## 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 A 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...

## Inflation And Unemployment In Discrete Time

The interaction of inflation and unemployment, discussed earlier in the continu-ous-time framework, can also be couched in discrete time. Using essentially the same economic assumptions, we shall illustrate in this section how that model can be reformulated as a difference-equation model. The earlier continuous-time formulation (Sec. 15.5) consisted of three differential equations (15.33) p a - T - fiU + hir expectations-augmented (15.34) j(p - 77) adaptive expectations (15.35) -k(m-p) monetary...

## Limitations Of Comparative Statics

Comparative statics is a useful area of study, because in economics we are often interested in finding out how a disequilibrating change in a parameter will affect the equilibrium state of a model. It is important to realize, however, that by its very nature comparative statics ignores the process of adjustment from the old equilibrium to the new and also neglects the time element involved in that adjustment process. As a consequence, it must of necessity also disregard the possibility that,...

## JJT JJT0 T or fl

Example I Let H Q2, where Q AaabSince Q(a, b) is homogeneous and h' Q) 1Q is positive for positive output, H a, b) is homothetic for Q > 0. We shall verify that it satisfies (12.60). First, by substitution, we have H Q2 (Aaabfi)2 A2a2ab2(3 Thus the slope of the isoquants of H is expressed by n 6n A22aa2a-'b _ _ ab 1 ' This result satisfies (12.60) and implies linear expansion paths. A comparison of (12.61) with (12.56) also shows that the function H satisfies (12.59). In this example, Q(a, b)...

## Market Model With Price Expectations

In the earlier formulation of the dynamic market model, both Qd and Qs are taken to be functions of the current price P alone. But sometimes buyers and sellers may base their market behavior not only on the current price but also on the price trend prevailing at the time. For the price trend is likely to lead them to certain expectations regarding the price level in the future, and these expectations can, in turn, influence their demand and supply decisions. In the continuous-time context, the...

## 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...

## Algebraic Definition

The geometric characterization above can be translated into an algebraic definition for easier generalization to higher-dimensional cases A function is . iff, for any pair of distinct points u and v in the (convex) domain of , and for 0 < 6 < 1, To adapt this definition to strict quasiconcavity and quasiconvexity, the two weak You may find it instructive to compare (12.20) with (11.20). From this definition, the following three theorems readily follow. These will be stated in terms of a...

## Yt yP yc

But since there will be a total of n arbitrary constants in this solution, no less than n initial conditions will be required to definitize it. Example 3 Find the general solution of the third-order difference equation 7 1 1 By trying the solution k, the particular integral is easily found to bey 32. As for the complementary function, since the cubic characteristic equation

## Iko

(10.9) Rate of growth of V y r Several observations should be made about this rate of growth. But, first, let us clarify a fundamental point regarding the concept of time, namely, the distinction between a point of time and a period of time. The variable V (denoting a sum of money, or the size of population, etc.) is a stock concept, which is concerned with the question How much of it exists at a given moment As such, V is related to the point concept of time at each point of time, V takes a...

## Secondorder Linear Difference Equations With Constant Coefficients And Constant Term

A simple variety of second-order difference equations takes the form (17.1)___r 2 + a yt+l + c You will recognize this equation to be linear, nonhomogeneous, and with constant coefficients a2) and constant term c. As before, the solution of (17.1) may be expected to have two components a particular integralj representing the intertemporal equilibrium level of and a complementary function yc specifying, for every time period, the deviation from the equilibrium. The particular integral, defined...

## Optimization A Special Variety Of Equilibrium Analysis

When we first introduced the term equilibrium in Chap. 3, we made a broad distinction between goal and nongoal equilibrium. In the latter type, exemplified by our study of market and national-income models, the interplay of certain opposing forces in the model e.g., the forces of demand and supply in the market models and the forces of leakages and injections in the income models dictates an equilibrium state, if any, in which these opposing forces are just balanced against each other, thus...

## Rules Of Differentiation For A Function Of One Variable

First, let us discuss three rules that apply, respectively, to the following types of function of a single independent variable y k (constant function), y x, and y cxn (power functions). All these have smooth, continuous graphs and are therefore differentiable everywhere. The derivative of a constant function f(x) k is identically zero, i.e., is zero for all values of x. Symbolically, this may be expressed variously as In fact, we may also write these in the form where the derivative symbol has...

## General Market Equilibrium

The last two sections dealt with models of an isolated market, wherein the Qd and Qs of a commodity are functions of the price of that commodity alone. In the actual world, though, no commodity ever enjoys (or suffers) such a hermitic existence for every commodity, there would normally exist many substitutes and complementary goods. Thus a more realistic depiction of the demand function of a commodity should take into account the effect not only of the price of the commodity itself but also of...

## Further Applications Of Exponential And Logarithmic Derivatives

Aside from their use in optimization problems, the derivative formulas of Sec. 10.5 have further useful economic applications. When a variabley is a function of time, y f(t), its instantaneous rate of growth is defined as* (10 23) r dy dt marginal functlon * ' ' r-v y f(t) total function But, from (10.20), we see that this ratio is precisely the derivative of In f(t) In y. Thus, to find the instantaneous rate of growth of a function of time f(t), we can instead of differentiating it with...

## From a Marginal Function to a Total Function

Given a total function (e.g., a total-cost function), the process of differentiation can yield the marginal function (e.g., the marginal-cost function). Because the process of integration is the opposite of differentiation, it should enable us, conversely, to infer the total function from a given marginal function. Example 1 If the marginal cost (MC) of a firm is the following function of output, C'(Q) 2e -2Q, and if the fixed cost is CF 90, find the total-cost function C(Q). By integrating...

## The Meaning Of Equilibrium

Like any economic term, equilibrium can be defined in various ways. According to one definition, an equilibrium is a constellation of selected interrelated variables so adjusted to one another that no inherent tendency to change prevails in the model which they constitute.* Several words in this definition deserve special attention. First, the word selected underscores the fact that there do exist variables which, by the analyst's choice, have not been included in the model. Hence the...

## Market Model With Inventory

In the preceding model, price is assumed to be set in such a way as to clear the current output of every time period. The implication of that assumption is either that the commodity is a perishable which cannot be stocked or that, though it i> stockable, no inventory is ever kept. Now we shall construct a model in which sellers do keep an inventory of the commodity. 1. Both the quantity demanded, Qdr and the quantity currently produced, Qsr are unlagged linear functions of price Pt. 2. The...

## Comparative Statics Of Generalfunction Models

When we first considered the problem of comparative-static analysis in Chap. 7, we dealt with the case where the equilibrium values of the endogenous variables of the model are expressible explicitly in terms of the exogenous variables and parameters. There, the technique of simple partial differentiation was all we needed. When a model contains functions expressed in the general form, however, that technique becomes inapplicable because of the unavailability of explicit solutions. Instead, a...

## The Nature Of Mathematical Economics

Mathematical economics is not a distinct branch of economics in the sense that public finance or international trade is. Rather, it is an approach to economic analysis, in which the economist makes use of mathematical symbols in the statement of the problem and also draws upon known mathematical theorems to aid in reasoning. As far as the specific subject matter of analysis goes, it can be micro- or macroeconomic theory, public finance, urban economics, or what not. Using the term mathematical...

## Levels Of Generality

In discussing the various types of function, we have without explicit notice introduced examples of functions that pertain to varying levels of generality. In certain instances, we have written functions in the form V 7 y 6x 4 y x2 - 3x 1 etc. Not only are these expressed in terms of numerical coefficients, but they also indicate specifically whether each function is constant, linear, or quadratic. In terms of graphs, each such function will give rise to a well-defined unique curve. In view of...

## Exact Differential Equations

Given a function of two variables F y , its total differential is When this differential is set equal to zero, the resulting equation is known as an exact differential equation, because its left side is exactly the differential of the function F y, t . For instance, given 14.16 2ytdy y2dt Q or f 0 is exact. is exact if and only if there exists a function F y t such that M dF dy and N dF dt. By Young's theorem, which states that d2F 8t dy d2F 8y dt, however, we can also state that 14.17 is exact...

## Absolute versus Relative Extrema

A more comprehensive picture of the relationship between quasiconcavity and second-order conditions is presented in Fig. 12.6. A suitable modification will adapt the figure for quasiconvexity. Constructed in the same spirit and to be read in the same manner as Fig. 11.5, this figure relates quasiconcavity to absolute as well as relative constrained maxima. The three ovals in the upper part summarize the first- and second-order conditions for a relative constrained maximum. And the rectangles in...

## Mathematical Economics-equilibrium On A Market With A Price Ceiling Using Phase Lines

Tion models, there exists an easy method of analysis that is applicable under fairly general conditions. This method, graphic in nature, closely resembles that of the qualitative analysis of first-order differential equations presented in Sec. 14.6. Nonlinear difference equations in which only the variables yt x and y appear, such as gt 'r , 1 gt r3 5 0r Sin gt In V- 3 can be categorically represented by the equation 16.16 gt ,-1 v, where can be a function of any degree of complexity, as long...

## Leontief Inputoutput Models

In its static version, Professor Leontiefs input-output analysis deals with this particular question What level of output should each of the n industries in an economy produce, in order that it will just be sufficient to satisfy the total demand for that product The rationale for the term input-output analysis is quite plain to see. The output of any industry say, the steel industry is needed as an input in many other industries, or even for that industry itself therefore the correct i.e.,...

## Mathematical Versus Nonmathematical Economics

Since mathematical economics is merely an approach to economic analysis, it should not and does not differ from the non mathematical approach to economic analysis in any fundamental way. The purpose of any theoretical analysis, regardless of the approach, is always to derive a set of conclusions or theorems from a given set of assumptions or postulates via a process of reasoning. The major difference between mathematical economics'1 and literary economics lies principally in the fact that, in...

## The Concept Of Limit

The derivative dy dx has been defined as the limit of the difference quotient Aj gt Ajc as Ajc - gt 0. If we adopt the shorthand symbols q A Ax q for quotient and v Ajc t for variation , we have lim lim q ax a - gt o ax u o In view of the fact that the derivative concept relies heavily on the notion of limit, it is imperative that we get a clear idea about that notion. Left-Side Limit and Right-Side Limit The concept of limit is concerned with the question What value does one variable say, q...

## 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 xQ lt 0. b. A...

## 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...

## Ji J J2

Example 3 Let the production function be Q Q K,L,t where, aside from the two inputs K and L, there is a third argument t, denoting time. The presence of the t argument indicates that the production function can shift over time in reflection of technological changes. Thus this is a dynamic rather than a static production function. Since capital and labor, too, can change over time, we may write Then the rate of change of output with respect to time can be expressed, in line with the...

## Second Order Total Differential

It has been mentioned that, inasmuch as the constraint g x, y c means dg gx dx gYdy 0, as in 12.10 . dx and dy no longer are both arbitrary. We may, of course, still take say dx as an arbitrary change, but then dy must be regarded as dependent on dx, always to be chosen so as to satisfy 12.10 , i.e to satisfy dy gx gY dx. Viewed differently, once the value of dx is specified, dy will depend on gx and gv., but since the latter derivatives in turn depend on the variables x and y, dy will also...

## Exercise

Find P and Q by a elimination of variables and b using formulas 3.4 and 3.5 . Use fractions rather than decimals. 2 Let the demand and supply functions be as follows a Qd 51 -3P b Qj 30- 2P find P and Q by elimination of variables. Use fractions rather than decimals. 3 According to 3.5 , for Q to be positive, it is necessary that the expression ad be have the same algebraic sign as b d . Verify that this condition is indeed satisfied in the models of the preceding two problems. 4 If f 0in the...