## Objectives

At the end of this section you should be able to Identify and sketch linear consumption functions. Identify and sketch linear savings functions. Set up simple macroeconomic models. Calculate equilibrium national income. Analyse IS and LM schedules. Macroeconomics is concerned with the analysis of economic theory and policy at a national level. In this section we focus on one particular aspect known as national income determination. We describe how to set up simple models of the national economy...

## Y x

If a point P lies above the line then the y coordinate is greater than the x coordinate, so that y > x Similarly, if a point Q lies below the line then the y coordinate is less than the x coordinate, so that Of course, the coordinates of a point R which actually lies on the line satisfy y x Hence we see that the inequality y > x holds for any point that lies on or above the line y x. It is useful to be able to indicate this region pictorially. We do this by shading one half of the...

## Y3 y1 y31 y2 y

It is not possible to simplify this any further, because x and y2 have different bases. However, if you because negative powers denote reciprocals. (c) An obvious first step in the simplification of is to apply rule 4, treating x2 as the value of a and y 1 3 as b to get (xV1 3)3 (x2)3( y 1 3)3 Rule 3 then allows us to write As in part (b), if you think it looks neater, you can write this as because negative powers denote reciprocals. (a) (x3 4)8 (b) (c) (x2y4)3 (d) Vx(x5 2 + y3) Hint in part...

## Getting started with Maple

The second computer package that will be used in this book is Maple. This is a symbolic algebra system. It not only performs numerical calculations but also manipulates mathematical symbols. In effect, it obligingly does the mathematics for you. There are other similar packages available, such as Matlab, Derive and Mathcad, and most of the Maple examples and exercises given in this book can be tackled just as easily using these packages instead. This is not the place to show you the full power...

## Getting started with Excel

Excel is the Microsoft spreadsheet package that we shall be using in some of our worked examples. If you are already familiar with this product, you may be able to skip some, or all, of this introductory section. A spreadsheet is simply an array of boxes, or cells, into which tables of data can be inserted. This can consist of normal text, numerical data or a formula, which instructs the spreadsheet package to perform a calculation. The joy about getting the spreadsheet to perform the...

## Y

(0, 0) 100 200 300 400 500 600 700 (450, 0) (0, 0) 100 200 300 400 500 600 700 (450, 0) You may have encountered this difficulty when solving Practice Problem 8 in Section 8.1. If desired, we can always find the exact coordinates by treating the corresponding equations as a pair of simultaneous equations and solving them algebraically. The variable x can be eliminated by multiplying equation (2) by 30 and subtracting from (1) to get Equation (3) gives y 150 and if this value is substituted into...

## Practice Problem

5 The supply and demand equations of a good are given by The government decides to impose a tax, t, per unit. Find the value of t which maximizes the government's total tax revenue on the assumption that equilibrium conditions prevail in the market. We conclude this section by describing the use of a computer package to solve optimization problems. Although a spreadsheet could be used to do this, by tabulating the values of a function, it cannot handle the associated mathematics. A symbolic...

## Practice Problems

4 Consider the supply and demand equations We conclude that stability depends on the relative sizes of a and c, which govern the slopes of the supply and demand curves. Bearing in mind that we have chosen to consider supply and demand equations in which Q is expressed in terms of P, namely we deduce that the system is stable whenever the supply curve is flatter than the demand curve when P is plotted on the horizontal axis. Throughout this section we have concentrated on linear models. An...

## ExampleEXCEL

Consider the supply and demand equations (a) Assuming that the market is in equilibrium, write down a difference equation for price. (b) Given that P0 1, find the values of the price, Pt for t 1, 2, , 10 and plot a graph of Pt against t. Describe the qualitative behaviour of the time path. 12 - Pt P 4 which rearranges to give Notice that this difference equation is not of the form considered in this section, so we cannot obtain an explicit formula for Pt in terms of t. (b) We are given that P0...