Transcription
Hello and welcome to another lecture of this course, mathematics for economics, part one. So, over the last few lectures, we have been discussing this particular module of difference equations and how we can use difference equations in the analysis of economics. Now, at present, we are talking about second-order difference equations.
So, as you can see on your screen, this is the general form of a second-order difference equation, which is linear, non-homogeneous, with constant coefficients, and on the right-hand side, you have a constant term. So, this is yt plus 2 plus a1yt plus 1 plus a2yt is equal to small c. And we have seen how to solve this. We have seen that there are three subcases. If we look at the complementary function, then if we want to find the complementary function, then we will get a characteristic equation, and from that characteristic equation, we will get characteristic roots. Now, these roots can be of three natures. The first one is real and distinct roots. Right, under this condition, here the complementary function will take this form. Secondly, it is possible that you have this condition which has been satisfied. In that case, the complementary function will have this form. And finally, you can have complex roots where this condition is satisfied: A1 square is less than 4 A2. In this case, the complementary function will be given by this: R to the power t, A3 cos theta t plus A4 sin theta t, where theta and R are defined by these two relations.
Alright, now what about the convergences? Under what conditions can we say that this yt is going to follow a stable path and it will move towards yt as t goes to infinity? Now, here we need to know that we have two characteristic roots here, b1 and b2. If in the case of first-order difference equations, we had just one root and convergence was attained when the absolute value of the root is less than one. But here there are two roots. So, how to deal with that? So, first let us take the case of real and distinct roots, where b1 is not equal to b2. We know this is the complementary function. Now, if the absolute value of both are less than one, then it is clear that as t goes to infinity, both these terms will tend towards 0. Both these terms, A1 b1 to the power t and A2 b2 to the power t, both of them will converge towards 0. That means yc will converge towards 0. So, this is the, this is just one case. What happens if in the opposite case where the modulus of both of these roots is greater than one? In that case, both b1 to the power t and b2 to the power t will be explosive. And in that case, yc will diverge. It can be like, you know, oscillatory divergence or it could be monotonic divergence, whatever it is, but there will be divergence. But what about the intermediate case where suppose modulus of b1 is greater than 1 and modulus of b2 is less than 1? So, how to conclude that case? In this case, obviously where you are talking about A1 b1 to the power t, that will be explosive because b1 is, the absolute value of b1 is greater than 1. But the second term that is A2 b2 to the power t will die down because the modulus of b2 is less than 1. And, so, what happens as the summation, as the net result of these two? The first term will eventually dominate yc because you know one part is going to 0, that is fine, but the other part is explosive. So, therefore, the summation will also be explosive, so yc will diverge.
Now, out of these two characteristic roots, let us call the larger root as the dominant root. So, from the above what we can conclude is this: if the time path will diverge or converge, it depends on the absolute value of the dominant root. Dominant root means the root which is larger in the absolute terms. If the larger root is greater than one, then we have a problem; there is divergence. If it is less than one, there is convergence. If it is equal to one, then we know there is not going to be any convergence. What about the role of the non-dominant root? The non-dominant root, that is the smaller root in an absolute value, plays a role in terms of affecting the configuration of the time path at least in the initial periods. Because you know it has a kind of additive influence on yc. Maybe it will not be able to affect the nature of yc as t goes to infinity because that is going to be determined by the dominant root, but this non-dominant root affects the configuration of the time path.
What about the stability in case of repeated roots? This was the case of repeated roots, the complementary function. Now, as far as the first term is concerned, this term, like in the case above, the absolute value of the characteristic root b will determine if a3 b to the power t is explosive or not. The second term that is A4 tb to the power t has t as a multiplicative factor. If b is greater than or equal to one, it is clear that as t goes to infinity the second term is explosive. So, if b is greater than or equal to one, then this term will either explode or it will remain constant. But as t is rising, so the entire term is definitely going to be explosive. What happens if b is less than one? If b is less than one, by the presence of A, that is t, the second term A4 t b to the power t rises, but b to the power t dies down as t goes to infinity. It should have been written as absolute value of b less than 1. If the absolute value of b less than 1, then b to the power t dies down. But this is being multiplied by small t, and t is rising. So, in this case, what will happen to the entire term? In this case, the latter effect will dominate, that is b to the power t. Hence, there is going to be convergence of yc to 0, as t goes to infinity. So, one way to understand this is this that b to the power t, this goes to 0, as t goes to infinity. So that pulls down the entire term, that is A4t multiplied by b to the power t. Thus, as in the previous case, the absolute value of the root should be less than 1 for convergence. Previous case means the case where we talked about repeated roots, distinct and real roots; that was the previous case. So, there we saw that it is the absolute value of the dominant root that determines the characteristic of the time path. Here also the same thing is being concluded. It matters as to what is the absolute value of b. If it is less than 1, then we have convergence.
What about the complex roots? So, in the complex root, we have this as the complementary function. The time path will be cyclical, we have noticed this before, with stepped fluctuations because t can take only discrete values. For that, there is convergence of yc, depends on what happens to rt. So, this within the brackets term is just fluctuating within certain limits. So, the convergence or divergence is purely determined by R to the power t. Thus, the value of R, that is capital R, is critical. If it is less than unity, there is convergence. But what is R? R is given by this, root over h square plus v square; it is the absolute value of complex conjugate roots, h plus minus vi. So, it is the absolute value of complex conjugate roots. So, in some, in this case also, the absolute value of the roots needs to be less than unity to ensure dynamic stability. So, there is a common thread. Therefore, in all these three results, the absolute value of the roots has to be less than 1.
Now, we talk about one important application of the second-order difference equation. And this is the multiplier-acceleration interaction model, proposed by the famous economist Paul Samuelson. So, some of these things that we are going to see have been introduced earlier. So, we start with the national income identity. The national income at a particular period t is given by Yt is equal to Ct plus It plus Gt. That is the national income is the summation of consumption expenditure, investment expenditure, and government expenditure. So, we are assuming a closed economy; there is no export and import. Now, the consumption expenditure in this model is assumed to be proportional to the income of the previous period, that is the national income of the previous period. So, in terms of mathematics, Ct is equal to gamma multiplied by Yt minus 1, where this is important, gamma lies between 0 and 1. So, what is gamma? Gamma is the marginal propensity to consume, but we have to be careful here because we are talking about the dependence of the consumption expenditure on the income of the previous period. Secondly, what about the investment expenditure? What is the form of the investment expenditure? We are assuming that the investment expenditure is induced. It is a proportion of the change in the consumption expenditure. So, this is the form that it takes: It is equal to alpha, multiplied by Ct minus Ct minus 1, where alpha is greater than 0. So, what is the justification for this? This is something new. It, that is investment in a particular period, it depends on the change in the consumption expenditure. Ct minus Ct minus 1, that is the change in the consumption expenditure. Now, what is the rationale for this? The rationale could be the following that the people who do the investment, that is let us suppose the producers, they look at how much consumption expenditure has been changing over the last period. So, that tells the investors as to whether the people who are doing the consumption are feeling confident about the economy. So, if the consumption expenditure is rising, then the investors feel that it is better to invest more money to build capacity so as to cater to the rising expenditure by the consumers. So, that is the logic.
Now, let us talk about the last element in the aggregate expenditure, which is the government expenditure. The government expenditure is assumed to be given at a fixed level. Let us suppose Gt is equal to G0. So, this is just to make the model simple. We are assuming that government expenditure is fixed. The notations gamma and alpha, they are called marginal propensity to consume and accelerator. This is called accelerator alpha. This alpha, alpha is positive, and the logic I have just described to you that as there is more consumption expenditure, more investment will be forthcoming. So, that is the reason why alpha is greater than 0. So, this is the reason why this model is called the multiplier-acceleration interaction model. We have already introduced the consumption marginal propensity to consume, that is gamma. That we shall see is an element of the multiplier. And we have introduced the accelerator. Subsequently, we shall see that these two will interact to throw up interesting configurations of the time path of income.
Now, we are going to use these two relations: Ct is equal to gamma Yt minus 1 and It is equal to alpha Ct minus Ct minus 1. We can combine these two things together, and we are going to get It is equal to alpha multiplied by gamma yt minus 1 minus gamma Yt minus 2. Basically, we have just used this and that is the consumption function of the income of the previous period. We have substituted that here and here. So, we have got it, and now what we are going to do is I am going to substitute it and the consumption function in the national income identity, which is this. And we are going to get this. You can see it is only a function of Yts' alone, and there are three Yts' here: Yt, Yt minus 1, and Yt minus 2. And so, this is a difference equation of the second order. G0 is constant. I can change the time period a bit. I can write the same relation as in terms of Yt plus 2, Yt plus 1, and Yt. So, this is a second-order difference equation; we know how to solve this. First, we look for the particular integral. So, we, I am just keeping some steps. So, yp the particular integral is found out to be G0 divided by 1 minus gamma. This is a familiar expression of the multiplier. If there is an exogenous expenditure x, in this case that x is G0. The equilibrium income is given by x multiplied by 1 minus gamma. Here the same thing is happening: G0 divided by 1 minus gamma. x is multiplied by this term, and this term is called the multiplier, 1 divided by 1, minus gamma. And what is gamma? Marginal propensity to consume. So, that is what I said that MPC is an element of the multiplier.
Now, let us concentrate on the complementary function. Now, as we know, there could be three subcases. In the first case, the roots could be distinct and real. This condition is a1 square is strictly greater than 4a2. In this case, it is translated into this relation: gamma square multiplied by 1 plus alpha whole square is greater than 4 alpha gamma, and if I take the gammas on the left-hand side, it becomes gamma strictly greater than 4 alpha divided by 1 plus alpha whole square. This was the case of real and distinct roots. What about the repeated roots? That relation is this: gamma is just equal to 4 alpha divided by 1 plus alpha whole square. And for imaginary roots, the inequality sign goes the other way: gamma is less than 4 alpha divided by 1 plus alpha whole square. So, actually this function, gamma is equal to 4 alpha divided by 1 plus alpha whole square is useful to identify these three regions. One is the case of real distinct roots, second is the case of repeated roots, and the third is the case of imaginary roots. This diagram actually is showing this function. So, which is this function? This line. So, it is a concave to the origin line as you can see. It is going up, reaching a maximum at alpha is equal to 1. Along the x-axis, we are representing the alpha, and along the y-axis, I am representing the gamma. Gamma, remember, cannot exceed 1, so there is an upper limit here because gamma is a marginal propensity to consume. Alpha, however, does not have any upper bounds, so it can go along the x-axis. So, this line is a concave function; it is reaching a maximum at alpha is equal to 1. At that particular value, the value of gamma is equal to 1, so this point P. And then it is coming down. As we have just seen that there are three regions here, one is above this function, so that is A and D, A and D. So, at A and D, you have gamma is strictly greater than 4 alpha divided by 1 plus alpha whole square. So, this is the case of real and distinct roots, this relation. On the line, so if we are on the line, then these two things are just equal. So, this is the case of repeated roots. And if we are talking about these two regions, B and C, alright, then actually gamma is strictly less than 4 alpha divided by 1 plus alpha whole square. So, this is the case of imaginary roots. So, actually these four roots are represented in this diagram in a very neat manner. For the time being, ignore this line. There is another line that I have drawn which is alpha gamma is equal to 1. Now, the importance of that line will be made clear subsequently. Right now, we just note that this concave function which I have denoted here by the red line is the line which separates the three subcases of roots.
Alright, now the question is when do we have convergence, when do we have divergence, and what is the nature of convergence if there is convergence? These are the important questions that we are going to deal with. First, we look at the characteristic equation. We are trying to find the complementary functions; for that, we have to look at the characteristic equation. This is the characteristic equation; from this, we get these are the roots b1 and b2. Now, what we know is that if we have a quadratic equation like this and b1 and b2 are the roots, then the summation of the roots is this term: the coefficient of the b minus of the coefficient of the b, which is gamma multiplied by 1 plus alpha. So, this is high school mathematics, and the product of the roots is this term: alpha multiplied by gamma. That we know. Furthermore, let us look at this expression: 1 minus b1 multiplied by 1 minus b2, where b1 and b2 are the roots. Now, this simplifies into this: 1 minus b1 plus b2 plus b1 b2, and this can be written as this. Because b1 plus b2 we have just noted it is gamma multiplied by 1 plus alpha that we substitute here, and b1 b2 is alpha gamma that we substitute here, and we simplify, and it turns out to be 1 minus gamma. Now, gamma, as we know it, lies between 0 and 1, the marginal propensity to consume, so this thing also will lie between 0 and 1, 1 minus gamma. And therefore, this will be satisfied that 1 minus b1 multiplied by 1 minus b2, this thing it lies between 0 and 1 because 1 minus gamma lies between 0 and 1. So, these are the certain things that we will be using now.
Now, let us take the case of real and distinct roots. Now, in this case, we know this has to be satisfied: gamma is strictly greater than 4 alpha divided by 1 plus alpha whole square. So, we are talking about the regions A and D, A and D. Now, we know that alpha and gamma both are positive. Now, that basically means that this is positive: b1 multiplied by b2 is positive because b1 b2 is equal to alpha multiplied by gamma. Now, if b1 multiplied by b2 are positive, then there are only two possibilities: either both of them are positive or both of them are negative. Both the roots have the same sign. Now, can both of them be negative? We know also that the summation of the roots is given by this: gamma multiplied by 1 plus alpha, and which is positive? So, the summation of the roots is positive; therefore, the roots individually cannot be negative, which means there is only one possibility that the roots are positive. If the roots are positive, then we know there is not going to be any oscillations. So, roots are positive, but even if the roots are positive, actually there are five cases here. So, these are the cases that have been noted down. Let us take one or two to get the feel of it. So, here we are taking the roots to be, both the roots to be lying between 0 and 1. We are assuming without loss of generality that suppose b1 is the larger root. The roots are different, so one of the roots is higher than the other root. So, let us suppose b1 is that bigger root. So, here both of these roots are lying between 0 and 1. Now, the question is, is that feasible? That is going to be an important point. Now, if both of them lie between 0 and 1, then what is happening to this term? That is 1 minus b1 multiplied by 1 minus b2. That is, we know is going to be equal to 1 minus gamma. That is known to us. And if b1 and b2 both are lying between 0 and 1, then 1 minus b1 is going to be positive. 1 minus b2 is also going to be positive. Now, the product of these two, it is possible that it is lying between 0 and 1. And which will satisfy this condition. So, this is feasible. And furthermore, we note that if this condition is satisfied, then the product of these two roots is going to be less than 1, and the product of these two roots is known to be alpha multiplied by gamma. So, here alpha multiplied by gamma is going to be less than 1. Here what is the time path? It is going to be convergent because both the roots are less than 1 and greater than 0. So, it is convergent and obviously it is going to be monotonic. So, this is possible. Whatever the second case, the second case is the larger root is equal to 1. Now, that is not possible; the reason being that if b1 is equal to 1, then 1 minus b1 is equal to 0. So, this condition is no longer satisfied. The product has to be greater than 0. But here the product is turning out to be 0. So, this is not feasible. Similarly, this is not feasible; this is the case where the larger root is greater than 1 and the smaller root is less than 1. So that is not feasible. If the larger root is greater than 1 and the smaller root is equal to 1, that is also not possible. And the last case is however feasible, where both the roots are greater than 1. Here let us suppose the roots are 1.5 and let us say 1.4. In this case, 1 minus 1.5 is something negative, and here also there is something negative. But the product of these two might turn out to be satisfying this condition, 0 to 1; they lie between 0 to 1. So, that is feasible. However, in this case, alpha multiplied by gamma will be greater than 1 because b1 b2 will be greater than 1. So, this is the case where there is going to be divergence because both the roots are greater than 1. So, here from the first case of real and distinct roots, we are getting two possible cases. In one case, there is convergence, and in the other case, there is divergence.
Now we come to the repeated roots case. Now, in the repeated roots, the, what is the root? Root is equal to gamma multiplied by 1 plus alpha divided by 2. And that basically means that we are on the curve. We are neither above the curve nor below the curve. And the root is positive, so there is no oscillation. Now, there are three possible values of the root, however: less than one, there will be convergence in this case; equal to 1; and greater than 1. The first case of less than 1 is similar to case one in the last slide; b is lying between 0 and 1. In this case, obviously, gamma will be between 0 and 1. So, here this is the case: 1 minus b multiplied by 1 minus b. Since b is less than 1 greater than 0. So, 1 minus b whole square will also be greater than 0 and less than 1. And in this case, the product of the roots will be less than 1. So, this is the case of convergence because b is less than 1. The second case will be infeasible because that case gamma will turn out to be 1, which is not possible. The third case is however feasible, and this is similar to case five in the last slide; here b is greater than 1. So, this is the case, and we have seen that it is feasible because in this case, gamma can be greater than 0 and less than 1. In this case, alpha multiplied by gamma is strictly greater than 1, and that is a divergence time path. Now, notice one pattern that whether we are going to have convergence or divergence that is getting reflected as to what is happening to alpha multiplied by gamma. If alpha multiplied by gamma is less than 1, it is convergence, and if it is greater than 1, there is divergence. Okay, the third case of this: gamma is less than 4 alpha divided by 1 plus alpha whole square. So, this is the region B and C in the diagram. So, this B and this C, this region and this region, this is the imaginary roots case. Now, as we know, the absolute value of the roots, that is important. R has to be less than 1 for convergence. Here R is equal to root over a, where a is this coefficient. So, in our case, a is equal to alpha multiplied by gamma. So, R is equal to root over alpha multiplied by gamma, and here again like before there are going to be three cases: R is less than 1 means convergence in this case. Alpha multiplied by gamma has to be less than 1. R is equal to 1 means alpha multiplied by gamma is equal to 1, and R is greater than 1 means alpha multiplied by gamma greater than 1. Now, out of these three, only the first one will be convergence because in that case R is strictly less than one. The other cases will be cyclical movements, obviously. But here there will be uniform oscillation, and here there will be oscillation with explosive behavior. And generally, these are unstable cases. So, this diagram actually sums it up. We have talked about this figure before, this line before. What is the meaning of this line? Now we talk about this line. This line, as we have noted, it basically demarcates the stable cases from the unstable cases. So, if we are below this line, you have all the stable cases on convergence. If we are above the line, there are cases of explosive behavior. On the
Line: there is neither explosion nor convergence. Basically, uniform oscillation; or in the case of repeated roots, it is possible. At this point P, there is no oscillation, but it is not converging to the y peak.
So, that is what I have written here in summary: A and D are the case of real and distinct roots. Both the roots are positive. What we can say here is that A is stable and D is unstable. Second case: B and C; so B and C are imaginary roots that we have seen. B is stable, C is unstable; both step fluctuation, both A and B stable. So, this and this, both of them are below the alpha multiplied by gamma is equal to 1 line. Between D and C, that is, between D and C, you have unstable cases. So, this line: unstable, but repeated roots. And here, between A and B, this line: you have stable. So that basically sums it up.
Now, what is the economic import of all this? The economic import of all this is that in this model, what we have seen is that through the applications of second-order difference equations, we can actually work with a very simple (())(38.37) model, and we can find different cases of whether there is going to be fluctuation, whether there is going to be no fluctuation, and you know, whether there is going to be stability or not. So, all these things can be found out in a very simple framework. In real life, the world actually sees the fluctuations, or what are known as trade cycles in the national income. So, it is likely that since we are seeing actually there are fluctuations, so we are basically talking about these two cases, B and C, where there is fluctuation. In C there is instability, but in B there is no instability; there is convergence. So, it is possible that the real world is closer to B and C, where there is actually cyclical behavior of the national income. But that is for the empirical researchers to ascertain.
At this point, we are through with the course of what I intended to cover in this course. Before I finish, let us try to do a brief roundup of what we have covered in this course. So, this is the list of topics that we have covered in this course; you can see on your screen. So, we started this course with some certain basic ideas of the studying point of mathematics as such: real number system, logic, mathematical proof. We talked about what is deductive logic, what is inductive logic and mathematical proof, different ways to prove certain theorems. Then we talked about sets, the basic ideas of sets and sets operations. Then we went into the discussion of functions of one variable and different graphs of the functions. We talked about different types of functions: quadratic, cubic functions, polynomials; what are polynomials. All these things were discussed in this particular module, the third module. And then we talked about differentiation, the basic idea of differentiation. Then, what are differentiable functions, properties of differentiation, and importantly, what is partial differentiation. In the second part of this course, if there is one mathematics for economics part two, this partial differentiation will be used a lot because we are going to be talking about functions of multiple variables where partial differentiation will be very much relevant.
In the fifth module, we talked about differentiation of higher order. For example, you can take the second derivative or the third derivative, and what does it have to do with the nature of the graph of the function. Those were the things that are important in this discussion and linear approximation. So, if you have a sort of complicated function which is not linear, then actually you can imagine that function to be linear in the immediate neighborhood of a point. That linearization actually makes our task quite easy at times. We also talked about sequences and series. The idea of limits, the idea of limits was introduced in the case of differentiation itself. But here in this module, that is the sixth module, we dwell deeper into the idea of limits and also differentiability. I talked about that, continuity at greater length in this particular module. And this is very useful; it gets the basic ideas clear. We talked about convergence of the series and very importantly, exponential and logarithmic functions. Exponential functions or logarithmic functions are used extensively in economic and finance because we are often talking about growth over time. For example, you have your money in the bank and which is giving you some rate of interest over a period of time. So, on its own, the money will keep on growing. If it is growing, then how do you find out what is going to be the amount of money after a point of time? Or you can talk about the present value. What is the present value of certain money which will be obtained in the future? So, those things are very important when people do their calculations in finance.
Then we went to a very important module, actually two modules; it is about optimization: single variable optimization part 1 and single variable optimization part 2. So, we talked about the first-order and second-order conditions. What it means in terms of geometry, these conditions. And we talked about the different kinds of functions: convex functions, concave functions; and obviously, a plenty of applications from the field of economics and not necessarily only economics. We talked about applications in other fields also, for example, oil extraction, which has something to do with environmental economics. Also, the population growth; you know, all those things were there. Then in the ninth module, the area under the curve, indefinite and definite integrals, these were discussed; this comes under the module of integration part one. So, there were two modules on integration: integration part one and integration part two. We covered these in the ninth and the tenth module. And we talked about the economic applications of these also and integration by substitution, applications of integration in other fields; for example, when you are talking about the distribution of income in a country. So there the idea of definite integral is very important. So, those things, we talked about that.
And the final topic that we covered in this course is difference equations. Again, this was covered in two modules, the eleventh and the twelfth modules. So, here the time is discrete. And that was introduced in the eleventh module: discrete time. And first-order difference equations were used, and we talked about the applications of first-order difference equations in real life and how to solve first-order difference equations. So, we talked about the cowboy model, for example. So, that is one application. And in the last module, the twelfth module, we talked about higher-order difference equations. We also talked about phase diagrams. So, you have a first-order difference equation, but which is not linear. So, in this case, you can take the help of phase diagrams to understand the stability properties of the system. And in the twelfth module, we talked about the solution of second-order difference equations, and we talked about applications of it. For example, we talked about Samuelson’s multiplier accelerator model.
This concludes our discussion, and the course has been completed. I thank you for being with me in this course, and all the best in your future career. Thank you.