## Info

(b) The individual should rank alternative (ii) as better. Even if the individual wants the money well in advance of 5 years, she should rank alternative (ii) as the better one because at r 0.08 (8 ) she could borrow more money now and pay it back after 5 years with the 150 received at that point in time than she could from receiving the 100 in one year's time. EV lim,v *> EL 2*' linw 2 +oc I. (a) fix,,) 0 - 5 n, 1,2,3_____This suggests (b) fix,,) -2 -r 3 ii. n 1,2.3 This suggests (c) (* ) 2m...

## Intermediate Value Theorem

In this section we present a straightforward theorem, the intermediate-value theorem and show that it can be very powerful in the study of equilibrium, which is one of the most important concepts in economics. The particular application we make is a very simple one. The rtnge of applications, however, is in fact very broad. Suppose that the function )' -- f(x) is continuous on the interval fa, J. b > a. It follows that the function must take on every value between f(a) and ( > ). which are...

## Introduction

Almost for as long as economics has existed as a subject of study, mathematics has played a part in both the exploration and the exposition of economic ideas.' It is not simply that many economic concepts are quantifiable (examples include prices, quantities of goods, volume of money) but also that mathematics enables us to explore relationships among these quantities. These relationships are explored in the context of economic models, and how such models are developed is one of the key themes...

## Introduction to Comparative Statics

The Simple Keynesian Model of Income Determination Let ) denote the value of the aggregate supply of goods and services in the economy. Since this accrues as sales revenue to firms who then pay it out as incomes to suppliers of inputs, including labor, we also refer to Y as national income. The aggregate demand for goods and services has two components consumption demand C and investment demand I. We take as exogenous, but C is determined by Ihe consumption Junction where the constant c is the...

## Ioi

Where the determinant ) is given by Now the condition for a maximum to this problem is that the determinant of the Hessian is positive. This is exactly the determinant D . so D > 0 by the second-order condition. Next, consider the sign of the partial with respect to income, say dxf dm. The theory places no restrictions on the signs of iii2 and ii 2. Thus we have no way of saying whether the numerator is positive or negative. Likewise we have no way of saying whether the numerator in the...

## J

Derivative ', in the neighborhood of a stationary value. The curvature of the function f determines the slope of its derivative '. In figure 6.16(a). the function is strictly concave in the neighborhood of a . This implies that in figure 6.16(d). at x-values below a*, we have . (a) > 0. while at x-values above x*. f x) < 0. The slope of ' is the second derivative (a), and so we have J'(x') < 0. Since, in this case, x* yields a maximum of the function, we have Theorem 6.2 If '(x* i 0, and...

## J 10xeZxdx Vojc I 0dx

Ft is also useful to know how to differentiate with respect to some parameter that affects either the limits of an integral or the integrand. The economic applications of these techniques are primarily in the field of dynamic analysis (see chapter 25), and it is traditional to use the variable t. which represents time, as the variable of integration. We will usex as the parameter which may affect either the integrand or the limits of integration. First, consider the case in which only the upper...

## Ahm

Using jc2 kx in j and fz shows that marginal products are independent of a only if or + 0 I. 7. (a) Since .v,' ,rl is a homogeneous function and f is a monotonie transformation of it, then is homotheiic. (bi Since .vf.r is a homogeneous function and is a monotonie transformation of it, then f is homotheiic. is not strictly concave because f fu . ', . However, W* M, fa are both < 0 and I 'I fufn ,2 < Oand so is (weakly) concave, (c) is quasiconcave, since l l -fifn + 2 I i Ij -...

## Oiq

Figure 24.5 Phase diagram tor example 24.14 Figure 24.5 Phase diagram tor example 24.14 for which the solutions are -1 I 2 and r2 1 2. Since the roots are of opposite sign, the steady-state solution is a saddle-point equilibrium. Step I Determine the motion of yBegin by graphing the yi isocline setting y, 0 to find the isocline gives the horizontal line y 2. Next, we note that y i < 0 below this isocline (when v2 < 2) and v, > 0 above the isocline (when > '2 > 2). The appropriate...

## Qtd

And after canceling the dt terms, this becomes Since the integral of I jy is just In y 4- ci. where Ci is a constant of integration, we now have integrated both sides, giving To obtain an explicit solution for y. take the antilogarithm of both sides. This gives where C eC _C is still an arbitrary constant of integration. This gives the solution to the homogeneous form. To avoid confusion later on, we will use the notation 'h to refer to the solution to the homogeneous form (the It subscript...

## Wge

Use the information provided in order to determine the constant of integration. Fix) I(5x3 4- 2.v + 6)dx, FiO) i) (d) F(x) I Ixdx, F< 3) I0 4. Evaluate the following integrals. Use the information provided in order to determine the constant of integration. (c) F(x) (Zr 4- 4x ) a . F(0) 0 5. A firm uses one input, labor (L), to produce output ( ), The marginal-product function for the input is MP(Z ) I OA, 2. Find the production function, Qi L). Assume that Q...

## J2

A 2b2 a22b 11 22 2 21 ai b - u b2 11 22 12 21 Example 24.4 Find the complete solution to the following system of differential equations First, put the system into its homogeneous form This is the homogeneous system solved in example 24.3. The solutions obtained were yj' ( ) C,t'3, 2 + C2en V (r) - Icze Next, find the steady-state solutions to use as the particular solutions. Set y, 0 and y2 0 in the complete equations. This gives Now add the homogeneous and p irticular solutions together to gel...

## Jl

Example 12.5 Solve the competitive firm's profit-maximizing use of labor and capital for the case where v L p UK), w 10. and r 20. Show that the solution is a true maximum. rc(L. K) IOOLluA'06 - 10L - 20K The first-order conditions are 20 .-UilA'a6 -10-0 60L02K < iA 20 0 Before solving these, we check for a maximum. We have Wi -16-py < 0 for any (K. L) e E .

## JlU Unlli u

E(u u ) E U U2) ( ) where is the identity matrix of order n. The assumption above about the errors simply states that they are pairwise uncorrelated, since E(u,u ) 0, for all i j- They also have the same variances. If we further assume that the joint distribution of these n errors is normal, then they will be independent, sincc for the case of normality, lack of correlation implies independence, and vice versa. Then we say that the u,s are independently and identically distributed, or i.i.d and...

## JU tj F ritvij a 0 f2x Ai a XtA x2 a

If we assume that the functions and g possess continuous first and second derivatives, and that the determinant then we can apply the implicit function theorem. This amounts to saying we can solve for the endogenous variables as differentiable functions of the exogenous variable in the neighborhood of the optimal point, so the value of the function in the same neighborhood is and V is known as the value function for the maximization problem. The value function expresses directly the idea that...

## K

The reason for calling this function linear is obviously that its graph is a straight line. The steepness of the line is determined by the absolute value of a Taking two -values, we can write where Ay is read ihe change in y, and likewise for Ax. The ratio Ay Ax is called the slope of the line and so a is the slope coefficient. Note that (he line y ax is fully determined once a is chosen. In equation (2.7) y is often referred to as the dependent variable and x as the independent variable. This...

## K 61 F0064

Since the present value of building costs (5467,301 195) exceeds that of nel operating revenues ( 330,039,026), the utility should not proceed with the project. The following simple example explains clearly the economic rationale for discounting revenues and costs using net present-value formulas in order to decide on the economic viability of an investment. As the example illustrates, the present-value approach is an appropriate framework regardless of whether the funds needed 10 finance the...

## Bew

The determinant of the coefficient matrix of the linearized system is It is negative because < 5 > 0. 0 < a < J, and R 0. Therefore, the steady state is a saddle point. 7. (a) v, 0,(2)' - 2 x, C,(2)' - 1 (b) y, C,< -4)' +C (-1V +0.8 x, C ( 4)' + - ( 1)' + 0.7 9. Let Pi be the probability of high sales, and let q, be (he probability of low sales yj( ) 2v 3C,e' - 2v 3C er - 16 where n, rz 1 2 + V5 2 The steady stale is a saddle point. v2< n 2 (C, + CzDe1 + -Cit1 + 44 The steady state...

## K k

The cost function c in be factored into output multiplied by a unit-cost function w. r ) that depends only on input prices. A firm pays SI0 per unit for input x and S8 per unit for input xi. It has the CES production function y (0.4.v, 2 + 0.6x 2)'5 What is its cost-minimizing input combination to produce one unit of output'.' Solution C I Ox i +8x2 + - (Q.4xf2 + 0.6xj2)05 10 - 0.5 0.4.v,2 + 0.6A, l 30.8X J 0 8 - A.0.5 0.4xf2 + 0.6xJ2 '51.2xfJ 0 1 - 0.4xr2+0.6x2-2 - '5 0 Taking the ratio of the...

## Lcb

The determinant of the coefficient matrix of the homogeneous system is ( -< 5 a), which is negative. We therefore know immediately that the steady-state equilibrium is a saddle point. Bv theorem 24.2, the solutions to the system of differential equations in (25.7) and (25.8) aie where r and r2 are the eigenvalues or roots of the coefficient matrix in equation (25.9), Ci and C2 are arbitrary constants of integration, and k and K are the steady-state values of the system, and serve as...

## Km

The steady-state solutions are y ( 1 und yi - 2. The complete solutions then are Construct the phase diagram for this system. First, determine the motion for _V . Setting v, 0 gives the y isocline as the horizontal line yj - 2 in figure 24.7. If v > 2, then y, < 0. so above the isocline, yi is decreasing below the isocline, yi is increasing. Second, determine the motion for y2. The y2 isocline occurs along the line v V + I. This is a straight line in figure...

## L

Figure 5.39 Funciion f(x) x1, x R. and its lirsi two derivatives A linear function is convex according to definition 5.7. In many instances, however, we will want to consider linear functions separately from functions with positive or nonzero second derivatives. Thus we often use the concept of strict convexity to exclude linear functions. This is done by replacing the weak inequality (> ) in definition 5.7 with the strict inequality (> ). A twice differentiate function fix) is strictly...

## L5rw

1 ' 2f C2v T72cos(jt 4) - CrN T72sin T 4) - 2C When the solution is required to satisfy given initial conditions, the constants C and Co take on specific values. Example 24.23 Find the values of C and C2 that make the solution to the difference equation system in example 24.20 satisfy v0 2 and xq 1. Setting - 0 in the solutions given in example 24.20 and setting yo 2 and .Vu I gives Solving these two equations now tor Ct and C2 gives, alter some simplification, C, -8 and C2 8. In chapters 18 to...

## Lim V 1 1246858 r ri I fr

Which is 78 of the entire present value, compared to the error of 2.1 if the interest rate were 8 (from the calculation above). Example 3.19 Suppose that a stream of equal payments of amount I 0.000 per yea is to continue in perpetuity. At the interest rate of 6 compute (i) (he present value of this entire stream of benelils (ii) (he present value of the benefits beginning at the end of (he 50th year (iii) the present value of the first 50 years of benefits

## Limit Of A Sequence

A sequence is said lo have the limit L if, for any e > 0. however small, there is some value N such that L < e whenever n > N. Such a sequence is said to be convergent, and we write its limit as lini,,-. a L. In less formal language, the definition above states that a sequence has a limit L provided that all values of the sequence beyond some term can be made as close to L as one wishes (i.e., the condition a - L < t can be met for as small a positive number i as one likes by choosing a...

## Linear First Order Differential Equations

In the next three chapters we explain techniques for solving and analyzing ordinary differential equations. We do not attempt to provide exhaustive coverage of the topic but instead focus on the types of differential equations and techniques of analysis that are most common in economics. We begin in this chapter with linear, first-order differential equations. In the next chapter we turn to an examination of nonlinear, first-order differential equations, and in the chapter after that we examine...

## Linear Second Order Differential Equations

Until now we have confined our analysis of differential equations 10 those of the first order. In this chapter we will examine linear, second-order differential equations with constant coefficients. We focus our attention on the autonomous case in section 23.1 and consider a special nonautonamous case in section 23.2. We begin by explaining how to solve a linear, autonomous, second-order differential equation. The linear, autonomous, second-order differential equation (constant coefficients and...

## Linear Systems in nVariables

Although graphing solutions and ifnding solutions by simple substitution is tine for systems of equations with only two variables, we need to develop other procedures to find solutions for general systems of linear equations. These procedures often involve generalizations of some of the alternative solution methods that we have already referred to subtracting equations and multiplying equations by a constant. As we develop these methods, we will illustrate with 3-variable systems first and...

## Lk Kl

Since v Y L and using y f(k), this becomes This nonlinear differential equation for the capital-labor ratio describes the growth of the economy in this model. We want to analyze this differential equation to see what the assumptions of the Solow model imply about the properties of the growth path of the economy. Since the labor force is assumed to grow exogenously. does this mean that the capital-labor ratio, A. and output per person, y, will fall over time, rise over time, or reach a steady...

## Lm

We assume the latter is smaller than the former. The parameter p tells us how sensitive BP is to the interest rate differential. The larger is p the bigger are the induced capital movements when the interest rate differential changes. As p becomes very large, the slope of the BP curve becomes zero, which is essentially the situation we had in example 7.9. How the equilibrium is determined now depends on the assumption to be made about the exchange rate. We will assume that the exchange rate is...

## M

The reason for calling this a gradient vector is that each element. ,, indicates the rate of change in the function value with respect to the variable x,. This is analogous to signs indicating the grade or steepness that we find when driving a vehicle through hills or mountains. V 5 - 2.Vi 4- 3.vi in example 11 .S V 5 - 2.Vi 4- 3.vi in example 11 .S Find the gradient vector for the function Ui, .v ) 5 2.V + 3*2. Solution The first derivatives of the function are The meaning of this gradient...

## M injx m2x2 0 0pM wJ

- r X - r2x > 0, p* > 0. 0 The Lagrange multipliers are ususally referred to as dual variables in linear programming. The key point is their interpretation as the shadow prices of the input constraints. At the optimal solution, the value of A.*, p*, or p' gives the increase in revenue the firm would earn if it acquired a little bit more of the respective inputs and allocated that optimally between the outputs. The last three conditions also tell us that if a shadow price is positive, all of...

## Mathematics for Economics

Michael Hoy John Livernois Chris McKenna Ray Rees Thanasis Stengos The MIT Press Cambridge, Massachusetts London, England 2001 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. This book was set in Helvetica and Times Roman by Interactive Composition Corporation. Printed and...

## Matrix Arrays

We will be discussing matrix manipulations in greater detail in the next two chapters. However, it is useful to introduce the idea of a matrix here, because matrices are a very convenient way of summarizing certain types of information. For our immediate purposes, a matrix is just another way of writing out the information contained in a system of linear equations such as dial in example 7.10. Notice that as with the other examples, the key to finding the solutions involved identifying ways of...

## Matrix Transposition

A very useful operation in matrix algebra is that of transposition. The transpose of a matrix A, is the matrix in which the rows of the original matrix A become columns and the columns of A become rows. The transpose is denoted by Ar. The transpose matrix, AT, is the original matrix A with its rows and columns interchanged. This implies that A' will have its dimensions reversed when compared with A. Example 8.14 Find the transpose of the 2x3 matrix 4, given by Since A is 2 x 3. its transpose...

## Maxfxix st ati 2

Consists of deriving the following first-order conditions to find the critical point(s) of the Lagrange function C xuXi, X) (A-I, .vi) + .tiUi, A I (A-, A,') + > .> ,( A'f. Ai ) 0 h(xl x ) + X*glOtJ.JT ) 0 OX 2 How can we be sure that for some arbitrarily given problem there is always a unique solution point at a tangency of the Constraint curve and contour of , and that the Lagrange procedure always works in that it delivers this point Exactly what role is played by the assumptions on the...

## Mc Ac

Figure 13.11 Total-, average-, and niarginal-eost curves with constant returns to scale Figure 13.11 Total-, average-, and niarginal-eost curves with constant returns to scale Figure 13.12 Total-, average-, and marginal-cost curves for diseconomies of scale y > > . The cosl function is said to exhibit diseconomies of scale. Figure 13.12 illustrates. (Refer to section 11.5 for a more comprehensive discussion of returns to scale.) Cost-Minimization with the CES Production Function In chapter...

## Mpc

So the consumption ratio decreases as income increases. The Proportional Consumption Function Derive the functional form of the proportional consumption function, lor which die average propensity to consume is constant. For the average propensity to consume to be constant, we require that Now substitute the definition for APCl ) Thus the consumption function with a constant APC has a zero intercept, and in this case the average propensity to consume and marginal propensity to consume are the...

## Mrh HMHKJK

The second term of every pair combined with ihe hrst term of the subsequent pair is always zero. So we are left with The sum sn is itself a sequence. Since the sequence is the sum of the first n terms of some other sequence, it is called a series. A simpler example is that n is the series associated with the constant sequence a, 1. Thus we have the following definition If u,, t 1, 2,3, is a sequence, then sn - a,, n 1,2,3,, , is called a series. Since a series is just a specific type of...

## N

Thus, if we choose N to be the next integer greater than 1 e. we will satisfy the condition. Example 3.3 Show that the sequence f (n) (-1 ),n 1.2. 3____is divergent. According to definition 3.3 a sequence is divergent if it has no limit. L. It is easy to illustrate diagrammatically that there is no value L such that all the terms in the sequence (n) i 1), n > A', lie within distance e of L for every e > 0. no matter how large we choose N to be. The reason is that no matter how large we...

## N 123

It is quite easy to see that a sequence that grows without hound cannot have a limit and so doesn't satisfy definition 3.2 for convergence. The sequence a ( I) (figure 3.4) illustrates another type of sequence that is not convergent. The terms do not grow without bound, yet there is no limit because all terms of the sequence beyond the Nih term must satisfy definition 3.2 if the sequence is to have a limit. It is clear that whatever choice of L is made, terms ofa for n greatei than any N will...

## N a yWi

Combined with y' ' + Y I this gives implicitly die optimal outputs. 7. Hint L'se the first-order conditions to solve for Kir, u y), A'(r, w, v) and .(r. mj. y ) Insert these into die Lagrangean and differentiate with respect to y. Consider that the optimized Lagrangcan has the same value as the cost function for all v.

## Nonautonomous Equations

If the coefficient, a, or the term. h. in a linear differential equation are functions of time, the equation is nonautonomous. In that case, the solution technique used in the previous section will not work in general. In this section, we explain a general solution technique that works for any linear, first-order differential equation. The general form of the linear, first-order differential equation is where a(t) and ( ) are known, continuous functions of t. Notice that an equation of the form...

## Nonlinear First Order Differential Equations

In chapter 21 we saw that we could apply a single solution technique to solve any first-order differential equation that is linear. When the differential equation is nonlinear, however, no single solution technique will work in all cases. In fact only a few special classes of nonlinear, first-order differential equations can be solved at all. We will examine two of the more common classes in section 22.2. Even though solutions are known to exist for any nonlinear differential equation of the...

## Numbers

The most basic and familiar kinds of numbers are natural numbers, the elements of the set They arise naturally in counting objects of all kinds. What does it mean to count a set of objects, say a pile of dollar bills' When we count dollar bills, we lake each element in the set of dollar bills and pair it wilh an element of Z+, starting with 1 and moving successively through the sel. When we have exhausted the elements of the first set (of dollar bills), the element of Z, that we have reached in...

## O

Therefore the sequence is monotonically decreasing if r > 0, while it is monoionically increasing if 1 < < ( In the case with > 0, a, V ( I 4- r)' is bounded below by 0 and above by V, and so the sequence is convergent. In the case with I < r < 0, we have 0 < I + r < I and lim, oo V ( 14- r)' -t-oo that is, the sequence is not bounded and so i I is not convergent. 3. Steps of the proof are (i) For any (large) K. there exists an N , such that > > K for i > N . (ii) For any...

## Obtaining the Determinant and Inverse of a 3 x 3 Matrix

In section 9.1 we obtained the determinant of a 2 x 2 matrix. Below we will derive the determinant of a 3 x 3 matrix and then obtain the determinant of a general square matrix of dimension n x n. The determinant of a 3x3 matrix A will be composed of all the elements of A. However, as we will see, the expression of the determinant of A will be reduced to particular expressions involving the determinants of certain 2x2 submatrices 9.2 OBTAINING THE DETERMINANT AND INVERSE OF A 3 , 3 MATRIX 371 of...

## Obtaining the Inverse of a 3 x 3 Matrix

Having obtained the determinant of a 3 x 3 matrix, it is quite straightforward to obtain its inverse matrix. Below we will present the steps that one follows to obtain the inverse of a matrix of order 3 and illustrate the method bv means of an example. Step 1 Corresponding to the elements ( , of A, we obtain the cofactors C1(, ' 1.2, 3 j 1,2, 3. Then we form a matrix in which each element a,, is replaced by the corresponding cofactors C,,, given below as 9 2 OBTAINING THE DETERMINANT AND...

## Om

For example, we have just seen that cos(() ) I. cos(90 ) 0. cos( 180 ) -1. cos( 270) 0, and cos< 360 ) 1. A similar process generates the sine function. Extend a perpendicular from P to the vertical axis at N. Again, as P moves around the circle in the counterclockwise direction. ON lakes on values starting with zero at ,4. rising to one at B. falling to zero at A'. falling further to negative one at B' and rising back to zero at A. Formally we have

## On

We can see from (he unit circle that some particular values of the sine function are sin(0' ) 0. sin(90' ) I.sin(l80 ) 0,sin(270') -1. and sin(360 ) 0. Although we are all accustomed to measuring angles in degrees, it is actually easier (and customary) in theoretical work with angles to measure them in terms of the distance of the arc A P taken counterclockwise. The units of distance are called radians. We can calculate the relationship between I radian and I easily because we know the distance...

## Optimization of Functions of nVariables

The idea of optimization is fundamental in economies, and the mathematical methods of optimization underlie most economic models. For example, the theory of demand is based on the model of a consumer who chooses the best ( most preferred) bundle of goods from the set of affordable bundles. The theory of supply is based on the model of a lirm choosing inputs in such a way as to minimize the cost of producing any given level of output, and then choosing output to maximize profit. Rationality and...

## Optimization of Functions of One Variable

Many economic models are based on the idea that an individual decision maker makes an optimal choice from some given set of alternatives. To formalize this idea, we interpret optimal choice as maximizing or minimizing the value of some function. For example, a firm is assumed to minimize costs of producing each level of output and to maximize profit a consumer to maximize utility a policy maker to maximize welfare 01 the value of national output', and so on. it follows that the mathematics of...

## Optimization over an Interval

The discussion of first- and second-order conditions in sections 6 I and 6.2 has dealt exclusively with the unconstrained case, in which a solution to the problem can be anywhere on the real line. Often in economics, however, this is unacceptably general. For example, in problems in which firms choose outputs (as in several examples in the previous section ) or consumers choose goods, we cannot assume that negative quantities are possible. In other problems, it may be reasonable to place an...

## P

To expand this and determine the limit as n > oo would be a tedious exercise. However, it is clear that the limit of the numerator is a and the limit of the denominator is i. Thus one can apply result (iv) of theorem 3.1 to see that the limit of the expression is a p. The following two examples illustrate the lirst two results of theorem 3,1. Example 3.10 Use the result that lim,, 1 w 0 and theorem 3.1 (i) to find the limit of the sequence ( ) 2 n, n 1. 2, 3

## P Q

2.5 PROOF, NECESSARY AND SUFFICIENT CONDITIONS 61 There are several ways in which these statements can be read, and it is useful to spell these out P is a sufficient condition tor Q Q is a necessary condition for P > is sufficient for means that the truth of P guarantees the truth of Q. Q is always true when P is true. It follows that if Q is not true, then P cannot be true. Thus it is necessary that Q is true for P to be true. In other words, P can be true only if Q is true. We illustrate...

## P [ B paa

Which gives a system of reduced-form equations for the prices. The following 3x3 example illustrates and shows how to solve for a particular price (rather than the entire vector of prices) using Cramer's rule. Example 9.22 Consider the markets for coffee, tea, and sugar. These goods are related in demand. since the first two are often substitutes for each other while the third is often complementary with each of the other two goods. Ignoring any supply side links (which are. in any case,...

## P2

In figure 13.5 we graph the level curves of the utility function, the consumer's indifference curves. The line BB graphs the budget constraint, and so has slope pi pz and intercepts m p and m pz- Points on this line require an expenditure exactly equal to the consumer's income. The solution is a point of langency, and the expressions given above for Jt and '2 enable us lo calculate these solution values once we have numerical values for the parameters ur, and p2- The demand functions (13.4) and...

## Power Exponential and Logarithmic Functions

When a number a is multiplied by itself n times, we write a. where n is called the exponent. This leads to the rules of exponents Intuitively, we may think of n as an integer, but in fact n could be any real number. The power function takes the form (Note that the rectangular hyperbola is a special form of the power function with b I. The linear function is also a special case with b I. The quadratic may be thought of as the sum of two power functions. ) Figure 2.26 shows two power functions...

## Present Value of a Stream of Payments

Earlier we saw how the sequence P V, V ( 1 +r)' (see equation 3.3) represents the present value of an amount of money V received t periods into the future. In many economic settings we need to compute the equivalent present value of a series (i.e., the sum total) of such amounts. For example, a mortgage or other long-term loan represents a current sum of money loaned to ar, individual or institution in return for a stream of future payments. Thus, if an individual makes annual payments at the...

## Present Value Calculations

An important economic application of sequences is the determination of the present value of a sum of money to be received at some poini in the future. This computation is the inverse of determining how much money one would have in the future upon investing a certain amount now. Suppose, for example, that one had S90.91 to invest currently at an annual interest rate of 10 . Then the amount of money received at the end of one year would be 90.91 (I 4-0.1) 100. In general. investing X today at an...

## Proof Necessary and Sufficient Conditions

Why should one person ever accept as true a statement made by someone else The usual response would be, Prove it If the statement is a purely factual one, for example, prices have risen. then proof would lake die form of some factual evidence that substantiates the statement. Economics is more often concerned, however, with deductive statements such as the money supply increases, then the price level will rise. which is to say. increases in the money supply lead to inflation. A stronger...

## Properties Of Sequences

Example 3.13 Use the results that and result (iv) of theorem 3.2 to find ( + 3)( r-l) -f 3 n2 - 1 lira ---- --- hm - Iim - - -few n cc n ii oo tl n oc j Another useful application of this theorem concerns the present-value formula developed in section 3.3 Theorem 3.2 provides a proof of the claim that, if r > 0. then PV, - 0 as I -v oo, Since the denominator is a delinitely divergent sequence (if r > 0) and the numerator is a constant, then part (v) of theorem 3.2...

## Q 20 p 05p7 q2 100 2 p2

(a) Comment on the relationship between the three goods on the demand side. fb) What is the nature of any production externality on the supply side (c) Solve for the equilibrium prices and quantities of the three goods. 5. An economy has an IS curve given by r 210-2)' and an LM curve given by r m + y 4. The long-run equilibrium level of output must equal 100. What value of M makes the IS and LM curves intersect at Y - 100 What is the economic interpretation of a situation in which M exceeds...

## Qq1

The above implies that both the columns and rows of Q are orthogonal. Example 10.13 Show that the matrix A below is orthogonal. '-l v 3 0 V2 V3' 2 710 -2 s I I n 5 .2 713 3 n 5 N 2 N S. 0 -2 jt 3 5 .s lfS I n 5 v 2 yi5. 1 3 + 2 3 -2 n 30 + v 2v 5 .-2 v 45 + 2 v 45 -2 V3 + V2 5 4 10 + 4 10+ 1 5 4 V 50 - 6 VT5 + 2 V75 _ 2 v45 + 2 n 45 4 5 - 6 - 5C) + N 2 N 75 4 15+9 15 + 2 15 The orthogonal matrix of eigenvectors diagonalizes the symmetric matrix A

## QrQ QQr

An orthogonal matrix is a matrix for which its inverse equals its transpose. Theorem 10.3 For the problem in equation (10.8), where A is a symmetric matrix, the eigenvectors that correspond to distinct eigenvalues are pairwise orthogonal and if put together into a matrix, they form an orthogonal matrix. Let qi and q denote the eigenvectors corresponding to A and Then Aqi Aiqi qi Aq, A.,q qi A< 2 X2q2 > q 4q2 X2q q Since A is symmetric we have that f Ai Aj....

## Quadratic Functions

We can write a quadratic function in explicit form as As figure 2.24 shows, this is a useful function in economics because in its convex form, with a > 0. it could be used to depict a typical U-shaped average or marginal-cost curve, while in its concave form, with a < 0, it could depict a typical total-revenue or total-profit curve. (Note that in these examples the domain of the function must be restricted to R+, since negative outputs arc not allowed.) The unique minimum (< n the convex...

## Quantities Magnitudes and Relationships

We can slart by thinking of how we measure things in economics. Numbers represent quantities and ultimately i( is this circumstance that makes it possible to use mathematics as an instrument for economic modeling. When we discuss market activity, for example, we are concerned with the quantity traded and the price at which (he trade occurs. This is so whether the quantity is automobiles, bread, haircuts, shares, or treasury hills. These items possess cardinality, which means lhat we can place a...

## R

Since any vector in R' may he expressed as a linear combination of c linearly independent vectors that form a basis. Reversing the argument, we can form a matrix C of order r X n by retaining die set of r linearly independent rows and discarding the remaining m r. Matrix C will be of order r x n. The maximum number of linearly independent columns of A. namely c. will be the same as that of C It follows that since C has r-element column vectors. Using equations (10.5) and (10.6), we see that r...

## R 106006

(in as of the beginning of the 50th year the present value of 10,000 per year in perpetuity is 166,666.67, as computed in part (i). Since this is in effect received at the end of the 50th year, its current ( i.e as of now) present value must be discounted so that it becomes (iii) The present value of the first 50 years' worth of payments is simply the answer in (i) less that in (ii) 166,666.67 - 9,048.06 157,618.61 The examples above all deal with the problem of determining the present value of...

## R E V I E W

Characteristic equation complete solutions direct method eigenvalues eigenvectors general form global stability homogeneous solutions improper stable node improper unstable node isocline isoscctors local stability particular solutions phase diagram phase plane saddle path saddle point saddle-point equilibrium simultaneous system stable focus stable node steady state substitution method trajectory unstable focus unstable node 1. Explain how the characteristic equation can be derived and how it...

## R i rTi tfluv r i

The rule for multiplying matrices is essentially a generalization of the above reasoning. To multiply matrices, it is not necessary that they be of the same order. The requirement is that the number of columns of the first matrix be the same as the number of rows of the second matrix. Matrices that satisfy this requirement are said ro be conformable for matrix multiplication. Before we present the formal definition of matrix multiplication we will illustrate the idea by means of some examples....

## R1 11 v

If this were not true, then y,+ l would not depend on .v,7 and we could then solve it directly as a single, lirst-order difference equation. Using this to substitute for.v, gives +1 11 y, Vi+2 11 V H-t 4-0 2 2l> 'i -FapiJ22( - Alter simplifying and rearranging this becomes yl+2 - (fl -F 22)Jin-1 + (fl ifl22 - Oi2 21 ).V, 0 which is a homogeneous, linear, second-order difference equation with constant coefficients. Equations like this were solved in chapter 20 theorem...

## Rank of a Matrix

The concept of linear independence or dependence of vectors is closely linked to the concept of nonsingularity or singularity of matrices discussed in chapter y. We now investigate further the nonsingularity (singularity) properties of matrices by introducing the concept of the rank of a matrix. Consider any arbitrary matrix of order m x n. U consists of it column vectors with m elements each, and w row vectors with n elements each. Therefore the n column. m-element vectors belong to R',...

## Review

Rectangular hyperbola slope coefficient relative difference universal set t. How does a Venn diagram help to illustrate the possible relationships between sets and subsets 2. What is meant by the real line 3. What is a supremum What is an infimum 4. What is a point set What is a convex set 5. Distinguish between closedness and boundedness of a point set. fi. Distinguish between concavity and convexity of a function. 7. Distinguish between quasiconcavity and concavity.

## Review of Fundamentals

In this chapter we give a concise overview of some fundamental concepts that underlie everything we do in the rest of the book. In section 2.1 we present the basic elements of set theory. We then gv> on to discuss the various kinds of numbers, ending with a concise treatment of the properties of real numbers. End the dimensions of economic variables. We then introduce the idea of point sets, beginning with the simplest case of intervals of the real line, and define* their most important...

## Rule 1 Derivative of a Constant Function fx c

The reason that fix) 0 when fix) c is easy to see intuitively by looking at the graph of the function fix) c (see figure 5.18). Regardless of which poini A' is chosen. Ay 0 for any value of Aa. Here Ay - 0 for any size of A.v. Example 5.4 Marginal Revenue Function for a Competitive Firm A competitive firm believes that if it sells more output there will not be a reduction in the market price. The extra revenue generated by producing and selling one more unit...

## Rule 3 Derivative of a Power Function fx xn

For example, the derivative of the function f(x) ,r is fix) 2x, as we denved1 from the definition of the derivalive in section 5.2. The following list of examples illustrates various types of results using this rule. For fix) x3 2,x > 0. we get f'(x) 0 2)x 2' Q 2)xt 2. The function fix) x 2 has derivative fix) 2x 3. The derivative of fix) .v. -v 0. is fix) Ox'1 0 x 0 for .v 0. Since for x 0, x 1. this is clearly the correct result. (Recall that 0 is not...

## Rule 6 Derivative of the Sum of an Arbitrary but Finite Number of Functions hx gx

Ifh(x) v ' l g,tx), then h'(x) S'M) This result also applies to the case where some or all of the operations involve subtraction rather than addition. Rule 6 is a straightforward generalization of rule 5. That is. since he derivative of the sum of two functions is simply the sum of the derivatives of the functions taken separately, then doing this iteratively allows one to establish rule 6. Thus, for example, if h(x) v4 + 8.v2 + 2a. we can treat h(x) as the sum of two functions, fix) (a-4 +...

## Rules of Differentiation

In section 5.2 several examples were presented showing how to derive the derivatives of some simple functions by using the definition of the derivative. It would be tedious, however, to have to do this every time we wanted to find the derivative of a function. Since such derivations have been done for various general classes of functions, we can use these results and so avoid repeating the exercise each time. These rules or methods of finding derivatives are collected below in table format and...

## Rules Of Differentiation 185

Figure 5.25 Marginal revenue foi u monopolist facing the inverse demand function p 40 - 2q Figure 5.25 Marginal revenue foi u monopolist facing the inverse demand function p 40 - 2q Notice the change in sign to positive for the second term. This simply recognizes that in the expression for AR Aq. we treat the reduction in price as a positive value, while in the expression lim. - o AR Aq. die term dp dq is the slope of the inverse demand function which is itself negative (i.e Ap Aq is the slope...

## S

Give the amounts of labor and capital required to produce, respectively, one unit of agricultural, mining, and manufacturing output. (i) Find the primary input requirements ofthe economy when it wishes to produce the final demand vector (ii) Is this final demand vector feasible for this economy if it has available a maximum of 1,200,000 units of labor and 1,700,000 units of capital (hi) Find the set of final demand vectors that are feasible for the economy given the primary input availability...

## Second Order Conditions

We saw in section 6. that the condition f'(x) 0 does not in itself tell us whether x* yields a maximum, a minimum, or a point of inflection of the function . Since this condition is staled in terms of the first derivative of the function it is usually referred to as the first-order condition. We now go on to examine how conditions on the second derivative of a function, namely second-order conditions, can be developed to help us distinguish among the three kinds of stationary value. In figure...

## Second Order Conditions for Constrained Optimization

We saw in the previous section that the first-order conditions for a maximum and a minimum of a constrained problem are identical, as in the unconstrained case, and so it again becomes necessary to look at second-order conditions. One approach to these is global assumptions are built into the economic model to ensure that the objective function and the constraint function(s) have the right general shape. As we will see in section 13.3, it is sufficient for a maximum (minimum) that the objective...

## Secondorder Partial Derivatives 475

The exercise of determining the economic interpretation of first- and seeond-ordcr partial derivatives is illustrated in the following example Example 11.12 Find and interpret the second-order partial derivatives of the Cobb-Douglas production function with two inputs. The general form of the Cobb-Douglas production function with two inputs is > (je,, *-> ) Ax' 4, vi. *2 > o wliere .rj and x2 are input levels, y is the output level, and a, P, A > 0, are technological parameters. We...

## Sequences Series and Limits

Studying sequences and series is the best way to gain intuition about the rather perplexing notions of arbitrarily large numbers (infinity) and inrtnitesimally small (but nonzero) numbers. We gain such understanding by using the idea of the limit of a sequence of numbers. Thus, from a mathematical perspective, this chapter provides very useful background to the important property of continuity of a function, which we will explore fully in chapter4. There are also some interesting economic...

## Sets and Subsets

5 (x x is an even number between I -nd 11 The first way of writing 5 corresponds to definition by property (the should be read as given that) the second to definition by enumeration. The key aspects of this notation are as follows A capital letter denoting the set, here S. A lowercase letter denoting a typical element of the set, here x. Braces, ( ), that enclose the elements of the set and emphasize that we treat them as a single entity. In general, a lowercase letter such as x denotes items...

## Simultaneous Systems of Differential and Difference Equations

It is common in economic models for two or more variables to be determined simultaneously. When the model is dynamic and involves two or more variables, a system of differential or difference equations arises. The purpose of this chapter is to extend our single equation techniques to solve systems of autonomous differential and difference equations. 24.1 Linear Differential Equation Systems We begin with the simplest case a system of two linear differential equations and solve it using the...

## Single Equation Models and Multiple Equation Models

Although sometimes the problem we are trying to analyze may be captured in a single-equation model, there are many instances where two or more equations are necessary. Interactions among a number of economic agents or among different sectors of the economy typically cannot be captured in a single equation, and a system of equations must be specified and solved simultaneously. We can extend our earlier example to illustrate this. Consider first the demand and supply of two goods. We denote the...

## Sma iUj A it

Next, choose a set of w, e .v, _ i, .v, values to generate the largest possible value for the Riemann Sum that is. choose ft), such that (< u,) > (x) for all x e xi-1, a, . Let us refer to these values as & > , and the sum they generate as the upper For our example, it is easy to see that Lire values ( are found by choosing the leftmost point of each subinterval, while the values < > , are found by choosing the rightmost point of each subinterval (see figure 16.4). The result is...

## Solution

From example 3.6 we can see immediately that the answer to this question is 2,401,831.27. To see that this is so, suppose that we start with S2.40I.83I.27. invested at 12 for one year, thus generating 2,401,831.27 x 1.12 2,690,051.02 at the end of the first year. After spending Si million, w c have 1,690,051.02 left over to invest for the second year, thus generating 1,690,051.02 x 1.12 1,892,857.14 at the end of the second year. After spending SI million, we have 892,857.14 to invest for the...

## Solution by Row Operations

The idea behind finding solutions by row operations is to transform a given system of equations into another with the same mathematical properties and hence the same solution. The aim is to transform a system in such a way as to produce a simpler system which is easier to solve. Three types of operations are permitted to transform a system 1. Multiply an equation by a nonzero constant. 2. Add a multiple of one equation to another. In practice, not all of these operations may be required to...

## Some Advanced Topics in Linear Algebra

In this chapter we consider diree important advanced topics in matrix algebra vector spaces, eigenvalues, and quadratic forms. All play important roles in a variety of contexts in economic theory and in econometrics. Vector spaces enable us to talk about distance between points, and linear dependence between vectors. They are therefore closely linked to die study of systems of linear equations of chapter 7. Eigenvalues play an important role in determining the stability properties of dynamic....

## Some Properties of Point Sets in R

In section 2.2 we saw diat a point on the real line always corresponds to a real number. We now place two real lines at right angles so that they intersect at the number 0, as in figure 2.9. Any point in the coordinate system formed by these two lines can be defined as an ordered pair of numbers (.r, y) by assigning to x the real number vertically below it on the horizontal axis, and assigning to y the real number horizontally across from it on the vertical axis. The figure shows some examples....

## Some Special Matrices

A number of matrices have particular properties that are often found to be useful in studying systems of equations. A square matrix A of any order is idempotent if Idempotent matrices play a very important role in statistical distribution theory. We will see examples of them in later chapters when we discuss quadratic torms. Example 8.18 Verify that A below is idempotent It is only necessary to verify that A A A. (Why ) 1 6 -1 3 1 6 -1 3 2 3 -1 3 1 6 -1 3 1 6 1 6 -1 3 1 6 -1 3 2 3 -1 3 1 6 -1 3...

## Statics and Dynamics

In introductory treatments of economics the time dimension c> t the problem is often ignored, or supressed for simplicity. In reality, the time dimension is always present. In the examples of market equilibrium already discussed, we should think of the quantities as flows per period of time, so that qD is the quantity demanded of a good per period, however short or long that period may be in terms of calendar time. Models where we explicitly or implicitly consider a situation within a single...

## Su

Assuming an initial condition of kid) kn, we must set as the explicit solution for capital in this model Although we would not expect k(t) to converge to a steady state, given the nonautonomous nature of the model, it is reasonable to ask whether it converge . to any particular growth path. Inspection of the solution reveals that this is not the case in this model because the exponential term in the solution grows without limit.

## Systems of Linear Equations

In chapter 2 we defined a linear function as one lhat takes the form for known constants a and b. and where x is the independent variable which takes on values over some specified domain, and y is the resulting value of the function at each .v-value. We also know that by taking specific values of .v, we can draw the graph of .v and y in a two-dimensional picture. The graph is a straight line- hence the phrase linear function. There are many examples of functions in economics which can be...

## T1 8

And so v 0.625 vields a local minimum of the function. d2y __ ( (x4 + 2)272x3 (12 - 18x4)& r3(*4 + 2)) dx2 ' (x4 + 2)4 Since, x4 + 2 is always positive, we can ignore it in determining the sign of d2y dx2. Atx 0.9, 72v7 - 240x3 -140.5 < 0. where there is a local maximum. At x 0.9,12.x1 240x3 140.5 > 0. where there is a local minimum. At x 0.277. 6x2 - 30x + 4 -3.85 < 0, where there is a local maximum. At x 7.222,6x2 - 30x + 4 100.28 > 0. where there is a local minimum. At x 0, y' 0,...

## Taylor Series Formula and the Mean Value Theorem

Recall (hat the differential, dy f' x)dx. can be used to provide an approximation to the change in the y variable, dy Ay, for a given change in the x variable, dx Ax (see figure 5.45). As we saw in section 5.2, the percentage error from using dy as an approximation to the actual change in y. Ay, can be made arbitrarily small if we are willing to consider changes in a that are made arbitrarily small. However, we are not always satisfied with ihe restriction that A.v be small and for nouitv...

## The Completeness Property of K

We have already indicated that the set of real numbers R has the property that there are no gaps. That is, between any two points (numbers) on the real line, every point is occupied by a number that is either a rational number or an irrational number. In other words, corresponding to each point on the real line there is a real number, and vice versa. This is known as the completeness property, and we can express this property formally by considering the concepts of the greatest lowest bound and...