## Complex Numbers And Circular Functions

When the coefficients of a second-order linear differential equation, y(t) + aly'(t) + a2y - b, are such that a < 4a2, the characteristic-root formula (15.5) would call for taking the square root of a negative number. Since the square of any positive or negative real number is invariably positive, whereas the square of zero is zero, only a nonnegative real number can ever yield a real-valued square root. Thus, if we confine our attention to the real number system, as we have so far, no...

## The Realnumber System

Equations and variables are the essential ingredients of a mathematical model. But since the values that an economic variable takes are usually numerical, a few words should be said about the number system. Here, we shall deal only with so-called real numbers. Whole numbers such as 1,2, 3. are called positive integers', these are the numbers most frequently used in counting. Their negative counterparts 1, 2, 3. are called negative integers these can be employed, for example, to indicate subzero...

## Differential Equations With A Variable Term

In the differential equations considered above, the right-hand term b is a constant. What if, instead of b, we have on the right a variable term i.e., some function of t such as bt2, ehl, or sin The answer is that we must then modify our particular integral . Fortunately, the complementary function is not affected by the presence of a variable term, because yc deals only with the reduced equation, whose right side is always zero. We shall explain a method of finding y , known as the method of...

## O

For depicting the general qualitative character of the phase diagram, however, a few representative streamlines should normally suffice. Several features may be noted about the streamlines in Fig. 18.2. First, all of them happen to lead toward point E. This makes E a stable (here, globally stable) intertemporal equilibrium. Later, we shall encounter other types of streamline configurations. Second, while some streamlines never venture beyond a single region (such as the one passing...

## Info

Since the determinant of the coefficient matrix in (8.23') is nothing but the particular Jacobian determinant which is known to be nonzero under conditions of the implicit-function theorem, and since the system must be nonhomoge-neous (why ), there should be a unique solution to (8.23'). By Cramer's rule, this solution may be expressed analytically as follows By a suitable adaptation of this procedure, the partial derivatives of the implicit functions with respect to the other variables, x2, ,...

## Continuity of a Function

When a function q g(v) possesses a limit as v tends to the point N in the domain, and when this limit is also equal to g(N) that is, equal to the value of the function at v N the function is said to be continuous at N. As stated above, the term continuity involves no less than three requirements (1) the point N must be in the domain of the function i.e., g(N) is defined (2) the function must have a limit as v -* N i.e., lim g(v) exists and (3) that limit must be equal in value to g(N)-, i.e.,...

## Chain Rule

If we have a function z f(y), where y is in turn a function of another variable x, say, y g(x), then the derivative of z with respect to jc is equal to the derivative of z with respect to y, times the derivative of y with respect to jc. Expressed symbolically, This rule, known as the chain rule, appeals easily to intuition. Given a Ax, there must result a corresponding Ay via the functiony g(x), but this Ay will in turn bring about a Az via the function z (> ') Thus there is a chain reaction...

## Simultaneous Differential Equations And Difference Equations

Heretofore, our discussion of economic dynamics has been confined to the analysis of a single dynamic (differential or difference) equation. In the present chapter, methods for analyzing a system of simultaneous dynamic equations are introduced. Because this would entail the handling of several variables at the same time, you might anticipate a great deal of new complications. But the truth is that much of what we have already learned about single dynamic equations can be readily extended to...

## Extreme Values Of A Function Of Two Variables

For a function of one choice variable, an extreme value is represented graphically by the peak of a hill or the bottom of a valley in a two-dimensional graph. With two choice variables, the graph of the function z f(x, y) becomes a surface in a 3-space, and while the extreme values are still to be associated with peaks and bottoms, these hills and valleys themselves now take on a three-dimensional character. They will, in this new context, be shaped like domes and bowls, respectively. The two...

## J 2x2 dx 2 fx2 dx 2 y C J yr3 c

In this case, factoring out the multiplicative constant yields j3x2 dx 3 jx2 dx 3( y + c'i) + Note that, in contrast to the preceding example, the term x3 in the final answer does not have any fractional expression attached to it. This neat result is due to the fact that 3 (the multiplicative constant of the integrand) happens to be precisely equal to 2 (the power of the function) plus 1. Referring to the power rule (Rule I), we see that the multiplicative constant (n +...

## L Yx and s Ty

Since f is itself a function of x (as well as of y), we can measure the rate of change of fx with respect to x, while y remains fixed, by a particular second-order (or second) partial derivative denoted by either fxx or d2z dx2 Th notation Xi has a double subscript signifying that the primitive function has been differentiated partially with respect to a twice, whereas the notation d2z dx2 resembles that of d2z dx2 except for the use of the partial symbol. In a perfectly analogous manner, we...

## Rules Of Differentiation Involving Two Or More Functions Of The Same Variable

The three rules presented in the preceding section are each concerned with a single given function f(x). Now suppose that we have two dijferentiable functions of the same variable x, say, (x) and g(x), and we want to differentiate the sum, difference, product, or quotient formed with these two functions. In such circumstances, are there appropriate rules that apply More concretely, given two functions say, f x) 3x2 and g(x) 9x'2 how do we get the derivative of, say, 3x2 + 9x'2, or the...

## Total Differentials

The concept of differentials can easily be extended to a function of two or more independent variables. Consider a saving function where S is savings, Y is national income, and i is interest rate. This function is assumed as all the functions we shall use here will be assumed to be continuous and to possess continuous (partial) derivatives, which is another way of saying that it is smooth and differentiable everywhere. We know that the partiaL derivative dS dY (or SY) measures the rate of...

## Twovariable Phase Diagrams

The preceding sections have dealt with the quantitative solutions of linear dynamic systems. In the present section, we shall discuss the qualitative-graphic (phase-diagram) analysis of a nonlinear differential-equation system. More specifically, our attention will be focused on the first-order differential-equation system in two variables, in the general form of Note that the time derivatives x'(t) and y'(t) depend only on x and y and that the variable t does not enter into the and g functions...

## Secondderivative Test

Returning to the pair of extreme points b and e in Fig. 9.5 and remembering the newly established relationship between the second derivative and the curvature of a curve, we should be able to see the validity of the following criterion for a relative extremum Second-derivative test for relative extremum If the first derivative of a function at x x0 is f'(x0) 0, then the value of the function at x0, (x0), will be a. A relative maximum if the second-derivative value at x0 is f(x0) < 0. b. A...

## M P M p rM TP rMP

Thus (15.35) stipulates that dll dt is negatively related to the rate of growth of real-money balance. Inasmuch as the variable p now enters into the determination of dU dt, the model now contains a feedback from inflation to unemployment. Together, (15.33), (15.34), and (15.35) constitute a closed model in the three variables it, p, and U. By eliminating two of the three variables, however, we can condense the model into a single differential equation in a single variable. Suppose that we let...

## Exercise

1 Use the rules of differentials to find (a) dz from z 3x2 + xy 2y3 and (b) dU from U 2xi + 9x,x2 + x . Check your answers against those obtained for Exercise 8.2-1. 2 Use the rules of differentials to find dy from the following functions Check your answers against those obtained for Exercise 8.2-2. 3 Given y 3x,(2x2 l)(x3 + 5) (b) Find the partial differentia of y, if dx2 dx3 0. (c) Derive from the above result the partial derivative dy dx 4 Prove Rules II, III, IV, and V, assuming u and v to...

## Leastcost Combination Of Inputs

As another example of constrained optimization, let us discuss the problem of finding the least-cost input combination for the production of a specified level of output Q0 representing, say, a customer's special order. Here we shall work with a general production function later on, however, reference will be made to homogeneous production functions. Assuming a smooth production function with two variable inputs, Q Q(a, b), where Qa, Qh > 0, and assuming both input prices to be exogenous...

## Finding The Stationary Values

Even without any new technique of solution, the constrained maximum in the simple example defined by (12.1) and (12.2) can easily be found. Since the constraint (12.2) implies we can combine the constraint with the objective function by substituting (12.2') into (12.1). The result is an objective function in one variable only U xt (30 - 2xt) + 2xj 32x, - 2xf which can be handled with the method already learned. By setting dU dx, 32 4x, equal to zero, we get the solution 3c, 8, which by virtue...

## QdD[PtPtPt Qs S[PtPtPt

If we confine ourselves to the linear version of these functions and simplify the notation for the independent variables to P, P', and P, we can write Qd a - fiP + mP' + nP (a, fi > 0) 7 Qs -y + 8P + uP' + wP (y,8> 0) where the parameters a, fi, y, and 8 are merely carryovers from the previous market models, but m, u, and w are new. The four new parameters, whose signs have not been restricted, embody the buyers' and sellers' price expectations. If m > 0, for instance, a rising price will...

## Ingredients Of A Mathematical Model

An economic model is merely a theoretical framework, and there is no inherent reason why it must be mathematical. If the model is mathematical, however, it will usually consist of a set of equations designed to describe the structure of the model. By relating a number of variables to one another in certain ways, these equations give mathematical form to the set of analytical assumptions adopted. Then, through application of the relevant mathematical operations to these equations, we may seek to...

## Exercise 164

1 On the basis of (16.10), find the time path of Q, and analyze the condition for its convergence. 2 Draw a diagram similar to those of Fig. 16.2 to show that, for the case of 8 , the price will oscillate uniformly with neither damping nor explosion. 3 Given demand and supply for the cobweb model as follows, find the intertemporal equilibrium price, and determine whether the equilibrium is stable 4 In model (16.10), let the QdI Qst condition and the demand function remain as they are, but...

## P P P Pp P

Curve, the price P, will lead to Q2 as the quantity supplied in period 2, and to clear the market in the latter period, price must be set at the level of P2 according to the demand curve. Repeating this reasoning, we can trace out the prices and quantities in subsequent periods by simply following the arrowheads in the diagram, thereby spinning a cobweb around the demand and supply curves. B comparing the price levels, P0, Px, P2, , we observe in this case not only an oscillatory pattern of...

## App Expressed By Capital-labor Ratio

The function g is homogeneous of degree one (or, of the first degree) multiplication of each variable by j will alter the value of the function exactly7-fold as well. Example 3 Now, consider the function h(x, y, w) 2x2 + 3yw w2. A similar multiplication this time will give us Thus the function h is homogeneous of degree two in this case, a doubling of all variables, for example, will quadruple the value of the function. In the discussion of production functions, wide use is made of homogeneous...

## The Cobweb Model

To illustrate the use of first-order difference equations in economic analysis, we shall cite two variants of the market model for a single commodity. The first variant, known as the cobweb model, differs from our earlier market models in that it treats Qs as a function not of the current price but of the price of the preceding time period. Consider a situation in which the producer's output decision must be made one period in advance of the actual sale such as in agricultural production, where...

## The Nature Of Comparative Statics

Comparative statics, as the name suggests, is concerned with the comparison of different equilibrium states that are associated with different sets of values of parameters and exogenous variables. For purposes of such a comparison, we always start by assuming a given initial equilibrium state. In the isolated-market model, for example, such an initial equilibrium will be represented by a determinate price P and a corresponding quantity Q. Similarly, in the simple national-income model of...

## Dynamics And Integration

In a static model, generally speaking, the problem is to find the values of the endogenous variables that satisfy some specified equilibrium condition(s). Applied to the context of optimization models, the task becomes one of finding the values of the choice variables that maximize (or minimize) a specific objective function with the first-order condition serving as the equilibrium condition. In a dynamic model, by contrast, the problem involves instead the delineation of the time path of some...

## Quasiconcavity And Quasiconvexity

In Sec. 11.5 it was shown that, for a problem of free extremum, a knowledge of the concavity or convexity of the objective function obviates the need to check the second-order condition. In the context of constrained optimization, it is again possible to dispense with the second-order condition if the surface or hyper-surface has the appropriate type of configuration. But this time the desired configuration is quasiconcavity (rather than concavity) for a maximum, and quasiconvexity (rather than...

## Nonlinear Differential Equations Of The First Order And First Degree

In a linear differential equation, we restrict to the first degree not only the derivative dy dt, but also the dependent variable y, and we do not allow the product y(dy dt) to appear. When y appears in a power higher than one, the equation becomes nonlinear even if it only contains the derivative dy dt in the first degree. In general, an equation in the form where there is no restriction on the powers of y and t, constitutes a first-order first-degree nonlinear differential equation. Certain...

## Domar Growth Model

In the population-growth problem of 13.1 and 13.2 and the capital-formation problem of 13.10 , the common objective is to delineate a time path on the basis of some given pattern of change of a variable. In the classic growth model of Professor Domar, on the other hand, the idea is to stipulate the type of time path required to prevail if a certain equilibrium condition of the economy is to be satisfied. The basic premises of the Domar model are as follows 1. Any change in the rate of...

## Conditions For Nonsingularity Of A Matrix

A given coefficient matrix A can have an inverse i.e., can be nonsingular only if it is square. As was pointed out earlier, however, the squareness condition is necessary but not sufficient for the existence of the inverse A1. A matrix can be square, but singular without an inverse nonetheless. Necessary versus Sufficient Conditions The concepts of necessary condition and sufficient condition are used frequently in economics. It is important that we understand their precise meanings before...

## Solow Growth Model

The growth model of Professor Solow is purported to show, among other things, that the razor's-edge growth path of the Domar model is primarily a result of the Robert M. Solow, A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics, February, 1956, pp. 65-94. particular production-function assumption adopted therein and that, under alternative circumstances, the need for delicate balancing may not arise. In the Domar model, output is explicitly stated as a function of...

## The Supply Function Of A Certain Commodity Is Q A Bp2 R1

3 The supply function of a certain commodity is Q a bP2 B 2 a lt 0, gt gt 0 R rainfall Find the price elasticity of supply eqP, and the rainfall elasticity of supply EqR. 4 How do the two partial elasticities in the last problem vary with P and R1 In a monotonic fashion assuming positive P and R 5 The foreign demand for our exports X depends on the foreign income Yf and our price level P X 1 2 P 2. Find the partial elasticity of foreign demand for our exports with respect to our price level.

## Partial Derivatives

Where the variables x, i 1,2, , n are all independent of one another, so that each can vary by itself without affecting the others. If the variable x, undergoes a change Ax, while x2, , x all remain fixed, there will be a corresponding change inj gt , namely, Ay. The difference quotient in this case can be expressed as If we take the limit of A y Ax, as Ax, - gt 0, that limit will constitute a derivative. We call it the partial derivative of y with respect to xx, to indicate that all the other...

## Logarithmic Functions

When a variable is expressed as a function of the logarithm of another variable, the function is referred to as a logarithmic function. We have already seen two versions of this type of function in 10.12 and 10.13 , namely, t logft v and t logt, v In gt which differ from each other only in regard to the base of the logarithm. Log Functions and Exponential Functions As we stated earlier, log functions are inverse functions of certain exponential functions. An examination of the above two log...

## Secondorder Conditions

The introduction of a Lagrange multiplier as an additional variable makes it possible to apply to the constrained-extremum problem the same first-order condition used in the free-extremum problem. It is tempting to go a step further and borrow the second-order necessary and sufficient conditions as well. This, however, should not be done. For even though Z is indeed a standard type of extremum with respect to the choice variables, it is not so with respect to the Lagrange multiplier....

## R Pq

The AR curve can also be regarded as a curve relating price P to output Q P f Q . Viewed in this light, the AR curve is simply the inverse of the demand curve for the product of the firm, i.e., the demand curve plotted after the P and Q axes are reversed. Under pure competition, the AR curve is a horizontal straight line, so that ' gt 0 and, from 7.7' , MR - AR 0 for all possible values of Q. Thus the MR curve and the AR curve must coincide. Under imperfect competition, on the other hand, the...

## The Difference Quotient

Since the notion of change figures prominently in the present context, a special symbol is needed to represent it. When the variable x changes from the value x0 to a new value xt, the change is measured by the difference x, x0. Hence, using the symbol A the Greek capital delta, for difference to denote the change, we write Ax x, - x0. Also needed is a way of denoting the value of the function x at various values of x. The standard practice is to use the notation x to represent the value of f x...

## Market Model

First let us consider again the simple one-commodity market model of 3.1 . That model can be written in the form of two equations Q a - bP a, b gt 0 demand These solutions will be referred to as being in the reduced form the two endogenous variables have been reduced to explicit expressions of the four mutually independent parameters a, b, c, and d. To find how an infinitesimal change in one of the parameters will affect the value of P, one has only to differentiate 7.14 partially with respect...

## Finding The Inverse Matrix

If the matrix A in the linear-equation system Ax d is nonsingular, then A ' exists, and the solution of the system will be x A 'd. We have learned to test the nonsingularity of A by the criterion A 0. The next question is How can we find the inverse A 1 if A does pass that test Expansion of a Determinant by Alien Cofactors Before answering this query, let us discuss another important property of determinants. Property VI The expansion of a determinant by alien cofactors the cofactors of a wrong...

## Comparativestatic Analysis Of Generalfunction Models

The study of partial derivatives has enabled us, in the preceding chapter, to handle the simpler type of comparative-static problems, in which the equilibrium solution of the model can be explicitly stated in the reduced form. In that case, partial differentiation of the solution will directly yield the desired comparative-static information. You will recall that the definition of the partial derivative requires the absence of any functional relationship among the independent variables say, x,...

## Digression On Inequalities And Absolute Values

We have encountered inequality signs many times before. In the discussion of the last section, we also applied mathematical operations to inequalities. In transforming 6.7' into 6.7 , for example, we subtracted 1 from each side of the inequality. What rules of operations apply to inequalities as opposed to equations To begin with, let us state an important property of inequalities inequalities are transitive. This means that, if a gt b and if b gt c, then a gt c. Since equalities equations are...

## Economic Models

As mentioned before, any economic theory is necessarily an abstraction from the real world. For one thing, the immense complexity of the real economy makes it impossible for us to understand all the interrelationships at once nor, for that matter, are all these interrelationships of equal importance for the understanding of the particular economic phenomenon under study. The sensible procedure is, therefore, to pick out what appeal to our reason to be the primary factors and relationships...