exercises-17-and-18-concern-a-simple-model-of-the-national-economy-described-by-the-difference-equat

Question

Exercises 17 and 18 concern a simple model of the national economy described by the difference equation 
$$
Y_{k+2}-a(1+b) Y_{k+1}+a b Y_{k}=1
$$
Here $Y_{k}$ is the total national income during year $k, a$ is a constant less than $1,$ called the marginal propensity to consume, and $b$ is a positive constant of adjustment that describes how changes in consumer spending affect the annual rate of private investment.

Find the general solution of equation $(14)$ when $a=.9$ and $b=\frac{4}{9} .$ What happens to $Y_{k}$ as $k$ increases? [Hint: First find a particular solution of the form $Y_{k}=T,$ where $T$ is a constant called the equilibrium level of national income. 1

Michael Jacobsen · Accepted Answer

in this video, we&#x27;re dealing with a model, which is a difference equation displayed here and critically. This is a non homogeneous difference equation due to the equals one on the right hand side, we&#x27;re going to work with this model specifically when the coefficient days 0.9 and B is 4/9. And so the difference equation then becomes why K plus two minus eight times one plus B in this case becomes 1.3. Then it&#x27;s multiplied by why K plus one and a times B becomes point for sweet right plus 0.4 times y que equals one. So next our goal here is to find the general solution to the model for the specific constants A is 0.9 and B s four nights. And so to that end, we need to find a particular solution. Since the right hand side is equal to a constant one. We suppose that see, or we can say, let see be a constant and assume that this constant C solves the difference equation. If we make that assumption and insert, see in place of y to the Power K plus two or twice up K plus two and so on. We obtained the equation that C minus 1.3 times see plus 0.4 times C is equal to the constant one. And if we collect all like terms, we have 0.1. Time C is equal to one, and this finally implies that the constant C must be equal to 10. What this work shows so far is that why K se equal to the constant 10 is a particulate er solution of this difference equation displayed here. So now that we have a particular solution, we can obtain all solutions by converting to the homogeneous equation. It&#x27;s why K plus two minus 1.3 times y que plus 0.4 times y que should be a k plus one here equals zero. The auxiliary equation for this is going to be r squared minus 1.3 times are plus 0.4 equals zero. If we plug this auxiliary equation into the quadratic formula, we find then that the solutions are r equals 0.5 and pointing. But what this then means is that this set of solutions which are going to be 0.5 to the power of K and 2.8 to the power of K, contains the solutions to the homogeneous equation that is displayed above. Let&#x27;s pause for a minute and considered this set here. Notice that the Homo genus difference equation is off degree, or rather, order to. That means the set of all solutions should be a dimension to, and this set of vectors that&#x27;s displayed here does in fact contain two vectors, or signals, which are not multiples of each other, so that their linearly independent. This means that the set forms a basis for the solution set of this equation provided here. So now that we have the complete basis for the homogeneous equation as well as a particular solution, we can now combine both sets of answers to get the general solution. So it&#x27;s right here that the general solution is we&#x27;ll call it Y Que. And first I&#x27;m going to go to this set of vectors, will take a scaler C one and multiply by 0.5 to the power of K at a scaler C two and multiply it with the second solution to the homogeneous equation, which is 0.8 to the power of K. Then go to our particular solution found here and tack it on. So this is our solution Set where C one and C two are in. Are so there there any scale? Er&#x27;s So this is our general solution and notice due to the 0.5 here and the point A found here. This implies that why k converges to 10 when K goes off to infinity since 0.5 to the parquet and point H to the power of K converged to zero as K increases without bound. So this is our general solution. But only in the case where a is 0.9 and B is four nights.

Nikolaos Alexandrakis · Answer

So when you find that general solution for this this critical ation. Now, unlike the previous problems, this education is not homogeneous, since it has one on the right cut side and not not zero. So it is a little bit more difficult, but nothing. Nothing too much difficult. No. Yes. So, uh, we see that we can apply possibly a trick toe, overcome this problem. And, um, we proceed by noticing that if some solution a particular solution off this aggression is constant. Let&#x27;s name it. See, then we will have constant times something constant here in constant here in constant. It&#x27;s here. Um, that will equal one with soldier constant. So it is reasonable toe. Assume that one particular solution is indeed a constant. And we proceed juristic early by substituting Cito the situation on dhe way rates. Tow the situation for C C minus a times one plus they times soon. Bless a be times C equals one. And after standard manipulations way are its toe the situation. See, time&#x27;s one minus a equals one. Wits gives us that C should be won over one minds and fun or a depends. Um, hold you say it. It&#x27;s for a equals. Oh, going nine. These c evaluates. But then so we found a particular solution with equals then and now, in order to find the general solution we need to solve, we need to find his owner solution off the homo genus problem, which is this equals zero instead, off one, we proceed by standard theory a calculating the axle, a vacation for the homogeneous problem which is this one. We simplify it, Andi, since we&#x27;re looking for nontrivial solutions where search for solutions says that our is known is not zero Um, and we present by solving face second order occasion. So we have that discriminate off. This situation is all point Oh, cool toe five After standard mullet money populations off Santa Calculation since is open to nine and the equals one over two and the square root off the discriminative is oh, going 15. And, uh, these gives us two solutions solutions with from standard theory, I mean castrated with, like, a short list. We can&#x27;t verify that these two solutions are linearly independent. Um, we find one solution which is all 10.6 to the K and the other solution, which is I mean okay are 1.1 or two to the killing. The other solution is or buoyant 75 for the game. So the general solution off. Why Kay is this one C one all 0.6 does a K plus Sito full point 75. So they k plus then which is the particular solution city to the K and we are done.

Alex Loukas · Answer

in the final question, Chapter 34. We have a set of equations that describe the state of the economy of the magma economy and in party. We need to explain the meaning of each of these equations. All right, so let&#x27;s start with the first equation the main one, which is the national income identity. Why it will see plus tie plus CI, which simply means that angry output or national income why or GDP you can call a GBP is equal to its components. Consumption. That&#x27;s investment plus government spending. Usually, you might have seen net net exports added to this equation as imports minus exports or experts minus imports. But here we&#x27;re dealing with a closed economy, so there&#x27;s no international trade, all right. The second equation is a consumption function, which reads consumption equals to some fixed level of consumption. You can think of it up subsistence level of consumption, which is equal to 100 pleasure 1000.75 times. Why minus T. Let&#x27;s think about what, why, minus the represents, it is income minus taxes, so it&#x27;s disposable income. This is again a very simple consumption. That&#x27;s a simple function that says that consumption is equal to some fixed level times parameter that describes how strongly consumption reacts to changes in income or taxes. So if we think about it graphically, this represents a linear function that has an intercept of 100 and ah and up and its upward sloping with the slope of the line equals 0.75. So we don&#x27;t know it&#x27;s a positive relationship between consumption and income or negative relation between consumption in taxes. All right, The third equation is an investment function in aggregate investment function, which states that investment equals to some fixed level of investment, which is equal to 500 minus 50 times the interest rate again here. This is a linear function with negative slope, which is equal to minus 50. And we can think about this minor 50 as the sensitivity of investment of interest rate, which means that as the interest rate goes up, invest by 1% investment will go down by 50 units. So there&#x27;s negative relation between investment and the interest rate, as we already know. And finally, we&#x27;ve been given some family values for government spending and taxes, which are usually exogenous. Leo, given in a model. All right, hard. Be asked us to determine what is the marginal propensity to consume in this economy. Well, margin pretends to consume, has something to do with consumption, right? So if we look at the consumption function, what would that be? We know that the the MPC is the slope of the consumption function, and here the soap is 0.75. So the embassy is going to be equal to 01 75 or two or 3/4. And this means, of course, that one income increases by one unit Consumption will increase by 0.75 units or um, unfortunately, if Texas, uh, are reduced by one unit again, consumption will increased by 0.75 units. So it&#x27;s it&#x27;s the responsiveness of consumption to changes in disposable income. In this case, part seem, says suppose Essential bank&#x27;s policies to adjust the money supply to maintain the interest rate at 4% so are equals four. So for the GBP, how does it compare to the full employment level? We know that the full employment level is 2000. This is given. So what we need to do for this question is first of all, weigh in to work backwards and turning what consumption investment will be at an interest rate, which is equal before. So we substitute the Ari pose for into our investment function, which there&#x27;s going to be able to 500 minus 50 times, for it was just 200. So it&#x27;s about 2 300 On the other hand, consumption will be equal to 100 pleasure upon 775 times. Why my 100? Now, remember that wise are, um all right, primal adventure someone into sulfur. Why? So if we distribute The Apprentice is this is a 100 0.75 Y minus 75. Now we put all those equations into our national income identity, and we get why it was 100 plus tio 0.75 y minus 75 plus 300 plus 1 25 which is the exceptionally given government spending. If we distribute why on the left and the and the fixed values on the right, we can sell for why, which is equal to 1,801,800 which is less off course in 2000. And this means that when the value of the interest rate is evil before the economy will underperform will be in a recession lower than the natural rate. Natural level. All right, and party, we assume no change in monetary policy. What change in government purchases with restore full employment. All right, so we need to do something that is similar to what we didn&#x27;t part see. But now our unknown variable is not why, but it&#x27;s G. Let&#x27;s call it G prime here. We know that we want the eye grade level of output to be equal to 2000. So we substituted everything that we have. But we way we&#x27;d let our g prime to be our unknown variable. And once again, once we&#x27;ve perform very basic operations, we console for G prime, which is equal to 1 75 So what does this mean? That one wearing the recession government precious isn&#x27;t to go up by 50 from 1 25 1 75 you know, interest in or distantly. Yeah, great demand and restore for employment. Finally, party says, assuming no change in fiscal policy now what changing the interest rate would restore full employment. Well, let&#x27;s see, we do exactly what we didn&#x27;t party but now are variable. Interest would not be will not be G, but are the interest rate Let&#x27;s goi are prime again with substance are known valleys and we leave our prime as our unknown value. We can again performs in simple algebraic manipulations. And so for our prime, which may well be equal to three. So what does this mean that won the interest rate? It is equal to four. This is too high. The interest rate need soup decline down 23 in order to stimulate once again I greet demand, consumption goes up, investment goes up and the economy goes back to its natural level thout Bush.

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.

Erwin Antoni · Answer

for this question. We&#x27;re looking to increase the economy by $0.3 trillion we get that by very simply just taking the difference between where the economy is at now and where its potential is. In the first question, we have to use the spending multiplier because we&#x27;re trying to figure out how much government spending needs to increase in order to after the multiplier effect increased the economy by 0.3 trillion. So we have a couple of equations. We have an equation for calculating the spending multiplier, and we have an equation for calculating the tax cut multiplier. So all we&#x27;re going to do is, since we have two of the three pieces of information, we&#x27;re gonna plug those pieces of information into each equation and then solve for the remaining variable. So in the first instance, we have the spending multiplier. That&#x27;s to the change in real GDP. That we&#x27;re looking for is 0.3, in this case, $0.3 trillion and the change in spending is what we&#x27;re trying to find. And if you just do simple algebra, nothing fancy. You&#x27;ll find that it&#x27;s 0.15 again in this case $0.15 trillion. We do the exact same process for the tax cut multiplier equation, and we get 1.6 is equal to 0.3, divided by ex Solving for X is $0.1875 trillion in tax cuts, so we can either spend $0.15 trillion or we can cut taxes by 0.1875 trillion dollars. Now, for the third part, they&#x27;re asking you to create an example using a combination of both. Probably the easiest way to do this is to just take half of each, because if you take half of each, you&#x27;re getting half the effect. And if you add those two halves together, you get the whole effect, which is $20.3 trillion. Maur, innit? Economic output. So in that case, spending is half of the original Mount, so it&#x27;s $0.75 trillion tax cuts would be half of the original amount, which would be $0.9375 trillion

Charles Carter · Answer

okay. System starts us with the national income formulas. We have C plus I. Let&#x27;s g um, plus the quantity X minus at. Okay. So playing in the values that they gave us, we have Why is equal Teoh 60 plus 0.8 cough line plus I, which is 90 time plus the G, which is 1 45 plus X, which is 50 minus, um three minus 35 Tu minus zero point to you. Why? Okay, So, combining like terms. We&#x27;re gonna have 16 plus 95 plus 1 45 plus 50. What is 35? It&#x27;s gonna be Why is equal to 315 and then committing are wide terms. We&#x27;re gonna have 0.65 Why? Subtracting 0.65 white from both sides. We&#x27;re gonna have 0.35 Why is equal to 315. So, dividing 3 15 by 0.3 times. It&#x27;s going to give us a in national income level of 900. So you say 900 units of national income, They read a word problem. Try to answer it in a sentence. So we like to think about it. Might just from a math teacher perspective. If I give you a word problem, I want you to make your answer in words. Okay? Thank you very much.

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Exercises 17 and 18 concern a simple model of the…

Problem 17 Hard Difficulty

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as $k$ increases, $Y _ { k }$ approach to $10 .$

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

Linear Algebra and Its Applications

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as $k$ increases, $Y _ { k }$ approach to $10 .$

Linear Algebra and Its Applications

Heather Z.

Caleb E.

Kristen K.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Heather Z.

Caleb E.

Kristen K.

Joseph L.

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

Linear Algebra and Its Applications