Transcription
Welcome to the ninth lecture of this course called Mathematics for Economics Part 1. So, today we are going to start with a new topic; this topic is called differentiation. Earlier, what we have discussed? We have discussed functions, and this differentiation is related to the idea of functions. Differentiation is a very old tool used in mathematics and many other scientific inquiries. And it is a commonly used tool in many fields of science, such as physics, mechanics, mathematics of course. It was discovered—this tool of differentiation and calculus as such—was discovered by two prominent mathematicians, Isaac Newton and Leibniz, separately and independently.
So, what does this idea of differentiation look at? The differentiation looks at the rate of change of a function. And so, in economics also, that idea is often used; we want to find out how a particular function changes—maybe population, for example—changes over a period of time, how it is changing, at what rate, what will be the total amount of population after a period of time. Or, for example, national income; so that is also changing with respect to time. So, those are the things that one can look at. Here, time is the independent variable. We are talking about changes of population with respect to time or national income or per capita income with respect to time, but it is not necessary that we always take time to be the independent variable. One can think about the change in the production of a firm with respect to change in the inputs. In that case, the inputs are the independent variables. Or you can think about the change in the consumption of all the consumers of the economy with respect to change in the national income. In that case, the national income is the independent variable. So, in each case, one tries to find out what is the rate of change of the dependent variable.
Let me start with this idea of differentiation. Derivative—or this is closely linked with the idea of differentiation as we shall see—derivative is the rate of change of the value of a function. Geometrically, let us first study the graph of a function. So, before going into the mathematics of it, let us try to see how it can be interpreted in a geometric manner. We have defined what the graph of a function is. So, suppose you have a graph like this: so y is equal to f(x), suppose, and suppose you have a linear function y is equal to ax + b. Now, here the slope of this function we know is given by small a, and what was the interpretation of this small a, the slope? It is the steepness of the graph, how fast the line rises from the left to right if a is greater than 0; and if a is less than 0, then this a measures the steepness of the fall of the function, because we know that if a is negative, then the function will be downward sloping. But in that case also, the magnitude of a will measure how steeply the function is declining. Now, a is also called the derivative of this function ax + b. So, this is basically the slope of this function; if we have a linear function, then the slope is the derivative of that function. So, it measures how much the y is changing with respect to x, and that is the idea of slope, and that is also the idea of derivative.
Now, this was the case where the function is a linear function of the form ax + b, but as we know, this is a very simple sort of function. In general, a function may not be linear, and if you do not have a linear function—if it is not of the form ax + b—then how do you define the derivative of a function? For a general function of the form y is equal to f(x), the derivative at a point P is defined as the slope of the tangent at that point. So, suppose you have this function which is given in the diagram here, and here is that function f(x), and here is the point P. Now, we want to find out what is the derivative at point P; we basically draw a tangent to the graph at that point P, and then we say that the derivative of the function at point P is the slope of the tangent. And at this point, we are wondering what is the tangent? A tangent at a point of a function is a straight line which touches the function or the graph at that point. So, here, if I have to draw the tangent at P, I draw a straight line which is touching the graph at point P. So, PT is here the tangent. PT is a straight line and which is touching the graph at point P; so therefore, PT is the tangent to the function at point P. At point P, suppose your value of the argument is given by a, then the derivative at point x = a is given by f'(a), which is the slope of the tangent at the point (a, f(a)). So, here at P, suppose the value of x is a; so what is the value of y? This is given by f(a). So, the coordinates of the point P is (a, f(a)); and therefore, if we want to find the slope at point P, we draw the tangent at P, which is PT, and then we find out the slope of that line, which is PT.
Now, this is a kind of heuristic way to explain what the slope is and what the derivative of any function at a point is. It is a geometric sort of explanation, all right, but we are not exactly defining it in a very rigorous manner. So, let us try to do that, and here you have another diagram, and here you have this dark line as f(x), and I have taken a particular point A. So, point A is taken on the graph of the function y is equal to f(x), and on the same graph I take another point here B, and I draw a straight line through A and B. This line which is passing through A and B, two points on the graph, is called a secant. So, it is a Latin name; it is like the idea of a chord. Now, one can measure the slope of the secant AB; it is a straight line; you can just take two points, and these two points are A and B, and you can measure the slope of this line AB. Now, suppose point B approaches point A; the secant AB, then we will approach the line AC, which is called the tangent at A. The slope of AC is called the derivative of the function at A. So, here I have taken another point B which is quite close to A, and I am assuming that point B is approaching point A and it is staying on the graph. Now, as it is approaching point A, then this line—the secant line AB—in the limit it becomes the line AC, and so AC will be called the tangent of the curve at point A, and the slope of AB will become the slope of AC, and the slope of AC is called the derivative of the function at point A.
So, let me do it a little bit mathematically. Now, let the coordinates of A be (a, f(a)), because capital A point is on the function, so f(a) will be the value of the function, and let the x coordinate of point B be a + h; h is a small number, and since a is the x coordinate—a small a—and a + h is the coordinate of B, that means that the horizontal distance between A and B is h. So, here if you draw a perpendicular, this is h, and this is a. So, I have taken a point on the curve, and the horizontal distance between the original point and the new point is given by small h. Now, what is the vertical distance between A and B? It is given by f(a + h) – f(a). So, in terms of this diagram, so this is f(a + h) – f(a). Therefore, the slope of the secant AB is the familiar expression for slope, which is (y1 – y2) / (x1 – x2), or perpendicular / base. So, here the perpendicular is f(a + h) – f(a), and the base is h. So, therefore, this is the slope of AB. This ratio, which is f(a + h) – f(a) / h, is called the Newton or the differential quotient of f, obviously at point A. So, this is named after Isaac Newton because he was one of the proponents of differential calculus.
Now, as B approaches A over the graph, a + h approaches small a, or in other words, h approaches 0. So, one is thinking about this h becoming 0, which means B is approaching A over the graph, over the function. Simultaneously, the secant approaches the tangent at A, as we have just discussed that the secant was a chord; A and B were points on the curve. Now, as h approaches 0, B is approaching A, and the secant no longer remains a secant; it becomes the tangent to the curve at point A; that is why I wrote that the secant approaches the tangent at A. Hence, the slope of the secant AB approaches the slope of the tangent AC, and which we denote by f’(a); f’(a) is the derivative of the function evaluated at point A. In short, the derivative of f(x) at point a in its domain—that we have to be careful about; this x = a should belong to the domain of the function—this is given by f’ or f’(a), which is equal to lim h→0, and then you have this Newton quotient, which is f(a + h) – f(a) / h. Now, here I have used this new symbol lim h→0, and this is a new symbol; I understand it; basically, it is trying to capture the fact that in this quotient we are taking h to be as close to 0 as possible. It does not mean that h is equal to 0. This is very important to understand because if you take h = 0, then what happens to the Newton quotient? It becomes equal to 0 because h = 0, and in the numerator you have f(a) – f(a), because h = 0; 0 / 0, and which is undefined. So, we are not saying that h = 0, because that will give us a quantity which is not defined. What we are saying is that B is approaching A. So, in the limit, h is tending to 0; we are not saying h = 0. We are going to talk about limits a little bit in detail later on. Now we are just introducing the idea of derivative.
So, here is an example: suppose the form of the function is given f(x) = x³. We have to find f’(a); a is any constant which belongs to the domain of the function, and then we have to find out f’(0) and f’(-1). So, once we can find f’(a), then I will just substitute a = 0 for a and -1 for a, and we will get these two other quantities. So, what do we do? We go by that formula. This is the formula of derivative: lim h→0 f(a + h) – f(a) / h. Now let us first concentrate on the numerator. Now, what is the numerator in this case? So, the form of the function is x³. So, instead of x, I am writing (a + h)³ – f(a); so that means a³. So, (a + h)³ – a³. And if I just expand (a + h)³, then I will get a term a³ which will get cancelled with this –a³. So, I will be left with this part: 3a²h + 3ah² + h³. Now, the Newton quotient is this thing divided by h, and so if I divide this thing by h, I am left with this expression: 3a² + 3ah + h², and then I take lim h→0, because this is the derivative, and if I do so, then this part will vanish, this part will vanish. So, I will be left with only 3a². So, 3a² is the answer; f’(a) = 3a². So, the first part is answered. Now we have to find out what is f’(0) and f’(-1); for that I just have to substitute 0 for a in this expression. So, if I put a = 0, I get 3 multiplied by 0, which is 0, and f’(-1); so here again I put a = -1; (-1)² is +1. So, this becomes +3; so these are the answers: this one, this one, and this one: 3a², 0, and 3.
Now, this above method of finding the Newton quotient first and taking lim h→0—this method can be used to compute the derivatives for relatively simple functions. Notation-wise, there are some other ways to express derivatives. So, what we have used is f’; so that is the notation to denote derivatives, but there are other notations also. So, if y = f(x), then the derivative of f at x is written as y’ also. It is also written as differential notation, which is written as dy/dx. So, this was used by Leibniz, and he used this particular notation dy/dx. dy/dx in this case will be written as d/dx of f(x), because y = f(x). So, all these things are the same things; they mean the same thing. So, in other words, you are given a function, let us suppose y = x³, then dy/dx = d/dx of x³, and that we have seen is equal to 3x².
Here is an example from economics. The cost of producing x units of a commodity is given by the formula C(x) = a + bx². Find C’(x). C’(x) is what? This is the derivative of the cost function; this is the cost function with respect to x; what is the x? This x is the units of the commodity that is being produced. To find the derivative of the function, we calculate the differential quotient—the Newton quotient or differential quotient. Now, what is the numerator? f(x + h) – f(x). So, that will be—if I use the formula that f(x) = a + bx²—it becomes this much: a + b(x + h)² – a – bx²; a and a will get cancelled; so I can take b common, and x² will also get cancelled, and what we are left with: b multiplied by h² + 2xh, and I have to divide it by h to get the Newton quotient or the differential quotient. And so we are getting this term: b multiplied by h + 2x, and then I take lim h→0; so this term will drop out, and we will be left with 2bx, and this is our C’(x) or C’(x). This C’(x) is also called the marginal cost.
Now, if I go back to the idea of derivative—what is the notion of derivative—then how do we interpret the marginal cost? So, derivative is the rate of change of the dependent variable. So, here marginal cost is the rate at which the cost is changing with respect to change in the output level, because here the independent variable is the output. So, this is the rate of change of the cost function with respect to change in the output level. The derivative of a function at a point is interpreted as the slope of the tangent to the graph at that point. So, this was a geometric interpretation. In economics, it is interpreted as the instantaneous rate of change at a particular value. This is important—instantaneous rate of change—because we are considering B to be in the immediate neighborhood of A, and then we are assuming that h is going to 0. So, that basically means that a and b are very close together. So, one is considering a very small change of the independent variable and then looking at the change of the dependent variable, which is why this is called the instantaneous change at a particular value.
Suppose, y = f(x) is a given function; we take x = a and take a change of x from a to a + h; h is a small number. Corresponding to this, the values of the function are f(a) and f(a + h). The change in the value of the function is given by f(a + h) – f(a). The rate of the change in the neighborhood of a is—on the numerator you have the change of the value of the function, and in the denominator you have the change in the value of the independent variable, which is h. This is the same as the Newton quotient. Once we take the limit h→0, we get the derivative of f at a; so this is what I just explained: the derivative of a function at a particular point a is the change—instantaneous change—rate of change at a particular value. It is sometimes denoted by the dot sign, especially if we take the independent variable to be time. So, t is time; so here in this particular function y = 4t, that means as time is changing, the dependent variable y is changing by this formula 4t, y = 4t. Now here I can find out the derivative of this function dy/dt, and this dy/dt is also written in this fashion: ẏ, and we can verify that here the derivative will be 4—it is a constant number.
So, here is an example: suppose India’s population at a time t is given by the function P(t) = 1.8t + 100, and suppose P(t) is the population, which is in crores, and we are given this information that at t = 0 we are considering the year 2000. So, this is called the origin; so if you put t = 0, then from this function what you get? You get P(0) = 0 + 100, which is 100. So, in the year 2000, the population was 100 crore, which is actually not off the mark; India’s population was around 100 crore in the year 2000, and then we are assuming that the growth of population or the population function is given by this formula: 1.8t + 100. Now, from this formula, we can find out what is the Newton quotient or the differential quotient, and it turns out to be 1.8. And obviously, if you take the limit h→0, this 1.8 remains 1.8 because there is no h term here. So, that means that the derivative of this function with respect to t is 1.8. So, India’s population rises at this rate: 1.8 crore each year. I am going to talk about why we should interpret—we can interpret this as—per year; each year means in 1 year. So, why do we say so? Why are you not saying that this is the instantaneous rate of change?
Another example is here: the cost function; we have already talked about this in a particular context; so the cost function of a firm is given by C = C(x). x is the level of output. Now we can take the derivative of this function and evaluate this at a particular point—at a point, suppose x = a—and then this becomes C’(a), and this is the definition of that. This is called the marginal cost at output level x = a. So, I do not want to spend more time on this because you have talked about this before. So, I am going through some of the examples where this idea of derivative is useful in economics. These are some other examples. Similarly, the derivative of the production function with respect to an input is called the marginal product of that input. So, here is the production function, suppose F; so you have labor, capital, land, etcetera, and suppose you take the derivative. Here I have taken a function which is a function of many variables, and we are going to talk about that also; we have multiple variables, but for the time being let us suppose these are constant; so these are parameters; capital and land are fixed. So, this becomes a function of only one variable, which is L—labor. So, you can now find out what is ∂F/∂L; this will be called the marginal product of this particular input, which is labor. So, once we do economics, we shall see plenty of examples of this marginal product.
The derivative of the aggregate consumption function with respect to the income level is called the marginal propensity to consume. We have again seen some examples of this. So, here is this consumption function: C is consumption; y is the income. Now we can talk about the derivative of this. So, dC/dY; this is called the marginal propensity to consume (MPC). Similarly, marginal propensity to save is the derivative of the savings function with respect to income. So, just as consumption of people depends on their income—so that is why you have C = C(y)—savings also depend on the income level of people. The more rich people are, the more they save. So, you have S, which is savings, is equal to S(y), and so you can now talk about the derivative; this is dS/dY, and this is called the MPS—marginal propensity to save.
Capital stock—again, this is an example from macroeconomics—capital stock at point t is denoted by K(t). The instantaneous change in the capital stock is called the rate of investment and which is denoted by I(t). So, K is capital stock; K is a function of t; the more time passes, it is conceivable that more capital is being accumulated. So, capital stock is a function of time. If one takes the derivative of this, this becomes the instantaneous rate of change of capital stock, and this is denoted by I. This itself can be a function of t.
So, I come to that point: why we are talking about change in the output level by one unit. Suppose I take h = 1, then this Newton quotient becomes equal to this: So, C(a + 1) – C(a) / 1; so it becomes C(a + 1) – C(a). So, the derivative becomes something very close to this amount, and what is C(a + 1) – C(a)? This is the change in the cost when the output level changes by one unit, or the extra cost of producing an additional unit of output. So, if the output level changes by one unit, then this C(a + 1) – C(a) gives you the change in the cost to produce that extra unit of output, and this we have seen is approximately equal to the derivative. So, it is useful to interpret the derivative of a function at a point as the change with respect to one unit change in the argument. So, practically speaking, in economics, we shall be dealing with discrete changes more often. And if we are talking about discrete changes, then the minimum amount of change can be assumed to be one unit; so therefore, the derivatives are often interpreted as the change in the value of the function with respect to one unit change in the argument.
Now, derivatives give you the instantaneous change, but compared to that, there is another rate of change which is called the rate of proportional change, and this is defined as f’(a) / f(a). Now, on the numerator you have the derivative, but you are dividing that by the value of the function; this is called the rate of proportional change—proportionately how much is the value of the derivative. In economics, this is often used to denote percentage change per year or per month. More on this later when we talk about exponential functions; so there is something called an exponential function; in an exponential function we shall see—I think we have discussed this before, but we have not talked about the derivative of exponential functions. So, in the case of exponential functions, the rate of proportional change is always constant.
Practically speaking, economic data are recorded at discrete time intervals, such as after a year, after a quarter, after a month, week, etcetera. So, for example, the GDP of a country is generally denoted with respect to a year; in a particular year, what is the GDP? In the next year, or if we go to smaller time intervals, it could be in a particular quarter. So, nonetheless, it is a discrete change in the time; it is not a continuous change. The data are not of the nature where one observes change in the value of the function with infinitesimally small change in…
Time is changing in a discrete manner. So, time is granular; it cannot be broken down further to very small, small grains. The function that is actually analyzed is thus an approximation derived from the empirical observations. So, empirically, what one finds is changes in the discrete manner. There you do not find change in the continuous manner, but from those discrete data, one constructs a function which is an approximation, and that function which one constructs is then analyzed as a function of a continuous variable.
Now we come to something that we have been talking about, but did not define it in a prescribed manner. This is the idea of limits. We talked about the fact that h going to 0 in the limit h is going to 0, but what is the idea of limits? Suppose a function f(x) is defined for all values of x near a, but not necessarily at a; then f(x) is said to have a limit equal to A as x tends to a if f(x) tends to A as x tends to a. So, you have a function f(x) which is defined for all values of x near a, but it is possible that it is not defined at x = a. Then we say that this function has a limit, and that limit is equal to A if the following thing is satisfied: that f(x) tends to A as x tends to a. So, this is, in mathematical terms, this is written as this: f(x) tends to A as x tends to a. And in a different manner, it is also written as this: limit of x goes to a, f(x) is going to A. If f(x) does not tend to a fixed number as x tends to a, then we say f(x) has no limit at x = a, or limit f(x), x going to a, does not exist.
So, what could be the visualization of this? That you have x going to A, but f(x) does not go to any finite or fixed number. So, you can imagine a function like this: that you have this x axis here, you have a, and the function is something like this. So, you see here the function is composed of two parts; this is f(x), this is also f(x). As x goes to a, the function from the left-hand side it goes to minus infinity; from the right-hand side, it goes to plus infinity. So, in this case, this limit at x = a does not exist.
Here is an example of how to find the limit. Here f(x) is, suppose, this: √(h + 1) - 1, the whole thing divided by h, and we want to find out the limit at h = 0 so that we have to find out. So, what do we do? We take values of h very close to 0. At h = 0, this function is undefined because the denominator becomes 0. So, here you have a dot; nothing is there. I am not taking h = 0; I am taking h very close to 0. For example, I am starting h at 0.5, and I am going to h = 0.01, and correspondingly, I can see that the value of the function from 0.449 it goes to close and close to 0.5; it approaches 0.5 because at 0.01 it takes the value 0.499. I can approach the value h = 0 from the left-hand side also. So, I take point h = -0.5; the corresponding value of the function is 0.506. Then I choose to take this will be -0.1; the corresponding value of the function will be 0.501, so it is very close to the value 0.5. So, one can guess that as h approaches 0, then the value of the function f(x) approaches 0.5 because from both the sides as h is approaching 0, that is the idea of limit; as it approaches 0, the value of the function becomes very close to the value 0.5.
This method of using calculators to find the limit is, however, ad hoc. So, what we are doing is that using calculators to find out the value of the function as we are taking the value of h very close to 0. This is ad hoc, why? Because we cannot find all the possible values of x close to a. So, in a more precise manner, how do we define a function or how do we define a limit is that suppose you have f(x) going to A as x goes to a. It means that f(x) can be made as close to A as possible as we want for all x sufficiently close to a. So, the value of the function can be made very close to the limit, which is A, if we take the argument, the value of the argument very close to that point, which is a.
Now, two points to note here: one is, firstly, when we say that the limit is A, when it is calculated, the value of x on both sides of A are to be considered, just as we have done before; here we are approaching the value 0 from the left-hand side here and from the right-hand side here, from the negative and from the positive, we are approaching the value 0. And the second point is a more conceptual point. One is not interested in the value of f(a); f(a) is not interesting to us, but how f(x) behaves close to x = a; that is the idea of limit. So, you are approaching x = a, and if we are approaching x = a, then how is this function f(x) behaving? So, these are rules of limits; I think we will stop here and take these up in the next lecture. So, we are calling it a day today, and I shall see you in the next lecture. Thank you.