Lecture # 1 - Introduction

Transcription

1 Lecture # 1 - Introduction Mathematical vs. Nonmathematical Economics Mathematical Economics is an approach to economic analysis Purpose of any approach: derive a set of conclusions or theorems Di erences: 1. assumptions and conclusions are stated in math symbols and equations 2. We use math theorems in the reasoning process. Advantage of Language of math in economics Precise, concise Draws on math theorems to show the way Forces declaration of assumptions Allow treatment of the n-variable case Language as a form of logic Logic per se (deduction/induction) Math as an extension of deductive logic From reality to rigor; math is a means not an end Critic: Too much rigor and too little reality But economic theory is an abstraction from the real world 1

2 Mathematical Economics vs. Econometrics Econometrics deals with measurement of economic data Study of empirical observations using statistical methods of estimation and hypothesis testing Math econ referes to the application of math to purely theoretical aspect of economic analysis Both are complementary Theories mus be tested against empirical data for validity Statistical work needs economic theory as a guide Deduction vs. induction Deduction: from general to speci c Induction: from speci c to general Weakness of deduction Correctness of initial assumptions Weakness of induction Correctness of nal results Hume s paradox: neither deduction or induction leads to the Truth So use both: the one is a check on the other 2

3 Mathematical Models A mathematical model consists of: Variables: something that can take di erent values (e.g: price pro t, revenue, cost savings, consupmtion). Endogenous variables: variables whose solution we seek from the model Exogenous variables: variables determined by forces outside the model Constants/Parameters: magnitude that does not change Parameters: represented by a symbol (symbolic constant) Equations De nitional equation (Identities): de nes an identity, 2 alternative expressions that have same meaning Examples: T R T C; Behavioural equation: explains how a variable behaves in response to changes in other variables Example: Y = a + bx Y : endogenous variable X: exogenous variable a: constant b: parameter (coe cient of exogenous variable X) Conditional equation: states a requirement to be satis ed (e.g. equilibrium conditions) Example: Supply-demand model Q d = a bp demand equation Q s = c + dp : supply equation Q d = Q s : equilibrium condition This leads us to two issues: the concept of function, and the concept of equilibrium 3

4 Functions of One Variable De nition: A function of a real variable x with domain D is a rule f(:) that assigns a unique real number to each x in D. Range of f(:): set of all possible resulting values of f(x) Domain: usually set of real numbers Example: Y = f(x) Y is dependent variable f () is the function or particular rule of mapping X into a unique Y X is argument or the independent variable More than one x can have the same y and still be a function If one x determines two or more y, then it is not a function, but a relation 4

5 Types of functions Algebraic functions Constant: y = a 0 Polynomial Linear: y = a 0 + a 1 x Quadratic: y = a 0 + a 1 x + a 2 x 2 Cubic: a 0 + a 1 x + a 2 x 2 + a 3 x 3 Rational y = x+2 x 2 3x+1 y = a x Transcendental/ non-algebraic Exponential: y = a x Logarithmic: y = ln x Examples Demand: Q = 100 2P (linear) Total cost: T C(Q) = Q + Q 2 (quadratic) 5

6 Linear Functions General form: y = ax + b Slope: b Examples: Consumption function, demand and supply Quadratic functions General form: y = ax 2 + bx + c Another way of writing it: y = a x + b 2 b 2 4ac 2a 4a Graph form: parabola When a > 0; then y has a minimum When a < 0, then y has a maximum Examples: cost function 6

7 Functions of Two or more (independent) Variables z = f (x; y) y = g (x 1 ; x 2 ; x 3 ; :::; x n ) Examples: Y = f (K; L) = AK L 1 P Expenditure = p 1 x 1 + p 2 x 2 + ::: + p n x n = n p i x i i=1 7

8 Systems of equations Example: (this example later) Y = C + I C = a + by Example: Supply-demand model Q d = a bp demand equation Q s = e + fp : supply equation Q d = Q s : equilibrium condition Which leads us back to second conept; equilibrium 8

9 Equilibrium De nition: Set of selected interrelated variables within the model adjusted to a state such that there no inherent tendency to change. Example: Equilibrium in market of cell phones: Q d = a bp demand equation Q s = c + dp : supply equation Q d = Q s = Q : equilibrium condition How to solve it? Several methods E.g.: eliminate one variable: ad bc b+d Q = a bp = c + dp! P = a+c b+d To nd Q: replace it in any of the equations before.q = a b a+c b+d = 1 b+d (ab + ad ab bc) = Elimination of variable method is OK only for simple models, e.g., three linear equation or two linear and one quadratic equation models Complexity increases geometrically with larger number of equations Need a better way to handle system of n linear equation! matrix algebra! 9

10 Properties of Functions If y = f (x) ; then z = y + c shifts the graph upward for c > 0 z = cy : graph is stretched vertically if c > 0 z = cy : graph is stretched vertically AND re ected about the x-axis if c < 0 Interesting for demand functions: Before: Q d = a bp demand equation, Q s = c + dp : supply equation Suppose now Q d = h bp, with h > a Graphically: shift demand upward: larger price and quantity In solutions: (previous page): same 10

11 More properties: Sum/di erence of functions: if f(x) and g(x) are functions de ned in the same set of real numbers, then m(x) = f(x) + g(x) is the sum and n(x) = f(x) g(x) is the di erence Example: interpret x as time Example in economics: cost function Inverse function De nition of one-to-one function: Let y = f(x) be a function with domain A and range B. If f(:) never has the same value at two di erent points in the domain A then it is a one-to-one function De nition of inverse function: Let y = f(x) be a function with domain A and range B. If and only if f(:) is one-to-one, then it has an inverse function g(:) with domain B and range A x = g(y) 11

12 Other concepts you need to review: 1. Real numbers 2. Inequalities 3. Intervals and absolute values 4. Integer and Fractional Powers 5. Summations 6. Set Theory 12