simple models only, e.g., three linear or two linear and one quadratic equation models Complexity increases geometrically w/ increase in # equations, e.g., six eq.. Linear model Need a[r]
(1)Mathematics for Economists (2) 2.1 Endogenous & Exogenous Variables; constants, parameters TR – TC (identity) Qd = Qs (equilibrium condition) Y = a + bX0 (behavioral equation) Y: endogenous variable X0: exogenous variable a: constant b: parameter and the coefficient of exogenous variable X0 (3) 2.4 Functions and Relations Function: a set or ordered pairs with the property that for (x, y) any x value uniquely determines a single y value, e.g., curves or , e.g., mpp or cost curves Relation: ordered pairs with the property that for (x, y) any x value determines more than one value of y e.g., curves or , (4) 2.4 General Functions Y = f (X) Y is value or dependent variable (w/ range, vertical axis) f is the function or a rule for mapping X into a unique Y X is argument or the independent variable (w/ domain, horizontal axis) (5) 2.5 Specific Functions Algebraic Y Y Y Y Y Y = = = = = = functions a0 (constant: fixed costs) a0+ a1 X (linear: S&D) a0 + a1X + a2X2 (quadratic: prod.) a0 + a1X + a2X2 + a3X3 (cubic: t cost) a/X (hyperbolic: indiff.) aXb (power: prod fn) Transcendental functions (Ch 10) Y = aX (exponential: interest) lnY = ln(a) + b ln(X) (logarithmic: easier) (Chiang & Wainwright, p 22, Fig 2.8) (6) 2.5 Digression on exponents Rules for exponents Xn = (X*X*X* *X) n times Rule I: Xm * Xnm= Xm+n X X m n Rule II: Rule Rule III: X = IV: X0 = Rule Rule Rule V: X1/n =nx VI: (Xm)n = Xmn VII:Xm * Ym = (XY)m Xn -n Xn (7) 2.7 Levels of generality Specific function 1: specific form and specific parameters Y = 10 - 5X Specific function 2: specific form and general parameters Y = a – bX General function: general form and no parameters Y = f(X) f maps X into a unique value of Y (8) 3.1 Find the equilibrium price (P*) and quantity (Q*) Given Qd quantity demanded Qs quantity supplied P price, where P* is the equilibrium price Assume Qd = Qs=Q* Qd a decreasing linear function of P Qs an increasing linear function of P P* > One equilibrium and two behavioral equations (9) 3.1 The meaning of equilibrium Qd a Qs = - c + dP (supply) Qd Qs Q * P ,Q * * Qd a bP P O -c (demand) P* ceterus paribus Equilibrium: a set of selected interrelated variables, e.g P, Q, within the model adjusted to a state such that there no inherent tendency to change (10) 3.1 Ingredients of a mathematical model A mathematical economic model will consist of: ◦ ◦ Variables, Parameters, Example: Constants Equations and Identities Supply-demand model Qd = a - bP demand equation Qs = -c + dP supply equation Qd = Qs= Q* equilibrium condition 10 (11) 3.2 Solving a linear market model Partial equilibrium model of supply & demand 1) Qd = Qs=Q* equilibrium condition 2) a – bP = -c + dP 3) (a + c) – (b+d)P* = where P=P* when Q=0 4) (a + c) – (b+d)P = Q more general form linear formulae 5) P* = (a+c)/(b+d) 6) Q* = (ad-bc)/(b+d) 11 (12) 3.2 Solving a Linear Market Model Partial equilibrium model of supply & demand Qd = Qs =Q* equilibrium condition Qd = 21 - 2P ; a=21, b=2 Qs = -4+ 8P ; c=4, d=8 linear formula P* = (a+c)/(b+d) = (21+4)/(2+8)=25/10=2.5 Q* = (ad-bc)/(b+d) = (21)(8)-(2)(4)/10 = 160/10 = 16 12 (13) Root of a Linear Market Model Let a 21, b 2, c 4, d 8 Qd a bP (blue) Qs c dP (black) Qd Qs Q * a bP c dP (a c) (b d ) P * 0 (a c) (b d ) P Q (root) (function, red) 13 (14) 3.3a Quadratic Market Model 1) Qd = – P2 demand equation 2) Qs = -1 + 4P supply equation 3) Qd = Qs = Q* equilibrium equation 4) - P2 = -1 + 4P 5) P*2 + 4P* -5 = root 6) P2 + 4P -5 = Q function How you solve for P*? ◦ Graphically? ◦ Factor? ◦ Quadratic formula? 14 (15) 3.3 eq.(3.6) Roots of a quadratic model Qd 4 P (black) Qs P (blue) Qd Qs Q * P P P P Q (red) P *2 P * 0 P * P * 0 P * 5, 1 Q * 21, 3 15 (16) 3.3b Deriving the Quadric formula Take ax2 + bx + c = 0, solve for x in terms a, b, c x2 + bx/a + c/a = x2 + bx/a + b2/4a2 = b2/4a2 - c/a (x + b/2a)2 = (b2-4ac)/4a2 x + b/2a = (b2-4ac)½/2a x = (-b (b2-4ac)½)/2a quadratic formula Solve the system of eq for P and set P = by moving all terms to one side, then apply quadratic formula 16 (17) 3.3b Solving a Quadratic Market Model Using the Quadratic Formula ax2 + bx + c = P*2 + 4P* -5 = b b 4ac P 2a * 16 (4)(1)( 5) 1 16 20 P * 1/ 2 6 1, 5 17 (18) Cubic formula: take ax3+bx2 +cx+d= solve for x in terms of a, b, c, d 18 (19) Finding the roots of a cubic function Find the rational roots of the following cubic equation 3 x x x 0 8 x x x 0 x c (1,1 8,1 ,1 2) let c 3 1 8 1 1 x x 0 4 8 4 2 ( 8) x x x 0 x x 1 1 x 1 0 x 1 x 1 0 x 1 x 1 x 1 0 x 1, 4,1 2 19 (20) 3.4 Two-commodity market model Qd Qs1 0 Q d Q s 0 Qd a a1 P1 a P2 Qs1 b0 b1 P1 b2 P2 Qd P1 P2 Qs 1 P1 P2 20 (21) Summary Elimination of variable method is ok for v simple models only, e.g., three linear or two linear and one quadratic equation models Complexity increases geometrically w/ increase in # equations, e.g., six eq Linear model Need a better way to handle the nequation linear models -> matrix algebra! 21 (22)