use a calculator for this because it&#x27;s not rational. We don&#x27;t know graphing calculator. Just a basic calculator will do. So we got X equals eight plus or minus the square of 4 48 And so we might have up to two solutions here because we have the plus or minus option. So the first option is X calls 1.82 for the plus option on then the minus option is X equals negative zero 0.8 to 2. We&#x27;re gonna reject this solution because this is a negative price, and that&#x27;s not in the domain of this problem. So though it is a solution to the quadratic, it&#x27;s not actually a solution to this real world application based problem. So this is not a solution that&#x27;s gonna work for us. This, however, is. And so this is saying, Ah dollar and 82 cents there not going to be making anything. But if we raise this, if we rounded up to the next nearest sense so $1.83 that&#x27;s when demand starts. So this is the lowest minimum price for oil that manufacturers will start making. Okay, So that&#x27;s the solution to our first part. We&#x27;ve got the price, the minimum price. The oil production will start up now as we&#x27;ve owned a part B. So I&#x27;m heartbeat were asked to use a system to find the equilibrium point and the system were using is the system of two equation that&#x27;s given at the beginning. Nothing&#x27;s different except for D. It means the same thing. Do you means most? Supply and demand. But equilibrium is one. Supply equals demand. So now D is just one variable. It just stands for oil. Um, what&#x27;s it gonna here? It stands for 10 thousands of gallons of oil in the marketplace. It&#x27;s the same variable. So we have to go about solving the system now for both P and D. We need to find the price at which will have equal of rib and also how much oil will be in the market place at that point. So to solve this system, we can go about it a few different ways. It&#x27;s gonna be hard to eliminate P because we have a piece squared and API as a linear term. So what would be easier? More doable is to eliminate D. So let&#x27;s go ahead and do that now it to eliminate D. It&#x27;s not quite ready to be eliminated because we have 60 a negative four deep. But if we multiply this whole equation by three halves, then this 40 will turn into a 60 so we can eliminate the So let&#x27;s go ahead and do that. When we do that, we will get 12 p squared minus 12. P minus 60 equals 18. So we&#x27;re dividing by two and multiply by three. Okay. And so now we&#x27;re going to add together this equation in this equation so we can illuminate D and keep going with our solving. So when we do that, we will get 22 piece weird minus 12 p because there&#x27;s just only not 1 12 p. And then the d eliminates, which is great, because that was our whole goal in the first place. Then that equals 1 62 All right, now this is a quadratic. Still, let&#x27;s go ahead and subtract 1 62 so we can move forward with solving it. Now this actually does factor. There are large numbers, but it does work out. So first, let&#x27;s take out, too. We got 11 p squared minus six p. Rinus eating one. Yeah, it takes a little bit of puzzling out. It&#x27;s totally fine to use the quadratic formula as well, but this does talk to really beautifully. Actually, we have to set up as 11 p and P because Elevens prime on the factors of 81 will use our 27 negative three. So, plus 27 here, minus three here because then we&#x27;ll get 27 p minus 33 p, which is negative. 60. So supercomm being it, Okay. And so then we have two possible solutions. Either p equals negative 27/11 dollars, which it doesn&#x27;t cause that&#x27;s a naked of dollar amount. So we gotta reject that friend goodbye or P equals $3 which that is going work. Okay, so we have found the price point at which were at equilibrium at $3 the amount of supply and demand will be equal. We actually have to assigned that amount, though, so we need to go ahead and plug in the $3 into one of our equations from earlier. It looks like this first equation will be a little simpler, so let&#x27;s go ahead and plug it in there. So let&#x27;s actually rewrite this equation. In terms of d, we&#x27;re gonna have 60 equals 1 44 minus 10 piece where it&#x27;s just subtracting the 10 p squared and then we plug in P goes three. So you got six d Because 1 44 that is 10 times nine just 90 this is 60 equals 54 and D is gonna equal nine. And then if we go back and look at the units it&#x27;s given in tends of thousands of dollars per gallon. Sorry, tens of thousands of gallons. So this nine actually represents 90,000 gallons, and that is our answer here. So at equilibrium, the market will be at equilibrium. $3 for gallon and 90,000 comments.

Wesley Hines · Answer

We&#x27;re looking to come up with a Matrix products that would give us production costs and in order to get total production costs related, promoted by the quantity that&#x27;s been produced by the cost it makes to produce that quantity. So we&#x27;re going to come with a matrix for Q and Matrix for Pete. But the quantity major X has already given to us through the table provided, so we can just copy that and just remove all of the numbers and all the everything under that is not a number. And I will give us the four by three matrix of 1923 27 17 2122 2118 and 20 and then for the final really of 27 25 22 that will represent the quantities of the products have been produced and then likewise, we need a price matrix. And again the price is also given to us for the cost. In this case, it&#x27;s going to us by the three by two matrix representing the day shift and the night shift, and for that were given 1.21 point 35 wait 758 fine and then 3.5 and 3.7. Now we want to come up with production costs. And will this give us production costs? Let&#x27;s think about what multiplying these two would do. So if we multiplied the first row together, you&#x27;ll be multiplying it by both the columns, respectively. And so we moved by the first row by the first calm that would represent the cost of producing during the day shift and then, likewise, that would also represents for the second column the cost of producing during the night shift. So since that is what we&#x27;re looking forward looking for total production costs and this will give us total production costs depending on what time of day it happens, we&#x27;re good to move by these together, and it should give us the answer. Now. It states in the problem that we already allowed to use a graphing calculator, so I&#x27;m going to be skipping the hard calculations. But it&#x27;s going to be equal to the four by two matrix of 1 34.55 1 45.1 1 13.15 1 22.2 one of 8.7 1 17.65 1 28.15 and finally won 39.1. And we&#x27;re going to interpret this matrix. Each row would represent the cost for the four factories of producing in the day shift for the first column and the night shift for the second column.

Ben Meadows · Answer

Okay, So the first part of the problem asked for us to calculate the profit for each quantity, Um And then ask us how much should affirm produced to maximize profit. Well, this is pretty simple. So what we can do is we&#x27;re going to just subtract they total revenue less the total cost, and that&#x27;s going to equal our profit. So for each one of these columns here, we&#x27;re going to go through and calculate the profit. So let&#x27;s do that right now. So for the first unit, if we produce zero units, then our total costs is eight. So we would get a profit of negative eight for that zero unit, but over here, all right. Total calls is nine. Total revenue is eight for still taking a loss. So that b negative one. All right, for our second unit. All right, now we&#x27;re turning a profit, which have a profit of six. Total revenues on the third unit is twenty four minus eleven. So in that case, we will be making thirteen dollars profit above over to the next unit, and we will be making nineteen on the five twenty one six units, twenty one at seven units nineteen. Now what? You jump out to you right here. Is that the move from six units two seven. Unit gets us less profit. So a rationally minded, uh, business owner would never want to make this jump. You&#x27;re losing money for every for the next unit you&#x27;re producing. So what&#x27;s the maximum amount of profit I could make? Twenty one dollars. So? Well, how much should have firm produced to maximize profit? Well, it shouldn&#x27;t matter. It should produce six units of whatever, uh, item it is making. Okay. All right, so that&#x27;s profit on being asked for marginal revenue and marginal calls for each quantity. So let&#x27;s go ahead and do that. Um, when we think about marginal revenue, right. That&#x27;s just saying, what is the, uh, revenue from producing the next output? Right. Next item. The marginal unit. So if we&#x27;re moving from zero to one. Well, how much? What is the marginal gain and revenue there? Well, so we were We used to be making zero revenue. Now we&#x27;re making eight, So the are marginal revenue in that case is eight. Okay, Well, what if we move from one to two? All right. That means we&#x27;re now going to be moving from revenue of eight to sixteen. Ah, well, again are marginal revenues. Eight. Well, you should pick up on a pattern really quickly that if you go through and do every single one of these sure enough, the marginal revenue is eight at every turn. So what does this tell us? Well, for every extra unit this firm is producing, it is still getting the exact same revenue from that next unit. So what does that tell us? Well, this is a price taking firm. And what does that mean? If it&#x27;s a price taking firm? Well, it&#x27;s in a competitive market, which is goingto answer next point in, Ah, part B. But real quickly. Let&#x27;s get back to it. Let&#x27;s get in to our marginal cultural All right. Margin are marginal costs, right? That&#x27;s what we&#x27;re doing next. All right, so here is just the same thing. Moving community, You know, what is the difference in cost? So as we go from zero units toe one unit Marshall calls is one nine to ten. One eleven. All right, eleven to thirteen. A way out. Just when we&#x27;re going from six units to seven notice that we have a big jump until on are marginal calls of ten. Okay, so that completes that part of the problem. Now we need to go over and graph this, and we&#x27;re going to see something that picks up on this. So if we go over to our graph, you&#x27;LL notice if you plot this out correctly, that for our marginal calls, remember? Over here. One one, one one. All right, so we have this flat curve here. Heard the instruction that says we need to plot this in a line. So as we, uh, move past, what do we say? A cz. We moved from three units to four units. You&#x27;LL see that our mortal call starts increase. This intersects with our margin revenue, which we said wass Ah, flat eight for every unit. And this intersects that six. What does this have to do with one another? Well, notice that marginal revenue and marginal cost intersect right at the point where we know where we know that profit is maximized right there. That&#x27;s where a profit was maximized. No surprise here that that is actually where these lines intersect. So like we mentioned earlier, it is in a competitive industry because we see this really flat marshal revenue curve. And then also, can we tell if this industry is in long run equal a gram when we actually can&#x27;t tell? Because we have no information on long term entry and exit, So we do not know the answer to that question.

Problem

Suppose an economy has four sectors, Agriculture …

Problem 3 Medium Difficulty

Answer

a. See solution
b. See Solution
c. 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 $p_{F}=91.7 .$

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Algebra Educators

Catherine R.

Alayna H.

Kayleah T.

Maria G.

Algebra Courses

Absolute Value - Example 1

Absolute Value - Example 2

Recommended Videos

Watch More Solved Questions in Chapter 1

Video Transcript

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

Linear Algebra and Its Applications

Catherine R.

Alayna H.

Kayleah T.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

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 $p_{F}=91.7 .$

Linear Algebra and Its Applications

Catherine R.

Alayna H.

Kayleah T.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

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

Catherine R.

Alayna H.

Kayleah T.

Maria G.

a. See solution
b. See Solution
c. 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 $p_{F}=91.7 .$

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

Linear Algebra and Its Applications