Transcription
Hello and welcome. So, this is another lecture of this course, Mathematics for Economics, part 1. The present module that we are going through is on difference equations. So, we have introduced what are difference equations, and presently we are trying to understand how to solve a difference equation of the first order, and in particular, these are linear difference equations. So, you can see on your screen the title slide. We have talked about the general form of a linear difference equation. So, this is the form: yt + 1 + ayt = c, where a and c are two known constants.
We have seen that there are two parts in the solution: one is the particular integral, yp, and the other is the complimentary function, yc. The complete solution is the summation of these two. And if we know the initial condition, then actually, we can find out the exact solution. And the exact solution will be of this form: yt = (y0 – c) / (1 + a) * (-a)t + c / (1 + a). However, this solution will be valid if a ≠ -1. Because you can see if a = -1, then the denominator becomes 0, and that is a problematic thing.
So, what happens if a = -1? Then the particular integral, we take a different solution to that; we take yp, that is, the particular integral, is equal to kt, and if we do so, then the solution is yt = y0 + ct, here ct is the particular integral, and y0 is the initial condition, that is, the value of y at the initial period. So, this is how we solve difference equations of first orders, and we have talked about one example.
Now, the question is, what about the dynamic stability? And that is what we were talking about as the last topic of the lecture. So, when we say dynamic stability, what is meant by that is that does yt, which is the complete solution, does it approach yp, the particular integral? yp, as we know, is the inter-temporal equilibrium value of y. So, does the solution, which is the complete solution yt, does it approach the inter-temporal equilibrium? Well, that depends on what happens to yc; if yc converges to 0, only then we can say that yt converges to yp. This is very simple.
Now, what happens to yc as t goes to, let us say, it goes on increasing; in general, it can go all the way up to infinity. If t goes to infinity, then what happens to yc? Well, here there are different cases that can come up. And in general, there are seven possible regions or cases, and these are enumerated here. So, the first case is: b is—what is b? Remember yc = Abt. So, here there is a b term; b is a constant, but constants can have different values. Now, if b > 1, then as t goes to infinity, this yc goes to infinity as well; this is the first case. Suppose b = 1, then bt remains constant at 1. So, in this case, it does not go to 0, that is what we want it.
Thirdly, suppose b is lying between 0 and 1; in that case, as t goes to infinity, what happens to bt? In this case, actually, it goes to 0. I mean, take an example: suppose b = ½; in this case, as t = 1, bt = ½; t = 2, bt = (½)2 = ¼; t = 3, b3 = (½)3 = ⅛. So, likewise, it will then become 1/16, 1/32. What is interesting to note is that the denominator, that is, the terms which are appearing here, they are going to infinity as t goes to infinity, and as we know, 1/∞ is close to 0. So, that is why I am saying that as t goes to infinity, if b lies between 0 and 1, bt goes to 0. If b = 0, then 0 to the power whatever power you put there, it will remain at 0.
If b lies between 0 and -1, then something interesting happens; then bt actually oscillates. So, start with t = 1, let us suppose; if you put t = 1, then -½, let us suppose b = -½; (-½)1 = -½; t = 2, (-½)2 = ¼. So, you can see from -½ it becomes +¼, then it becomes -⅛, then it becomes +1/16. So, in other words, the value of bt oscillates; it moves in a zigzag manner; it oscillates. And in the process, however, it approaches 0 because the denominator, like in this case, the denominator goes to infinity. So, the bt term approaches 0. This was the fifth case. What happens if b = -1? In that case also, it oscillates between 1 and -1. So, put t = 1, bt = -1; put t = 2, bt = 1; t = 3, bt = -1; t = 4, bt = +4. So, it oscillates between +1 and -1. This was the sixth case, and the last case, that is, the seventh case, is: suppose b < -1. Now, like before, in all these two cases, the bt was oscillating; here also it will oscillate, because b < -1. So, you can think about b = -2. In that case also, as t goes on rising, bt will go on oscillating, but in this case, it will be explosive. So, the oscillation will become explosive, that is, bt will become far and far away. It will become far and far away from the value 0. So, that is why I am saying that it oscillates and it explodes.
Now, let us come to certain observations from this table, these seven cases. First thing to note is that the movement of yc, that is, the complimentary function, is not smooth; time takes discrete values. So, the value of yc jumps between successive periods like a step-like function; that is obvious. Let us take this example that I have just constructed, bt; it is ½, then it is ¼, then it is ⅛, then it will be 1/16. So, you can see that between these two values, between ½ and ¼, for example, there are plenty of other values. For example, there could be 1/3. So, one can think of other values between ½ and ¼, but bt is not assuming those values. Similarly, between ¼ and ⅛, there are plenty of values which are left out, which means that the bt, that is, the complimentary function, is actually jumping from one value to another. So, you can think of something like this. So, this is ½, ¼, ⅛. So, all these values are coming down, and they approach 0, but there are gaps. So, that is what I mean that the movement of yc, this is the yc, depends on bt, but bt is not a smooth function; it takes the values of t which are discrete; therefore, bt also misses many values between the successive two values. It is like a step-like function. So, that is what I mean by this; you can imagine this to be steps. So, it is it is a step-like function.
Out of the seven regions that we have listed, in three, the value of yc converges to 0 as t goes to infinity; these are these three cases: b lies between 0 and 1, b = 0, and b lies between -1 and 0. So, these are the three cases where you have convergence. It is written here more clearly that it goes to 0, it stays at 0, and it oscillates, but again it goes to 0. So, these are the three cases where there is convergence.
Third observation: in the first, that is, this one, b lies between 0 and 1, yc converges to 0 monotonically; monotonically means it does not oscillate, non-oscillatory, that we have seen before. So, here is the case where it does not converge in an oscillatory manner; it goes to 0 but in a monotonic manner. In the last, there is oscillation where b lies between -1 and 0. We shall see the diagram in a minute.
Fourth observation is this: the common features of the three cases is |b| < 1. These three cases mean the cases of convergence. In these three cases, what is common is that the absolute value of b is strictly less than 1. This is the condition of dynamic stability. In such cases, the time path of yt converges to yp. So, this is the thing we wanted to figure out that, when do we have dynamic stability? And now we can answer that question. The time path of yt converges to yp, that is, we have dynamic stability if |b| < 1. So, this is the case; the three cases that we are talking about, these are the three cases of convergence.
And the last observation: whether the time path is oscillatory or monotonic depends on if b < 0 or b > 0. So, if b < 0, then it will oscillate, the time path; it will oscillate. Whereas, if b > 0, so, these are the three cases where b > 0, you do not have oscillation; you have monotonic movement; either it is explosive or it does not change at all; it stays at 1 or it converges to 0, but in all these cases, whatever be the case, convergence or divergence, the movement of yc is monotonic. So, this is something to remember that dynamic stability is ensured if the absolute value of b is less than 1; that is number 1. Number 2: if we have oscillation or monotonic movement depends on the sign of b, whether it is greater than 0 or less than 0.
Now, it may seem that we are focusing our attention only on b, but let us now talk about A. A, remember, was the coefficient of the bt term. Now, this A plays two roles: first, it has a scale effect; a high or low A raises or reduces the value of yc; it does not qualitatively affect the nature of the time path. So, what do we mean by saying that it does not affect qualitatively the nature of the time path? It means that if it is dynamically unstable, even if A changes, the path remains dynamically unstable. If it is dynamically stable, then A does not affect that particular quality of it. It basically gives it a scale effect; maybe the path shifts up a bit or shifts down a bit, but qualitatively the nature of the equilibrium does not change. The second role is it has a mirror effect; depending on the sign of A, the time path of yc can become the mirror of itself, and this is not difficult to understand. So, suppose you have a time path something like this, yt with respect to a particular value of A. Now, that A was suppose positive. Now, suppose the sign of A changes, then the time path can become just a mirror image of itself; that means, it becomes something like this. So, in that sense also, A can affect the time path; it has a mirror effect.
The initial condition has a bearing on A; it affects from which point one approaches yp or it does not approach yp. And this can be verified from this general solution here. So, this was a general form of the solution; y0 was the initial condition. That is the value of y at period 0. Now, remember this is my A, capital A. Capital A has this term y0 in it. So, if y0 changes, capital A will change, and depending on the value of y0, one can have a positive A, one can have a negative A. For example, if y0 > c / (1 + A), then capital A becomes positive; otherwise, if y0 < c / (1 + A), capital A becomes negative. So, that is why I have written: it affects from which point one approaches yp. So, the initial condition has a bearing on A.
The cases of b = 1, b = -1. So, these are the two borderline cases where b is taking certain specific values, +1 or -1. Now, these two cases cannot be considered to be cases of convergence. If b = 1, then what happens to Abt? It is equal to A, just A, because b = 1, 1t = 1. So, A + the particular integral, which is yp, this is my solution. Therefore, since A is constant, therefore, yt ≠ yp. In other words, yc does not become 0 as t changes, because t has no bearing on A unless A = 0, but that will be a very rare case of coincidence, where you have the first term dropping out. So, in general, if you have b = 1, then it cannot be considered to be a case of convergence. Similarly, if b = -1, then also it cannot be considered to be a case of convergence; then yt will go on oscillating. This was the role of capital A.
Now, I promised that I will show some diagrams about this movement of yt. So, here you have time on the horizontal axis; on the vertical axis, the yt is represented. Now, in both these diagrams, on the left-hand side panel or in the right-hand side panel, I have demarcated something like yp; yp is the inter-temporal equilibrium value of y. The question is: do we approach yt as t changes? In the first case, we do; you can see this is like t = 0, t = 1, t = 2, 3. And you can see the values of yt; it is here to begin with, then jumping down, then it is going up above the yp, and again it is coming down below yp; again it will go up yp, below yp; likewise, it will go on in an oscillatory manner. But the point to note is that the gap between the yt and the yp is declining over time. So, initially it was this much, but at this stage, you can see the gap has come down a lot. So, that is why it is called damped oscillation. And this is the case where b lies between 0 and -1. And the second case is a case of monotonic divergence. Here you have explosive movement of yt. It goes away. So, you are starting from 0. In period 1, the yt is farther away from yp. In period 2, it goes even farther away. In period 3, the gap rises. And as you can see, the gap between the successive yt, those gaps are rising. The gaps are rising, which means that the gap between yt and yp is also rising. Because yt was below y0 to begin with, and since the gap between the successive y's are rising and they are rising in an increasing manner, therefore, as time goes by, you have an explosive movement; yt is getting farther and farther away from yp in an increasing way. So, this is called monotonic divergence; you do not have oscillation here. So, this is the case of b > 1.
We have talked about the theory; now we are going to talk about certain applications. The first extremely well-known application of the first-order difference equation is the Cobweb model. So, what is this about? In the production of goods like farm products, it takes some time to complete the process of production. So, this is also known as gestation lag. Let me explain this; this time lag affects the price of the goods in an interesting manner captured by a difference equation of the first order. So, how does this lag come into being? Let the decision to produce output in period t is affected by Pt, the price prevailing in that period. But the good will not be available in the market until the next period, t + 1. Thus, one can define the supply function in period t + 1 as QSt + 1 as a function s of Pt. So, this is the function that captures the lag. So, in a particular time period t, let us say the farmers have decided how much they will produce. Now, when they decide how much they will produce in their farm, they look at the current market price; that is what they know off; they are sure of; based on that price, they decide how much they will produce. But, the decision to produce does not automatically translate into production; it takes some time for the production to actually materialize. So, the production will come into the market in the next period, whereas, the decision was taken in the previous period. So, therefore, I can write this as Q(S t + 1), that is, the quantity supplied in period t + 1 is a function of price that prevails in period t.
Now, what kind of function is this, this supply function? Now, we just assumed that it follows the law of supply, which is saying that S(Pt) is an increasing function of Pt, and this function can be written as this function: QS(t) = S(Pt – 1); that is, we have taken t to be one period back, but the lag obviously remains. Now, on the demand side, the demand in a period depends negatively on the current prevailing market price. So, there are no surprises here; quantity demanded, Qdt, is a function of price of that period, Pt. So, d(Pt), and since we have assumed that it negatively depends on the price, so, d(Pt) is a declining function of Pt. In order to keep the story simple, both S(Pt – 1) and d(Pt) are assumed to be linear. So, we take this particular form: Qdt = α – βPt and QSdt = -γ + δPt – 1. So, these are the two functions, demand function and the supply function. And thirdly, we assume that Qdt = QSt. These three equations complete the model; the last one is the market equilibrium condition; this one, as the quantity demanded is equal to quantity supplied; that is the market equilibrium condition. We are assuming that in each period, the demand should be equal to supply; that is, in each period, the equilibrium should prevail. What we want to find out is the time path of the equilibrium price. So, although the market is in equilibrium in each period, how does this equilibrium price in the market behave over time? So, that is what we are after. What about these parameters, α, β, γ, δ? These are all assumed to be positive. So, that is what makes the demand function a declining function of the price of that period and the supply function to be an increasing function of the price of the previous period.
Now, using these three equations, these three conditions, we can actually write this: demand = supply, and from this, I get a first-order difference equation, linear difference equation, non-homogeneous difference equation in price. And we know how to solve such a difference equation of the first order, linear difference equation; we can find what is the particular integral, yp, and this is—I am not going into the detail of that—it will be equal to (α + γ) / (β + δ). Similarly, we can jump certain steps and find out the complimentary function; it will be A, an arbitrary constant, multiplied by (-δ/β)t. And suppose we have certain initial conditions at t = 0; the price prevailing in the market is given by P0; let us suppose that is known to us; in that case, the price will have this time path. So, it looks like a cumbersome expression. But I mean this is quite logical; we are just replicating what we have seen before as the solution: Pt = [P0 – (α + γ) / (β + δ)] * (-δ/β)t + (α + γ) / (β + δ). So, this is the time path; in this case, as we have seen that this is the particular integral, and this thing is the complimentary function. And as we know, the particular integral itself is that inter-temporal equilibrium price, and let us suppose that is denoted by P*, to reduce the clutter; in that case, the time path can be written in this form: [P0 – P*] * (-δ/β)t + P*. So, this is the time path; as we know, all these parameters are positive; in particular, β and δ are positive, which means that this term, this is the b term, small b. That small b = -δ/β; δ and β are both positive, which means that this b is negative. And as we know, if b is negative, if its sign is negative, then there is bound to be oscillation in the time path. So, monotonic movement, whether you are talking about convergence or divergence, is not going to be there. So, that is number one; number two, stability. When do we have stability? We have stability if |b| < 1, which means this has to be satisfied: |-δ/β| < 1, and we know we—this can be written as δ/β. So, this is the condition of stability, which is implying that β > δ. So, this is the condition of stability. So, two conclusions we are getting from here: one, there is bound to be oscillation in the time path of the price, and number two, if we have to have stability, then β has to be greater than δ.
Now, what are β and δ? Just recall β and δ are the absolute value of the slopes of the demand and supply functions respectively. These are the two functions. So, β and δ are the absolute values of the slopes of these two functions, demand and supply. The stability condition implies that the slope of the demand curve or line, in this case, because we are assuming a linear function, the slope of the demand line is higher than the slope of the supply line. I mean, the absolute value of the slope; if this condition is satisfied, there will be damped oscillation towards the equilibrium. Because you know, in that case, the modulus of b is less than 1; on the other hand, if β = δ, there will be a uniform oscillation; the price will alternate between P0 and -P0 + 2P*, maintaining equal distance from P*. Let us try to see how this is happening; if β = δ, this becomes -1. So, suppose t = 0; if t = 0, then this part is just 1; in that case, Pt will be just equal to P0. So, that is one possible value of Pt; if t = 0, then you have this term becoming equal to just P0. So, that is mentioned here, P0. On the other hand, suppose this is equal to 1, t = 1; in that case, this becomes -1, in which case this entire expression becomes 2P* – P0, 2P* – P0. So, in this case, where β = δ, Pt will alternate between these two values, P0 and -P0 + 2P*, in every two successive periods. And these two values are actually equidistant from P*. And the third case is where you do not have a modulus of P less than 1 or equal to 1, but it is greater than 1. So, this is the case where β < δ; the demand line is flatter than the supply line; there is explosive oscillation; Pt will divert from P* by fluctuating around it. I have drawn…
The diagram for this is the case of stable equilibrium where beta is greater than delta. So, as you can see, the line Qd is quite steep, whereas the line Qs is not that steep; it is relatively flat. In this case, if you have noticed, I have represented price in the horizontal axis. And quantity along the vertical axis, and this is just to make sure that we are getting the slopes correct, that beta is greater than delta. Because when we are writing the demand function and the supply function in this manner, in this manner, and this manner, we are taking P as the independent variable, and usually independent variables are represented along the horizontal axis. Although in economics, the prices are generally represented along the vertical axis. So, in this case, to get the slopes correct, I have followed the convention of representing the independent variable in the demand and supply functions along the horizontal axis. They intersect each other at this point P* Q*. Here is P*, and here is Q*, and this P* Q* is the inter-temporal equilibrium; in particular, this P* is that value alpha plus gamma divided by beta plus delta.
Now, we can see how this model works. We start with some arbitrary value P1, and suppose this is period t = 1. Now, this is the price that is prevailing in period t = 1. As we know, the quantity supplied in this next period will be affected by this price. So, in period t + 2, the supply in the market is read from the supply curve, and this is given by this; so, this is Q1, this is the supply in period 2. For the demanders to buy this amount, that is Q1, the price has to be P2 less than P*. Let us suppose where is P2. So, P2 is here; if the demand function has to intersect this line, that is Q = Q1, then this is the intersection point; from this we read the price, which is P2; that means the price has to be equal to P2 in period 2. So, that the demanders, that is the buyers, they are willing to buy this much amount of good, which is Q1, that has been produced by the producers in period 2. So, the price, you can see, from P1 it drops to P2, and in particular P2 is less than P*. So, in t = 2, the price falls to P2. Due to this low price in t = 3, the supply also falls to Q2. If the price is P2, what is the supply that can be read from the supply line? So, this is the supply, and this is given by Q2. Mind you, this Q2 supply will come to the market in the next period, that is period 3. So, in period 3, to clear the market, the price must rise because the quantity supplied is very low. The price must rise, so the demanders demand less, and this price is P3, and in particular it is more than P*. So, I read the P3 from the demand function, and this is the P3 price. In particular, you can see that the P3 price is now greater than P*. So, we started with P1, which is greater than P*, then we came to P2, which is less than P*, and then again we go to P3, which is greater than P*.
In other words, we are oscillating around the P* price, which is the inter-temporal equilibrium price. Thus is the market clearing price, which is the equilibrium in each period, keeps fluctuating from below to above the inter-temporal equilibrium price P*. Each period the price inches to P*; in the limit it becomes equal to P*, and that you can see in the first period the P1 price is quite far away from P*; in the next period P2 is quite close to P*; in the third period it is even closer to P*, and in the next period will become even closer, and likewise we will go on. The fluctuation, or the amplitude of the fluctuation, is coming down in each successive period. So, in the limit it becomes equal to P*. This is the stable case; the path of the equilibrium price shown by the arrows resembles a Cobweb. So, this is the path you are going there, and here you are going, and then you are coming down, and in the next we will go up, and like this will go on; so in the limit it will approach the intersection point, but it looks like a spider Cobweb. So, it resembles; it is Cobweb; that is why it is called the Cobweb model.
We can also have the other cases. In the diagram here you have the uniform oscillation case where beta is equal to delta. As you can see, I can start from any arbitrary price; suppose this is the price I am starting from; going up, this is the supply in the next period; to clear the market the price has to come to this level; and if the price is this level, then the supply comes down, and to clear the market again the price has to go up, but you can see the price in the third period is the same as the price in the first period. So, therefore, actually you are going round and round in the same circle; there is no damped oscillation in this case at least. And here on the right-hand side of the panel you have the case of explosive oscillation. I am not going to explain it in detail, but you can see how it is happening; you take any arbitrary price; this is the supply. So, the market has to clear; so this must be the price in the next period, and this is the price in the next period; this is the quantity; then this is the price in the next one at that period. And you can see the price will go on fluctuating, but it is getting far and far away from the inter-temporal equilibrium. This was the case of explosive oscillation.
Now, so far, we have only focused on the time path of Pt, the market clearing price as defined by market equilibrium condition; the time path of equilibrium quantity. So, that we have not talked about. But as we know, in any sort of equilibrium market equilibrium, there are two sides to it; one is the quantity, the other is the price. What about the time path of the equilibrium quantity? That can be found out by plugging the time path of the price in the demand function because demand depends on the price of that period. So, let us look at how actually we can do that. So, this is the time path of price Pt. Now if we take this Pt and we plug it here, then we will get the time path of the quantity; this will be a function of t only, and it will have some P0 term, but as we know P0 is not a variable; it is a constant. So, in short, we are going to get Qd, that is quantity; I should not say Qd; I should say Qt. So, quantity in each period, the equilibrium quantity in each period will be a function of small t alone, and there will be some parameters. And that is what is meant by time path; in this case, the equilibrium time path. So, that is how it is done. Now, I said that it has to be plugged into the demand function, and the reason being that the demand in the Cobweb model in a particular period depends on the price of that period. So, there is no lag there. That is why the time path of the price has to be plugged into the demand function, not the supply function, and the stability of price will be reflected in the quantity as well. So, if the price path is stable or it is unstable, then that will be replicated in the quantity, stability or instability as well. So, this was the Cobweb model.
So, we can just talk a few words; you can say a few words about another model; this model is a market model with inventory, and then maybe you can close this lecture. If the market is not perfectly competitive, the producers can adjust price depending on how much they were able to sell in the previous period. So, here price is not determined automatically by equality of demand and supply; here price depends on how much the producers were able to sell in the previous period. In other words, it is the producers who determine the price. So, this is the market model with inventory; if the inventory is piled up, they may reduce the price so that the unintended inventory can be disposed of. Similarly, a price is revised upwards if the inventory has run down unexpectedly; inventory, in short, means stock. So, if the stock is running down unexpectedly, then the producers might jack up the price; or if the inventory is piling up, then they might think that it is a good idea to give some incentive to the buyers to buy, in which case they might reduce the price. Here both demand and supply functions are dependent on the current market price, unlike the Cobweb model, so there is no lag in the supply function as well as the demand function. The model is as follows. So, these are the three equations: Qdt = alpha - beta Pt, QSt = -gamma + delta Pt; you can see there are no lags there. And finally, Pt+1 = Pt - sigma multiplied by QSt - Qdt. So, this is the new thing that has come now; the price in the next period depends on the price of the previous period, but it also depends on whether the inventory is piling up or it is running down, which is captured by this term. We shall talk more about it in the next lecture. Thank you for joining me in this lecture, and see you in the next lecture. Thank you.