the-leontief-production-equation-mathbfxc-mathbfxmathbfd-is-usually-accompanied-by-a-dual-price-equa

Question

The Leontief production equation, $\mathbf{x}=C \mathbf{x}+\mathbf{d},$ is usually accompanied by a dual price equation,
$\mathbf{p}=C^{T} \mathbf{p}+\mathbf{v}$
where $\mathbf{p}$ is a price vector whose entries list the price per unit for each sector's output, and $\mathbf{v}$ is a value added vector whose entries list the value added per unit of output. Value added
includes wages, profit, depreciation, etc. ) An important fact in economics is that the gross domestic product (GDP) can be expressed in two ways:
$\{	ext { gross domestic product }\}=\mathbf{p}^{T} \mathbf{d}=\mathbf{v}^{T} \mathbf{x}$
Verify the second equality. [Hint: Compute $\mathbf{p}^{T} \mathbf{x}$ in two ways.

Michael Jacobsen · Accepted Answer

In this example, we&#x27;re going to be considering the Leon Tiff production equation, which is given here, as well as the dual price equation where P is equal to see transpose p plus v. Here he is a price vector and he is the value added vector. And we also have an equation that tells us that the GDP equation is GDP equals p transposed times D, which is also equal to be transposed times D. So our goal here is to show that these two quantities are in fact equal as the GDP equation tells us it should be. Let&#x27;s start off with the dual price equation that P is equal to see transposed p plus V. Now I&#x27;m going to look at P transpose coming from the left hand side of this equation we have P transposed is going to be the transpose of this entire group. So we&#x27;ll have see transposed p plus v transposed and we get next the following first note that when we take the transpose of this product, we have p transpose times see transposed transposed, which is C So we obtain the transpose is in the reverse order of the product. Then add be transposed together. So this is what P transpose equals. If we were to take, say, P transpose times a vector X, then this would result in p transposed times the production matrix c times the production vector X plus the value that it vector transposed times the production vector x again. So this is what P Transpose X would be equal to. But we also know that X is equal to this quantity. So let&#x27;s take another look at the equation that we had from here. We have P transpose equals p transposed time See plus V transposed. If we take p transposed times the vector X again and using the fact that X is equal to see X plus d, we&#x27;ll obtain that this is equal to p transpose times c x plus deep. Then this tells us altogether p transposed Times X is equal to If we distribute this vector on the inside p transposed times c x plus be transposed times deep. So now we have two expressions for P transpose x, and they both include this term here p transposed time. See Time&#x27;s X. That means if we set these two quantities equal to each other. We&#x27;re able to conclude immediately that V transpose X coming from the left hand side is equal to p transpose times d coming from the right hand side and this verifies the equation above.

Carson Merrill · Answer

so put fitness function. We&#x27;re going to begin with p evil. That&#x27;s what we&#x27;re given. And this is, you know, is equal to 200 hour plus can l squared on ground and in just a second spell you how squared. And this is l cubed and then were asked to find the average function. So the average function is, um, the same as PFL, divided by allies What we&#x27;re told. So every hour average function is going to equal 200. That&#x27;s trying to tell Divided by l Yes, l all right. And then lastly, the last something would ask. Defying is the marginal product function which is characterized by P primeval or the derivative of the product function. So remember, how looks you 100 honey minus three l So these functions are gonna be helpful, Um, and were able to interpret things of these functions. Thes three functions will be really crucial. Um, when dealing with product, we can look at the average here by dividing by l. And then marginal is taking the derivative. So we&#x27;re asked to show graphs for this, and this is what it looks like. So we have green is our PFL black is our average function are average product function, and M is the marginal product function and looks a little something like this. Obviously, depending on, um, the window size and stuff like that, it might look different, so you&#x27;ll have to adjust for that. But this is what d the graph should look like based on this window right here. All right, so after part B, it suggests that the peak of the average function and occurs at AL equals l not. And what that means is that a prime of L not is equal to zero. So we want to show that for the product function that Emma Velma is equal to A of AL got So to do this, it was Give me party. Let me start off with you now and we know it has to do something with the property function. So we know that the devil is related to pee about because a Veliko Pierrot invited by l next What we need to know is we need to get PFL by itself. That could be really important for this problem. So we multiply both sides by our miraculous aml times. L is equal to p a bell. All right, simple enough. Next we know that the we&#x27;re trying to show something with Emma Velle, and we know that a marvel is the derivative of pl So we are going to take the derivative of both sides giving us p prime equal to and here with me is the chain role. So it&#x27;s on me a prime about Oh, plus, yeah, well, right. Because we know, um, equals Emma. Well, we&#x27;re gonna subsequent at in middle equals this. Yeah, And now I&#x27;m is the crucial step. So we&#x27;re being asked specifically to do this for l not, but we know that this is true for all l possible. So we can input l not. And we&#x27;ll see why this is important. Just a second. Well, not is going to look like this, but for the sake of time, I&#x27;m just gonna type in a zero. So that&#x27;s I&#x27;m not right there. So we have and l not. And that is equal to a primeval nut. Oh, no, class. It&#x27;s pretty simple orders in putting all not infer out. So it&#x27;s a more specific value. We know that a prime of Elna is actually zero because That&#x27;s one of the things that tells us in part B. So we now see that I&#x27;m not equal with you your own time. Al ni plus a l not. And because we&#x27;re multiplying here by zero, we&#x27;ve actually proven marvel, not his equal to okay.

Wen Zheng · Answer

the problem is the total production. See off a certain product depends on the amount L of Labour used and mark k off Captain must mint in sections fourteen point one hand. Fourteen point three we discussed House Come Douglas model. He is therefore to be times out use the power of our far and skate to the power of one man. It&#x27;s Alfa follows from certain economic assumptions. Why b and afar Positive Constance and afar is less someone. If the cast off unit of library is M and it&#x27;s a cost of your native capital, his end and the company can spend on a P doors as it&#x27;s to goto part it, then maximizing the production he is subject. She&#x27;s a constrained ml class in k c showed that the maximum production occurs once. How is he? Culture half attempts p over. Um Andi Casey Goto one Manus Alphonse Towns people were in If we write genius, he caught you. I&#x27;m l okay. MBAs my third off the like range much players we have gry didn&#x27;t if they got London ham screening and yeah, genius you which become ah far times being comes out to the other half I Minus one times came. You want madness can fart. It&#x27;s the lambda. Come, Sam One minus alpha times B l too. F ar Kate off Are being too negative. Are far. Yes, the panda. I&#x27;m ill Plastic. Okay. It&#x27;s the people. The Christian one. Why the question too? We have fr ask a over one month off, have L. It&#x27;s because I am over him. So we have Kay did come true. One minus far comes now over in Tam&#x27;s offer, plug it into the Christian three self case. Okay, if they want to. One oneness offer comes You. All right, So So we&#x27;re plugging this one to really question three with self l. So Well, it&#x27;s equal to half a p over m. Okay. Is the old one minus alpha times L over now. Farm. Which is he called too? One minus offer comes. All right then. So if the maximum production he insists it can only or clear one l sicko too, as I appear for M K. Is your one minus alpha times. Pierre over there

Aamir Mithaiwala · Answer

we are, we are given is kind of scary looking polynomial to the fifth degree. And so we want to know. So since this is a model, since this is a model of GDP, for every year after 1933 we I want to know, Do you want no at at 1930 34 and at at 1942 1942 as well as 19 as well as 19 49. What was our GDP? And so since we already given the function and X is the number of years after 1933. So let&#x27;s set set our base case year to be 19 year 30 to be 1930 three. In other words, that is when X is equal to zero. And so, since that&#x27;s when X is equal to zero well, 1934 is when X is equal to one in 1942. That would be when X is equal to 1940 to minus 1933. That would be access equal to X is equal to nine and so 1949. That would be when X is equal to X is equal to 60. And so so with the With the nasty function, we can use our graphing calculators to plug these values. And so and when we plug in, only plug in acts is equal to one Begich, 5700 5000 700 and $78.1. And when we plug in X is equal to nine, we get we did nine 1003 100 and 78 0.2. When we plug in X is equal to 16 into the problem, we get yet 4000 or 1000 400 and to point $1. And so, since the problem looked really scary, we know that we can just use our our graphing calculator to to plug in these points and figure out what the value at this year this dysfunction gifts. And so since 1933 was are basically remember we said X equals zero. Then all we had to do is was just plug in. And so now let&#x27;s figure out, let&#x27;s figure out in this rain so we have from zero to this is this is 17 0 to 7 teen because we&#x27;re going from 1933 to 1950 so 17 year age range and our max we can clearly see is somewhere at this point right here. And so if we were to drop that down here somewhere, we would say, Okay, well, this is somewhere after 10 years, but before 11. And so when calculating years since sometimes. Well, even if this was let&#x27;s a 10 point, I&#x27;d say this is 10.7 10.7. If that was our X value, well, that would still count. That would still count as the 10th year, because if we add 10.7, this 0.7 is just how many months after we started 1/10 year. This 10.7 just means we&#x27;re still at X, is equal to 10. And remember, if we set 1933 1933 as our base year, we need to add 10 when we add 10. What we get is that at this year, 19 43 this was when we had Max two, in other words, the highest GDP. And so since this function looked really scary, it&#x27;s actually it&#x27;s actually very useful for economics for a calculating GDP. Um, and even though it&#x27;s a scary looking polynomial is very polynomial zehr very useful in all in all studies. So in this case, we use the real world application for a polynomial to figure out GDP within a given given your range.

Charles Carter · Answer

problem gives us the rather complicated at formula. But we&#x27;re just gonna be plugging in for each one of these values. So just I would replace each value with a set of parentheses. Um, I want to show you what that looks like. So we&#x27;re gonna have wise equal. Teoh, my see, value is 1 20 uh, plus 0.9. Why my I value is gonna be 100 and 40 my g value. It&#x27;s gonna be 150. Make sure you&#x27;re plugging into right variable to the right places. My ex value, this 16 and my him value is 20 plus 0.2 I Okay, so just be careful with your parentheses when you drop them. Eso will have. Why? Just combining some, like terms of 1 20 plus 1 40 That&#x27;s to 60 plus 1 50 4 10 uh, plus 60. That&#x27;s good. Before 71 in minus 20 is gonna be 450. Is there constant term? Let me have 0.9 wine. Uh, minus point to why? Which is 0.0.7? Why so just combining all my life terms the best track 2.7 y from both sides. I&#x27;m gonna have 0.3. Why is equal to 450. And if I divide both sides by 0.3, I have 15. Y is equal to 1500. So just go back to your problem and check all your units. So why would be our national income? Okay, so we have 1500. I don&#x27;t know what the units are. Units of national income. Good. Thank you very much.

Problem

Let $C$ be a consumption matrix such that $C^{m} …

Problem 11 Medium Difficulty

Answer

$\mathbf{p}^{\mathrm{T}} \mathbf{d}=\mathbf{v}^{\mathrm{T}} \mathbf{x}$

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

Linear Algebra and Its Applications

Catherine R.

Anna Marie V.

Heather Z.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

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$\mathbf{p}^{\mathrm{T}} \mathbf{d}=\mathbf{v}^{\mathrm{T}} \mathbf{x}$

Linear Algebra and Its Applications

Catherine R.

Anna Marie V.

Heather Z.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

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

Catherine R.

Anna Marie V.

Heather Z.

Kristen K.

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

Linear Algebra and Its Applications