Chapter 1: Mathematical Preliminaries

Transcription

1 Chapter 1: Mathematical Preliminaries Brackets -have different purposes: { 3 * [ 5 - ( 6 : 2 ) ] + 10 ]= 16 Here they are needed for grouping items from each other and serves a clarity purpose. ( 8 ) ( 3 ) ( 4 ) = 96 Here they are used to indicate multiplication. 2 * ( ) = 22 Here they indicate the order in which a series of operations should be carried out. Here they are used to indicate the independent variable. Independent variable is x. Powers -show how many times a number must be multiplied by each other ex. 3 3 = 27, which implies 3 x 3 x 3 Variables and letters -a given symbol (letter) for unknown quantity ex. (x) Square root ( ) -the inverse of the square of a number ex. 3 2 = 9 9 =3 Note: but Addition -if all terms in equation have the same sign, the answer is the sum of the absolute values of terms with the appropriate sign ex = = - 32 Subtraction -if terms in equation have different signs, the answer is sum of positive terms minus absolute value of negative terms, and the sign of the answer is the sign of terms which absolute value is the largest ex = ( ) - ( ) = Symbols ex. 2x + 3y + 4z + 4x - y - z = 6x + 2y + z a x a y b x ex. - =, where b and y can both be not equal to zero b y b y Number and symbols ex x - 4-5x = - 3-7x a

2 Multiplication/ Division outcome of sign is dependent on amount of negative signs of the quantities multiplied/ divided by each other (uneven number: negative outcome, even number: positive outcome) A x B, (A B), (A*B), (AB) A times B (the product of A and B) A, (A/B) B A divided by B Numbers ex. 4 x ( - 3 ) x 5 / ( - 6 ) = 10 (even negative terms cancel each other out) ex. - 8 x 6 / ( - 4 ) x ( - 2 ) = - 24 (uneven negative terms result in a negative answer) Symbols Ex.(x + y)² = (x + y)(x + y) = x x + yx + xy + yy = x ²+ 2xy + y² Ex. 2x/2 = x a x a x Ex. =, where b and y cannot be equal to zero b y b y a / b a y Ex. =, where b, x and y cannot be equal to zero x / y b x Other important identities (x y)² = x ² 2xy + y² (x + y)(x y) = x ² y² Fractions Please note that while dividing, the denominator cannot be equal to zero! x 1 x + 5x + 11 x = 1 2 = 0 because only zero divided by any number is zero substitute x into the denominator to check that it does not equal zero Equations and inequalities Linear equations An equation is defined as an x-dependent mathematical expression Example: 7x + 3 = 10 7x = 7 x = 1 Often we compactly denote an equation in the form f(x) Example: f(x) = 7x + 3 The general form of equation is: b

3 A linear equation can have: 1). A unique solution Example: x + 4 = 10 x = 6 2). Infinitely many solutions Example: x + y = 10 x = 10 y where y can take on any value 3). No solutions Example: 0 x = 10 Quadratic equations For quadratic equation b ± x = Inequalities A>B A<B A B A B b 2 4ac 2a A is greater than B A is less than B A is greater than or equal to B A is less than or equal to B the solutions are Example: x + x 1 4 (thus x cannot be equal to zero) X x X 2-4x (x 2) x x *because 2-3 is a positive number, x > 0 On a graph, there is difference between and Rules: a < b a + c < b + c a < b ac < bc, c > 0 ab > 0 a > 0 and b > 0 a < 0 and b < 0 ab <0 a >0 and b < 0 a<0 and b > 0 a < b ac < ab c > 0 ac > ab c < 0 -you can also use inequalities to define a certain range/ interval ex. 10 < x < 20 the variable x in this equation can thus take on all values between 10 and 20 (not including end points) c

4 Currency Conversions Exchange rate: 1 unit of given currency = y units of required currency To convert an amount (x) into another currency, use this equation, as follows x units of given currency = x(y units of required currency) For example, if the exchange rate for Us dollars is 1.32 British pounds and 1.23 euro, 1). Calculate the equivalent value of $850 in: a). British pounds: $1 = 1.32 $850 = 1.32 (850) $850 = 1122 b). Euro s: $1 = 1.23 $850 = 1.23(850) $850 = ). Now calculate the equivalent value of 400 in: a). US dollars: 1 1 = $ (400) 1 = (400) ۰ $ = $ b). British pounds: 1 1 = $ 1.23 $1 = (400) 1 = (400) ۰ = Percentages 20 20% of a number means: number 100 x If a number increases by x %, then the increase of the number is: number 100 The increased number (not the percentage but the value itself!) is thus: number number x x number + increase = + ( ) = number(1 + ) Basic knowledge on spreadsheets Electronic spreadsheet =computer equivalent of paper ledger sheet -involves easy number manipulation yet there is complex math behind it, it is an aid in mathematical and statistical problem solving -consists of a grid made from columns and rows -the intersections of these columns and rows are called cells d

5 o cells may contain the following data: 1. text (labels) 2. number data (constants) 3. formulas (mathematical equations that do all the work) -Using Excel, you can: open and save files enter data select cells move, cut, copy, and paste contents of cells copy a formula to a range of adjacent cells choose commands from the menu bar transfer files from the WWW into Excel enter formulas use functions and toolbar buttons create and edit different types of charts Chapter 2: The straight line and Applications Is defined by the equation y = ax + b where: 1). x is the independent variable 2). y is the dependant variable and value of the linear function f(x) = ax + b for a given x to find the vertical intercept (VI)/ y-intercept, you have to make x = 0 -in the equation f(0)= a(0) + b, VI is thus b to find the horizontal intercept (HI)/ x-intercept, you have to make f(x) = 0 b -in the equation 0 = ax + b, HI is a, where a 0 -thus in the equation y= ax + b, b is the vertical intercept and a will be the slope -if two points of an unknown linear equation are given, you can determine the values of a and b and thereby the linear equation as shown below: y 1 = ax 1 + b y 2 = ax 2 + b Subtract these two equations from each other and you get: Y 1 y 2 = a(x 1 x 2 ), where x 1 x 2 a = y 1 1 y 2 x x 2 Substitute this in the equation y=ax + b: y1 y2 y = x + b x x 1 2 e

6 You can therefore derive the slope (a) by using the coordinates of the two points In order to determine the value of b, we calculate in a similar manner: Y 1 x 2 = ax 1 x 2 + bx 2, y 2 x 1 = ax 2 x 1 + bx 1 Subtract these two equations from each other and you get: Y 1x 2 y 2 x 1 = b(x 2 x 1) b = y x 1 x 2 2 y x 2 1 x 1 -A straight line can also be written in the form y = mx + c here the slope is represented by m instead of a and the vertical intercept by c instead of b y2 y1 the same thing holds: m= x x 2 1 the slope can also be written as: m = y / x, where = delta = change in y and x Or it can be written as: slope = change in height / change in distance and the y- and x-intercepts also can still be derived as follows: vertical intercept = c c horizontal intercept = m -We can rearrange the equation ax + by + d = 0 to obtain the following: ax + by +d = 0 ax + by= -d by = - ax d 1 by b 1 = ( ax d) b 1 ( ax) 1 d y = + b 1 b 1 y = - a b slope(m) = x d b a b vertical intercept (b) = d b f

7 horizontal intercept (-m / b) = ( a/ b) d / b = a / b d / b Translations of y = f(x) -Vertical translations: by c units replace y with (y - c) by c units replace y with (y + c) -corresponding horizontal shift x = y / m Ex. Given: y = 2x + 2 by 2 units y - 2 = 2x + 2 y = 2x + 4 by 3 units y + 3 = 2x + 2 y = 2x - 1 -Horizontal translations by c units replace every x with (x c) by c units replace every x with (x + c) Ex. Given: y = 0,5x + 2 by 2 units y = 0,5(x + 2) + 2 y = 0,5x + 3 by 4 units y = 0,5(x+4) + 2 g

8 y = 0,5x+4 Mathematical Modeling - A dynamical system (process): any finitely delimited portion of our universe -where: the system input (I) is the action of the universe upon the system the system output (O) is the re-action of the system caused by the input - A (mathematical) model: a (mathematical) description of a dynamical system. -where: the model input (I) is the way we have represented the action of the universe upon the system the model output (O) is the result produced by the model when the model input has been applied - When is a model good? a model is considered good enough if the model output comes sufficiently close to the system output (given the predefined qualities) a model has to substitute the system itself -There are two methods to obtain a model, namely: (0) System Identification Techniques (SIT) (1) First Order Modeling Principles (FOMP) (1). System Identification Techniques (SIT) input/ output -you excite a system (input) and you get a response (output) (I 1,O 1 ) (I n, O n ) -modeling means determining a function O = f(i) which best represents those points O 1 f (I 1 )... O 2 f (I 2 ) (2). First Order Modeling Principles (FOMP) -is often used in engineering principles but not in business systems -involves energy conservation -certain universal laws of physics (ex. F = m*a) -is used for the application to the dynamical system which is being questioned. The model is the following relationship: g(i,o, p) = 0, where p collects the model parameters -Two main goals of modeling : 1) Analysis to try and forecast/ predict the future/ evolution of the dynamical system h

9 (based on the model I/O behavior) 2) Synthesis to try and find the input (I) that obtains the desired system output (O) (the system obeys us) Economic Models -can be divided into two types: 1) Microeconomic only looking at individual firms 2) Macroeconomic global (for instance, 10 6 times as big as a standard I/O model) Linear Models A supply-demand model (in which X is the product) 1) Demand-law: Q = f(p, Y, P S, A, ), where: Q is the quantity demand of product X P is the price of product X Y is the income of the consumer P S is the price per surrogate (substitute) of product X A is the amount invested in advertising product X 2) Supply-law: P = f(p,c,n, S, ), where: P is the price of product X C is the production cost N is the number of producers on the market S is the level of subsidies, etc. -opposite of subsidies is taxation These supply and demand laws are the most complex and realistic because they rely on a lot of factors and have un-constant coefficients (that change with time for example) The simplest demand-law is the linear one with constant coefficients alpha (α) en beta (β) Q = α + βp i

10 Demand function P = Q Inverse demand function Q = 20-2P Supply function P = -5 + Q Inverse supply function Q = 5 + P Other basic economic modeling ingredients; Elasticity of Demand, Supply and Income Price elasticity of demand (ε d ) - measures the sensitivity of demanded quantity with respect to changes in the price of the good which are being questioned. In other words, it shows by how many percent will the quantity demanded change if the price increases by 1%. ε d = % Q( d), where %ΔQ(d) is the change in demand, and %ΔP the change in price % P -by multiplying n by (N/100), you obtain n% of N. Likewise: ( f / f ) f x f ( x) = f = = x / x) f x j

11 and accordingly: % Q( d) ε d = = % P 100 ( Q / Q) Q = 100 ( P / P) P P Q -when the demand-law is linear P = a + bq then ΔP = bδq -this implies that the point elasticity demand in a certain point (P 0,Q 0 ) is: Example. ε d =(ΔQ/ γδq) x (P 0 /Q 0 ) or ε d = (1/b) x (P 0 /Q 0 ) P = 100-5Q Determine ε d when Q = 8 P = 100-5*8 = 60 ε d = 1/(-5) x (60/8) = -1.5 Quantity demanded will drop by 1.5% if price rises by 1% when Q=8 and P=60. The elasticity of demand changes in the range of - < ε d 0. It cannot be positive, because of the law of demand which states that if the price of a good increases, the quantity of this good demanded will decrease. The range in which price elasticity of demand changes can be divided into 3 intervals: < ε d < -1 - If elasticity of demand is lower than -1, we say that demand is elastic. It means that if price rises by 1%, the quantity demanded will drop by more than 1%. ε d = -1 -If elasticity of demand equals -1, we say that demand is unit elastic. It means that if price rises by 1%, the quantity demanded will drop by exactly 1%. 1 < ε d 0 -If elasticity of demand is more than -1 but still negative or equals zero, we say that demand is inelastic. It means that if price rises by 1%, the quantity demanded will drop by less than 1%. k

12 There are two special cases of demand elasticity. They are: ε d = - The demand is perfectly elastic. ε d = 0 The demand is perfectly inelastic. Factors Affecting the Price Elasticity of Demand 1). Availability of substitutes/products: the more possible substitutes/products, the greater the elasticity. 2). Degree of necessity or luxury: luxury products tend to have greater elasticity. 3). Proportion of the purchaser's budget consumed by the item: products that consume a large portion of the purchaser's budget tend to have greater elasticity. 4). Time period considered: elasticity tends to be greater over the long run because consumers have more time to adjust their behavior. 5). Permanent or temporary price change: a one-day sale will evoke a different response than a permanent price decrease. 6). Price points: decreasing the price from 2.00 to 1.99 may evoke a greater response than decreasing it from 1.99 to Arc price elasticity of demand As opposed to point elasticity, the arc elasticity measures the elasticity over an interval on the demand function Instead of using the price and quantity at a point as in point elasticity, arc elasticity uses the average of the prices and quantities at the beginning and end of the stated interval: l

13 Q 1/ 2( P(1) + P(2)) Q ε d = = ΛP 1/ 2( Q(1) + Q(2)) P P(1) + P(2) = Q(1) + Q(2) *note:here we use ½(P(1) + P(2)) and ½ (Q(1)+Q(2)) because those are the middle points This is called the midpoints elasticity formula or arc-price elasticity formula When the demand-law is linear, then: Q P(1) + P(2) 1 P(1) + P(2) ε d = =, since Q = 1 P Q(1) + Q(2) b Q(1) + Q(2) P b (the slope) P = a- bq bq = a P a 1 Q = P b b Q 1 = P b Ex. P = 90-6Q the slope of P is b = -6 b takes on an absolute value = Ex. Q = P the slope of Q is = 6 6 b 6 Price elasticity of supply is treated in a very similar manner as the price elasticity of demand the co-efficient will be positive because of the positive slope, and is given by the formula: % Q( s) Q P ε s = = % P P Q given the supply function P= c +dq, the slope = d = P/ Q 1/d is thus Q/ P the point elasticity of supply formula is: є s = Like in case of price elasticity of demand, coefficient range of elasticity of supply can be divided into 3 intervals. 1 < ε s < - If elasticity of supply is greater than 1, we say that supply is elastic. It means that if price rises by 1%, the quantity supplied will rise by more than 1%. ε s = 1 - If elasticity of supply equals 1, we say that supply is unit elastic. It means that if price rises by 1%, the quantity supplied will rise by exactly 1%. 0 ε s < 1 - If elasticity of demand is less than 1 but still positive or equals zero, we say that supply is inelastic. It means that if price rises by 1%, the quantity supplied will rise by less than 1%. 1 d P Q m

14 Chapter 3: Simultaneous Equations Systems of equations - The solution of a set of simultaneous equations is the value of x and y which satisfies all equations To solve the equations algebraically, eliminate all but one variable, solve for this one variable, then solve for the other(s) To solve the equations graphically, plot the graphs. The solution is given by the coordinates of the points of intersection of the graphs. Linear systems of equations - The general form is: (1) a 1 x + b 1 y = c 2 (2) a 2 x + b 2 y = c 2 -where: x and y are the variables a 1, a 2, b 1 and b 2 are system coefficients c 1, and c 2 are the free terms The method of substitution Example 4x - 2y = 14 (1) 2x +3y = 3 (2) We need to eliminate one of the variables from system of equations. For example, to eliminate y, we multiple equation (1) by 3 and equation (2) by 2 and then add them up: 12x - 6y = 42 4x - 6y = 6 16x = 48 x = 3 Now,we substitude x=3 into one of the equations: 4*3-2y = y = 14 2y = -2 y = -1 So,the set (3;-1) satisfies the system of two equations. Two lines can have zero, one or infinitely many points of intersection: One point of intersection the system has a unique solution 6x + 2y = 10 x - y = 1 Infinitely many points of intersection (THE LINES COINCIDE) the system has infinite many solutions (2 proportional yet not identical equations) x + y = 5 2x + 2y = 10 No intersection points (THE LINES ARE PARALLEL) the system has no solutions 3x + 2y = 10 n

15 3x + 2y = 5 This is similarly calculated for systems with three unknowns x, y and z described by: a 1 x + b 1 y + c 1 z = d 1 a 2 x + b 2 y + c 2 z = d 2 a 3 x + b 3 y + c 3 z = d 3 - yet the computation is more lengthy - a general solution is provided by MATRIX CALCULUS Equilibrium and break-even Goods market equilibrium Equilibrium is obtained when DEMAND = SUPPLY On the good markets, let: Q d be the quantity demanded Q s be the quantity supplied P d be the price that the consumer is willing to pay P s be the price the producer is willing to accept. - Then the goods market equilibrium condition reads out: P s = P d and Q d = Q s Labour market equilibrium - Labour demand: w d = a - bl : a negative relationship between the number of labor unit and the wage rate (price per unit) Labour supply: w s = c + dl: a positive relationship between the number of labor units and the wage rate (price per unit) Labour market equilibrium: L d = L s and w d = w s -Revise price ceilings, black market profits, price floors Governmental intervention When the balance P s = P d is reached at too high of a level, the government can intervene and apply ceiling prices which are below the market equilibrium. On the contrary, when the balance is too low, the government applies floor prices which are above the market equilibrium. Price ceiling When government introduces price ceiling, it means that companies cannot sell their product above certain price. In such cases, there is shortage of goods, since there are not enough suppliers who want to produce for this price. o

16 Here, the difference Q 2 - Q 1 is called a shortage. For a new price P 0 there are Q 2 goods demanded but only Q 1 goods that suppliers produce. Price floor When government introduces price floors, it means that producers cannot their goods below certain price. In such cases, there is surplus of goods, since there are not enough buyers who would by all goods that suppliers want to produce. Here, the difference Q 1 - Q 2 is called a surplus. For a new price P 0 there are Q 1 goods produced but only Q 2 goods that customers are willing to buy. p

17 The other form of governmental intervention is taxes and subsidies Taxes There are two types of taxes: Fixed tax. Whatever amount of goods supplier produces, he always pays a fixed amount of tax. Example: Original supply function is P = 2Q + 6. If there is a tax of 4 (euros) the supply function becomes P = 2Q Producers sell their goods for more money than they used to because they want to cover taxes. Percentage tax. For every unit produced, supplier pays certain amount above the good s original price. Example: Original supply function is P = 2Q + 6. If there is a tax of 10% for every good sold, the supply function becomes P = 2(Q+0.1Q) + 6 = 2.2Q + 6. It is important to see the difference between following graphs: Taxes There are two types of subsidies: Fixed subsidy. Whatever amount of goods supplier produces, he always receives a fixed amount of subsidy from government. Example: Original supply function is P = 2Q + 6. If there is a subsidy of 3 (euros) the supply function becomes P = 2Q + 3. Producers can sell their goods for less money than they used to because they will compensate it by money received from government. Percentage tax. For every unit produced, supplier receives certain amount of money. Example: Original supply function is P = 2Q + 6. If there is a subsidy of 5% for every good sold, the supply function becomes P = 2(Q Q) + 6 = 1.9Q + 6. It is important to see the difference between following graphs: q

18 Surpluses A balance is advantageous for everyone (consumer surplus is only beneficial for consumers, producer surplus only for producers) Consumer Surplus (C S ) is the difference between the expenditure a consumer is willing to make on successive units of a good from Q = 0 to Q = Q 0 and the actual amount spent on Q 0 units of the good at market price P 0 per unit. Producer Surplus (P S) is the difference between the revenue the producer receives for Qo units of a good when the market price is P 0 per unit and the revenue that the producer was willing to accept for successive units of the good from Q = 0 to Q = Q 0. National Income Model Equilibrium exists when income (Y) = expenditure (E) Expenditure may consist of: Consumption:C = C 0 + by, less tax Investment: I 0 Government expenditure, G 0 r

19 Exports, X 0 Less imports, M = M 0 + my E = C + I + G + X M To find the level of income (Y e ) at which equilibrium exists, solve the equation Y = E for Y. The solution is Y e. Equilibrium consumption: C e = C 0 + by e Equilibrium taxation: T e = ty e Revise the reduced expressions for Y e with multipliers. These are formulae from which T e may be calculated directly for various standard national income models. IS/ LM model Equilibrium in the goods market when Y = E (national income), but consider investment as a function of interest rate: I = I 0 dr, thus Y 1 = ( C(0) + I(0) dr G(0)) 1 b(1 t) + hence the equation r = f(y), which is the IS schedule. Equilibrium in the money market when money supply = money demand Money demand: M d = M + M + M =L 1 + L 2 = ky + (a hr) T d P d S d Money supply: M s = M 0 Equilibrium: M s = M d M 0 = ky + (a hr) This equation may also be written as r = g(y). This is the LM schedule. The goods and money market are simultaneously in equilibrium for the values of r and Y which satisfy the simultaneous IS and LM equations, for example: IS schedule: r = Y LM schedule: r = Y Chapter 4: Non-linear Functions and Applications Quadratic polynomial functions The general expression is P(x) = ax 2 +bx+c, a 0 The completion of squares formula We are interested in determining the roots of P 2 (x) and classifying their nature. ax 2 +bx+c = 0 ax 2 +bx = c since a 0 x 2 + (b / a)*x = (c / a) x 2 + (b / a)*x + (b 2 / 4a 2) = (c / a) + (b 2 / 4a 2) (x + (b / 2a)) 2 = (b 2 4ac) / (4a 2 ) ± x + (b / 2a) = b 2 4ac 2a s

20 x = b ± b 2 4ac 2a Note: b 2 4ac is called discriminant and is denoted as D. Root classification There are 3 possible outcomes of roots of quadratic polynomial equations: When b 2-4ac > 0 There are two real roots of the equation. x 1 = x 2 = b + b b 2 4ac 2a b 2 4ac 2a Since b 2-4ac > 0, the part other is negative. ± b 2 4ac is real and one of its values is positive, the Example: 3x 2-8x - 3 = 0 When b 2-4ac = 0 There are two identical real roots of the equation. In other words, there is only one real root. b x = 2a Since b 2-4ac = 0, the part ± b 2 4ac equals zero and therefore the sign in front of expression does not matter. Example: 9x 2-12x + 4 = 0 When b 2-4ac < 0 There are two complex roots of the equation. Since b 2-4ac < 0, the part ± b 2 4ac has no meaning (in real numbers) and therefore the roots are not real as well. Example: 2x 2 + 3x + 8 = 0 t

21 Two real roots. One real root. No real roots. The formulas of Viete A French mathematician Viete has discovered that: for any quadratic equation ax 2 +bx+c = 0 x 1 + x 2 = - b / a x 1 * x 2 = c / a This hint may be useful when one tries to guess roots: 2x 2-10x+12=0 x 1 + x 2 = -(-10)/2 = 5 x 1 * x 2 = 12/2 = 6 It is easy to guess that the roots are 2 and 3. Graphs translations u

22 For any function y = f(x): to shift function UP by c units substitute it by y = f(x) + c to shift function DOWN by c units substitute it by y = f(x) - c to shift function LEFT by c units substitute it by y = f(x +c) to shift function RIGHT by c units substitute it by y = f(x - c) Locating the maximum/minimum Any quadratic equation y = ax 2 +bx+c has either minimum or maximum. To find the values of x and y of maximum/minimum substitute a, b and c into formulas: x max/min = - b / 2a y max/min = (4ac - b 2 )/ 4a Rather than memorize second formula, you can substitute the x-value you obtained with the first formula, into the quadratic equation and get y-value of maximum/minimum. Example: y = 2x 2 + 8x - 6 x max = - b / 2a = -8/4 = -2 y max = (4ac - b 2 )/ 4a = (-48-64)/8 = -112/8 = -14 OR x max = - b / 2a = -8/4 = -2 y = 2*(-2) 2 + 8*(-2) - 6 = -14 Quadratic functions in economics v

23 - Recall we have introduced the linear supply-demand law as a basic dynamical model of a market. We shall extend now the model to be quadratic: P S (Q) = a 1 Q S 2 +b 1 Q S +c 1 P d (Q) = a 2 Q d 2 +b 2 Q d +c 2 - (Q 0 ;P 0 ) is called break-even point if P S (Q 0 ) = P D (Q 0 ). - How many break-even points admits the above model? Clearly, from PS(Q 0 ) = PD(Q 0 ) it follows that all break-even points are roots of the following quadratic equation a 1 Q 2 +b 1 Q+c 1 = a 2 Q 2 +b 2 Q+c 2 (a 1 - a 2 ) Q 2 + (b 1 - b 2 ) + c 1 -c 2 = 0 Accordingly to the positivity of discriminant (b 1 - b 2 ) 2-4(a 1 - a 2 ) ( c 1 -c 2 ), there are 0, 1 or 2 break-even points. (don t need to know the theoretical explanation for the exam) Cubic polynomial functions - The general expression is P 3 (x) = ax 3 +bx 2 +cx+d, a 0. - a polynomial function of third degree (cubic equation) can always be factorized as the product of two parts: ax 3 + bx 2 +cx + d = (Ex + F)*(Gx 2 + Hx + I) -Cubic equations are continuous curves, which may have: No turning points or three turning points One or three roots -How can we find the roots of the equation P 3 (x) = 0? Using the scheme of Cardan (Italian mathematician) 1. Normalize the equation by dividing the whole equation with a and denote the result by x 3 +ex 2 + f x+g = 0 2. Eliminate the x2-term by applying the substitution x (y e/3) and obtain the equation of the form y 3 + py + q =0 3. Introduce two new variables, t and u, defined by the equation u t = q tu = (p / 3) 3 4. Compute: 3q 3 ± 4 p^3 + 27q^2 u = 6 3 3q 3 ± 4 p^3 + 27q^2 t = 6 3 (where in the expression of t and u ± means in both either + or in both either ) 5. Then y1 = 3 3 t u is a solution of y 3 + py + q = Write down y 3 + py+q = (y y1)(ay 2 +By+C) and solve Ay 2 +By+C = 0 to get the other two solutions. don t have to memorize the derivation w

24 Remark 1 There are no algorithms to solve quartic equations P4(x) = ax 4 +bx 3 +cx 2 +dx+e = 0 -so which alternatives are left over? 1. Interval searching 2. Graphical methods 1. Interval searching -Being given n+1 real numbers coefficients a 0,a 1,,a n, a n 0 a real polynomial function of degree n is an expression of the following form: P n(x) = a n x n +a n 1 x n 1 + +a 2 x 2 +a 1 x+a 0 -Make a line, where P(a) is on the left hand side, c = (a+b)/2 is in the middle and P (b) is on the right hand side P(a) * P(b) = a negative number If P(a)* P(b) > 0, then the solution is between c and P(b) Keep repeating this step until you end up with a very small interval (very short line) The solution may come as close as to the actual solution Ε < C n X 2 The exponential function -the exponential function increases/ decreases faster than any other function -it takes on the general form: f (x) = a x - where: a is a real constant called the base x is the variable index or power of the equation f : R (0, ) - the basic properties of the exponential function are as follows: Semi-group property : a x+y = a x a y ; a x = 1 x = 0 Positivity : a x > 0, x R In particular, a x = 1/ a x, x R Monotonicity : Let 1 < a < b. Then a x < b x (increasing function) =growth curves Let 0 < a < b < 1. Then a x > b x (decreasing function) =decay curves all curves are continuous (no breaks) and pass through the point x = 0, y = 1 the graphs of y = a x and y = a x are always above the x axis, therefore y is always positive x

25 the graphs of y = - a x and y = - a x are always above the x axis -the basic rules of exponential functions are as follows: a m * a n = a m+n a m /a n = a m-n (a m ) k = a m * k -unlimited growth: -modeled by the equation: y(t) = ae rt, where a and r are constants -limited growth: -modeled by the equation: y(t) = M(1-e -rt ), where M and r are constants -logistic growth: -modeled by the equation y(t) = M / (1 + ae -rmt ), where M, a and r are constants The logarithmic function -The inverse function of the exponential function is called the logarithmic function, defined as: log a : (0, ) R such that (according to the definition of invertibility) log a a x = a loga x = x -the basic properties and rules of the logarithmic function are as follows: log a (xy) = log a x+log a y -this can be proved: use a x = a y x = y (injectivity) and get log a (xy) = log a x+log a y, a loga(xy) = xy = al oga x a loga y aloga x+loga y = log a x log a y = log a (x/y) log a x+log a y = log a (x*y) log a (x z ) z*log a (x) log a (x) log z (x) / log z (a) Change of basis: log a x = log b x / log b a Log(1) = 0 for any base There are no real logs of negative numbers Logs of numbers greater than one are always positive, less than one always negative Number =base power. Then log base (Number) = power (reversing the operation is called taking antilogs) The natural logarithm and the Euler exponential function - Define first the transcendental number: e = y

26 - a transcendental number van not be calculated / written in any calculation such as a fraction (irrational number, just like µ) -There are many mathematical and technical stories about how euler was originated -It is quite sure that it was introduced by Euler (German mathematician) who was studying the behavior of the values of the sequence e n = (1+ (1/n)) n for large values of n. -Taylor has shown that the Euler exponential function e x is well approximated by the following polynomial function of degree n e x = 1 + x + x2 / 2 + x3 / xn / n! where n! is the factorial function of the first n natural numbers n! = n -The inverse of the Euler function is the natural logarithm ln: (0, ) R, ln e*x = x=e ln*x e x = ln * x The hyperbolic function -can mostly be deduced from a fraction that has an unknown (variable) in its denominator -is a function defined by h : R { c / b} R, h(x) = a/ (bx + c) -The simplest hyperbolic function is : y = 1/x - The vertical line x = 0 is called the vertical asymptote of the hyperbolic function y = 1x, while the horizontal line y = 0 is called the horizontal asymptote. -If y = a/ (bx + c), revise the graphs and notice the vertical asymptote -If y = f(x), then Y*x = a constant Chapter 5: Financial Mathematics Arithmetic and Geometric Sequences and Series - A sequence is a list of numbers, which follow a definite pattern or rule. 1). Arithmetic sequence: T n = a + (n-1)d a is the first term of the sequence d is the common difference n is the n th term 2). Geometric sequence: T n = b*r (n-1) b is the first term of the sequence r is the common ratio n is the n th term - A series is the sum of the terms of a sequence. It is: finite if it is the sum of a finite number of terms of a sequence infinite if it is the sum of an infinite number of terms of a sequence 1). Arithmetic series (or progression denoted by AP) : S n = 2 n (2a + (n-1)d) z

27 2). Geometric series (or progression denoted by GP) : S n = If r is less than 1, the formula simplifies to S n = If r equals 1, the formula is simply S n = n*a a 1 r. a(1 r^ n) 1 r Example (Computers manufacturing) -Dell-Nederland produces N = 1200 computers/week. Due to an increase in the market demand in Rotterdam, the manufacturer has to increase its production. He can do that by: 1. increasing the production with 80 computers each week 2. increasing his productivity by 5% each week Then determine : a). The production output in week 20 in both cases b). The total output after 20 weeks (again both cases) c). In which week the production exceeds 8000 computers Solution -Option 1 leads to an arithmetic series with a = 1200, d = 80 -Option 2 leads to geometric series with a = 1200, r = 1.05 a). Indeed, let n denote the index of the week and let: f n denote the production in that week according to option 1 g n denote the production in that week according to option 2 Then: f n = a+(n 1)d = 1200+(n 1)80 f 0 = 1200 g n+1 = g n+(5/100)gn = (105/100) g n+1 = 1.05g n-1 = 1200(1.05)n 1 g 0 = Accordingly, when n = 20, then f 20 = 2720 while g 20 = 3032 b). The total output is the sum of the weeks outputs. This is given by the corresponding arithmetic and geometric series for n = 20 n A n = (2a+(n 1)d) = (20/ 2) (2(1200) + 19(80)) = r n G n = a* =1200 *{( ) / (1.05-1)}= r 1 c). Since f n and g n represent the productions in week with index n, we have to: -Find n, such that f n = a+(n 1)d > 8000 n > (8000- a)/d = 86 -Find n, such that g n = a*r n-1 > 8000, 8000/a < (1.05)n 1 n > 1+(log 8000/1200) / log 1.05= Hence n = 40 Simple Interest and Compound Interest aa

28 Simple interest -Let P 0 be a certain initial amount of one has put in the bank in an account with simple interest. How does the amount of money change in time? -Simple interest is a fixed percentage of the initial amount P 0 -This is modeled by arithmetic progression. -Let i% = i / 100 be the simple interest in question. Then after N-years, the owner of that account has accumulated an interest I = P 0 i N -Therefore, the total amount (end-sum) of euros he will have in the account (assumed no-withdrawals are made) is P N = P 0+P 0 in = P 0(1+iN) Related problems There are two types of problems associated with the simple interest: 1. The end-sum problem : Being given P 0, i% and N find the end-sum after these N years P N = P 0 +P 0 in = P 0 (1+iN) 2. The initial investment problem: How large should be the initial deposit P 0 such that for a given simple interest of i% so that after N-years we can count on an available amount of P N P 0 = P N / (1+iN) *so when you put money in a bank for a long period of time, don t think about simple interest, but select the bank with compound interest Compound interest interest applies not to initial amount but to amount of money at that moment, so it doesn t grow linearly like simple interest but exponentially, therefore, faster -Let P 0 be a certain initial amount of one has put in the bank in an account with compound interest. How does the amount of money change in time? -This is modelled with geometric progression. -Compound interest is a variable percentage of the initial amount P 0 -More precisely, compound interest applies to the interest of previous years as well as to the initial sum P N = P 0(1+i) N Multiple-compound interest Some banks have another kind of offers than to compound interest annually. They compound it several times a year, e.g every month or every quarter. -In reality the interest is compounded several times each year -Each period is an interest period -Let m be the number of periods of conversion each year bb

29 -Then the interest rate applied to each period is i/m -If t = m*n is the total number of interest periods (N is the number of years), then the value of the investment at the end of these periods is Continuous compound i Pt = P0 1+ = P0 1+ m t i m mn The last option of compounding is called continuous compound. Here, banks assume that interest is compounded every fraction of second, i.e constantly. The formula of continuous compounding is: P N =P 0 e i*n where P 0 is initial amount, i is annual interest rate and N is number of years for which compunding works. To sum up: Interest Future value Present value Simple P t = P 0 (1 + it) P 0 = P t (1 + I) -t Compound P t = P 0 (1 + i) t P 0 = P t (1 + i) t Compound m times annually P t = P 0 (1 + (i/m)) mt P 0 = P t (1 + (i/m)) - mt Continuous compounding P t = P 0 e it P 0 = P t e -it Annual Percentage Rates (APR) It is important to know the difference between Nominal Interest Rate (usually denoted simply as i) and Annual Percentage Rate (APR). For example, a bank has 10% annual interest rate and compounds the interest quarterly (4 times a year). It means that its Nominal Interest Rate i=0.1. APR shows by how many percent the initial amount will really increase. It is quite straightforward that if interest is compounded more than once a year, it will be more than 10%. To find the APR, we must substitute i=0.1 and m=4 into formula P t = P 0 (1 + (i/m)) mt Let s leave P t and P 0 as variables and t=1 since we need to find out ANNUAL percentage rate. We get: P t = P 0 (1 + (0.1/4)) 4 = P 0 It means that future value will be times initial amount. To find out by what amount the investment has risen, we subscribe initial amount P 0 from the equation: P t = P 0 - P 0 = P 0 cc

30 It means that investment rose by 10.38% during the year % is our APR. After some derivations, we can come up with the following formula for compound interest m times a year: We can also derive APR for continuous compounding: APR = e i - 1 For simple interest and interest compounded annually APR = i. Depreciation If you buy something new, in several years its value will be less than you have paid for it, since you have been using it during this time, i.e. you depreciated it. There are two most popular depreciation types: 1. Straight-line/ linear depreciation Over time, the value depreciated is the same for each year. The formula of future value A t in t years given depreciation rate i and present value A 0 is: A t = A 0 ( 1 - t*i) For example, you bought a washing machine for 2500 euros. If depreciation rate is 10% annually, then in 6 years the value of this machine will be 2500*(1-0.1*6) = Reducing-balance depreciation Over time the value by which a product depreciates decreases. Formula is: A t = A 0 (1 - i) t Taking the same example, A 0 = 2500, t = 6, i = 0.1. The value in 6 years will be: A t = 2500* (1-0.1) 6 = 1171 euros. NPV and IRR NPV = Net Present Value is the present value of future cash flow, discounted at a given discount rate r. Example: -The present value under compound interest is P 0 = P N /(1+i) N We compute: NPV = present value of inflow present value of outflows dd

31 We need to decide whether or not to invest into a project with following cashflows: Year Cash flow Interest rate = 10% To calculate NPV of cash flows, we find the present values of future cash flows P 0 = P N /(1+i) N and sum them up: NPV = /1.1-50/ (1.1) / (1.1) / (1.1) 4 = We should invest in a project whenever NPV>0. In this case, NPV=128.80, and therefore we should. IRR = Internal Rate of Return Internal Rate of Return is a rate by which NPV of cash flows equals zero. To find IRR, we must choose such two rates r 1 and r 2 (r 1 < r 2), by which one of the NPVs is positive, the other is negative (NPV 2 < 0 < NPV 1 ). Then we substitute these values into formula: Annuities and debt repayments -Annuity represents an extra fixed investment A 0 done at the end of each year for a period on N years in the top of the original investment P 0. Accordingly, the total investment is N (1 + i) 1 V N = P 0 (1+i)N + A0 i Where: P 0 (1+i)N = initial N (1 + i) 1 A0 = annuities i i% = the rate at which interest is compounded per time Interval V t = value P 0 = initial investment t = time periods (In the text, the basic time interval was assumed to be one year. Calculations for other time intervals are adjusted accordingly.) -The value V t for an initial investment of P 0 and t periodic investments of A 0 is: t (1 + i) 1 VANU, t = A0 i -The present value of an annuity is: V 0 = A 0 1 (1 + i i) t, where 1 (1 + i) t i is the annuity factor -The amount of periodic repayments on a loan L is ee

32 A Basic problem Li 1 = = L 1 1 (1 + i) 1 t (1 + i) 0 t 1, where t 1 (1 + i) is the capital recovery factor How much (V 0 ) should be invested now for the next N years at a given interest rate i% to cover a series of N annual equal payments A 0? N (1 + i) 1 V 0 (1+i) N = A0 i 1 (1 + i) t V 0 = A 0 i After N years what remains is 0 (the loan is paid). Hence N (1 + i) 1 0 = L 0 (1+i) N + A0 i A0 1 L 0 = 1 n i (1 + i) where : 1. L 0 is the amount borrowed 2. A 0 are the periodic debt payments 3. N is the number of paying periods Mostly we are interested in computing A 0 1 A 0 = L t 1 (1 + i) Sinking funds *The value of a sinking fund, payments A 0 made at the start of each year, is: V sk, t (1 + i) = A0 (1 + i) i t 1 Chapter 6: Differentiation and Applications Slope of the straight line y = mx + c is m Slope of a curve y = f(x) over a infinitely small horizontal distance x equals (=change in vertical height divided by change in horizontal distance) Slope of a curve at some point (x 0 ; y 0 ) equals dy/ dx. It also shows the slope of a chord which is tangent to a curve at the point (x 0 ; y 0 ). dy/ dx (also denoted as y ) is called a derivative of the function. Derivative shows the speed at which a curve increases or decreases at a certain point on the graph. If it is positive, the function increases, if it is negative, it decreases. y x ff

33 Here are some examples of functions and their derivatives. Function Derivative Y= f(x) dy / dx K 0 Kx K x n nx n-1 e x e x ln(x) 1/x a x ln(a) * a x Chain Rule for Differentiation: Sometimes we need to find the derivative of complex function. Complex function is a function that is expressed as y=f(g(x)). Its derivative is: y = (f(g)) *(g(x)) Thus, the derivative of complex function is derivative of the main function multiplied by the derivative of sub-function. For example, function ln(x 4 ). You can think of this function as two different ones combined in one: f=ln(g) and g(x).we get y=f(g(x)) According to the equation y = (f(g)) *(g(x)), we can write derivative as: y = ln(x 4 )) * (x 4 ) = 1/x 4 * 4x 3 = 4/x Sum Rule for Differentiation: y = f(x) + g(x) = u+v Derivative of sum of functions is: y = u + v y = 2x 2 - ln(x) + ex y = 4x - 1/x +e x Product Rule for Differentiation: y = f(x) * g(x) = uv Derivative of product of functions is: y =u v + v u y = 3x * e x y =3*e x + 3x*e x =3e x (1+x) Quotient Rule for Differentiation: y = f(x)/g(x) = u/v Derivative of quotient is: gg

34 y = 4x 2 /e x Higher derivatives: The so-called second derivative measures the speed of the speed of function, in other words, its acceleration. To find the second derivative means to find a derivative of derivative. Second derivative is denoted as d 2 y/dx 2 or y. y = 2x 7 y = 14x 6 y = 84x 5 Turning points and points of inflection If there is a point on a graph where first derivative equals zero, we say that this point is called a turning point. In this point: If second derivative is positive, there is a minimum of the function. If second derivative is negative, there is a maximum of the function. If second derivative is zero, there is neither a maximum nor a minimum of the function. The point at which second derivative equals zero is called a point of inflection. Curvature 2 d y -The curvature along an interval is concave up is > 0 along that interval 2 dx 2 d y -The curvature along an interval is concave down is < 0 along that interval 2 dx Marginal Function = the derivative of the total function: d( TC) d( TR) MC = MR = d( Q) d( Q) -Marginal cost (MC) is defined as the derivative of total cost with respect to output -Marginal revenue (MR) is often defined as the change in TR per unit change in output Average function TC AC= Q TR AR = Q -Average cost (AC) is total cost divided by the level of output produced -Average revenue (AR) is defined as average revenue per unit for the first Q successive units sold Marginal and average propensity to consume and save hh

35 C S MPC = MPS = Y Y -Marginal propensity to consume (MPC) = the change in consumption per unit change in income -Marginal propensity to save (MPC) = the change in savings per unit change in income C APC = Y S APS = Y -Average propensity to consume (APC) = the consumption per unit in income -Average propensity to save (APS) = the savings per unit in income Note: Since Y = C+S, then: dy dc ds = + 1 = MPC + MPS dy dy dy Similarly, APC + APS = 1 Any reference to consumption and saving assumes planned consumption and planned savings respectively C(0) APC > MPC since + b > b Y C (0) MPS > APS since 1- b > 1 b - Y Marginal and average funcions of product of labor d( Q) MP L = d( L) Q APL= L -Marginal product of labour (MPL) is the rate of change in total output, Q, with respect to labor input, L. -Average product of labor (APL) is a measure of the labor productivity (average output per unit of labor); it is equal to total output divided by the number of units of labor employed Optimization: particular rules to note 1. Maximum profit when MR = MC and d ( MR) d( MC) < Q dq 2. MC and AVC intersect at the minimum point on the AVC curve. 3. APL and MP L intersect at the maximum point on the APL curve. Elasticity: point elasticity of demand 1. Definition: If P = f(q), then ε % Q dq / Q dq = = = % P dp / P dp d 2. The function Q = a/p e has a constant elasticity of demand ε d = - c P Q ii

36 3. Relationship between MR and ε d : MR = P(1+ 1/ ε d ) d( TR) 4. Relationship between change in TR and ε d : = Q(1 + ε d ) dp Production and Labor -A short-run production function Q = f(l) TLC = wl -Costs: TLC ALC = L d( TLC) MLC = dl jj

37 Chapter 7: Functions of Several Variables Function of two variables z = f(x, y) E.g. z = 2x - 4y + 5 Isoquant is a set of two independent variables (L and K) in function Q (L;K) by which dependent variable Q remains the same. Q = 5KL 2 Given Q= 125 5KL 2 =125 KL 2 =25 K=25/L 2 Isoquant K (L) at Q = 125 L= (25/K) =5/ K Isoquant L (K) at Q = 125 First-order partial derivatives If we have function of 2 variables, there are two outcomes of finding the derivative. We differentiate with respect to one or another variable: Q=4L 3 K 2 Q/ L =Q L = 12L 2 K 2 (Here, we assume that K 2 is constant) Q/ K = Q K =8L 3 K (Here, we assume that L 3 is constant) These derivatives are called first-order partial. Second-order partial derivatives To find second-order partial derivatives, we differentiate both first-order partial derivatives with respect to both variables. Therefore, we get 4 outcomes (2 derivatives * 2 variables): 2 Q/ L 2 = Q LL = 24LK 2 Q/ K 2 = Q KK = 8L 2 Q/ L K =Q Lk = 24L 2 K 2 Q/ K L =Q kl = 24L 2 K Mixed second-order partial derivatives are equal in almost every function. kk

38 -The straight second order partial derivatives are calculated by differentiating the first-order partial derivatives again with respect to the same variable -The mixed second-order partial derivatives are calculated by differentiating the first-order partial derivatives with respect to the other variable The total differential of z is given by f f d z = ( )* dx + ( )* dy x y or for small changes dx x etc., the small (incremental) changes formula is given by Example: f f z ( ) x + ( ) y x y Given Quantity function Q = 1000L 0.4 K 0.6. Furthermore, we know that L incresases by 5% and K decreases by 3%: L= 0.05L K = -0.03K. We need to find Q. Q Q Q = ( ) L + ( ) K L K Q = 600K 0.4 L 0.4 = L Q = 400L 0.6 K 0.6 K Note: these two values are called Marginal product of labor andmarginal product of capital, respectively. Q = 600K 0.4 L 0.4 * 0.05L + 400L 0.6 K 0.6 * (-0.03)K Q = 0.6*1000K 0.4 L 0.4 * 0.05L + 0.4*1000L 0.6 K 0.6 * (-0.03)K Q = 0.6*1000K 0.4 L 0.6 * *1000L 0.6 K 0.4 * (-0.03) ll

39 Q = 0.03*1000K 0.4 L *1000L 0.6 K 0.4 Q = ( )*1000L 0.6 K 0.4 Q = 0.018*1000L 0.6 K 0.4 Since 1000L 0.6 K 0.4 = Q Q = 0.018Q The quantity produced will increase by 18%. Unconstrained optimization Step number: 1. Find the first and second derivatives z z z z z,,,, 2 2 x y x y x y 2. First order conditions: -At stationary point z / x = 0, z / y= 0 -Solve these two stationary points for the x- and y- coordinates of the turning point(s). -If required, find z for the x- and y- coordinates of the turning point(s). 3. Second-order conditions: -the point is a: -minimum if 2z / x2>0 and 2z / y2> 0 and provided > 0. -maximum if 2z / x2<0 and 2z / y2< 0 and provided > 0. -point of inflection if both second derivatives have the same sign but < 0 -a saddle point if the second derivatives have different signs and < 0, where Constrained optimization: Lagrange multipliers Constraint optimization is an equation like: 1) ak + bl = M 2) ak + bl < M 3) ak + bl > M z z z = 2 2 x y x y It means that sets of capital and labor must (1) equal M; (2) be less than M; (3) exceed B Given the function of two variables z = f (x; y). The constraint is given ax + by = M. To find optimum values for z and not to violate constraint, we introduce Lagrange function: L(x; y; λ) = L(x; y) + L(M- ax - by) Then we calculate the stationary points of the Lagrangian function by solving the system of equations: 2 mm

40 The values of x and y we got in this system of equation are optimum set of function z. Example: f(x; y) = 5xy x + 2y = 10 L(x; y; λ) = 5xy + λ(10 - x - 2y) = 5xy - λx - 2 λy +10 λ 5x - λ = 0 5y -2 λ = x - 2y = 0 We eliminate λ from the first and the second equation. 10x - 2λ = 0 5y -2 λ = 0 y = 2x Substitute y into third equation 10 - x - 4x = 0 x = 2 y = 2*2 = 4 Cobb-Douglas Production Q β = AL K where Q stands for production, L for labor and K for capital 1. Represent production functions in two-dimensional diagrams (isoquants) by fixing the value of Q, then express K = f(l) 2. The slope of dk/dl is called the marginal rate of technical substitution (MRTS) -if it exhibits a diminishing MRTS then ΔK / ΔL or the rate at which the amount of capital deceases for each unit in increase in labor diminishes as L increases 3. Slope is also expressed as dk MPL = dl MPK α K MRTS = β L nn

41 4. Returns to scale. When L and K are each replaced by λl and λk respectively in β β Q1 = AL K then Q 2 = A( λ L) ( λ K) =λ α+β AL α K β =λ α+β Q 1 If α+β = 1: constant returns to scale, i.e. Q 2 = λq 1 If α+β < 1: decreasing returns to scale, i.e. Q 2 < λq 1 If α+β > 1: increasing returns to scale, i.e. Q 2 > λq 1 5. A Cobb-Douglas production function exhibits diminishing returns to each factor. This is confirmed by the negative second derivatives: Law of diminishing returns to labor: Q LL < 0 curve concave to origin Law of diminishing returns to capital: QKK < 0 curve concave to origin 6. a). Production conditions for using labor: Q MP L = > L 2 d ( MP L ) Q = 2 dl L MP < APL L < 0 b). Production conditions for using capital: Q MPK = > 0 K 2 d( MPK ) Q = < 0 2 dk K MP < APK K 7. A Cobb-Douglas production function is considered homogenous, order r, if: r f ( λ L, λk) = λ f ( L, K) where r = (α + β) Utility functions -The analysis of utility functions parallels that of production functions in the previous section. -A utility function describes utility as a function of the goods consumed and may be written in the general form as: U = f (x, y) -A utility function that is widely used in economic analysis is the Cobb-Douglas utility function which is expressed in general form as: U = Ax α y β -An isoquant is a combination of inputs L and K which when used give a firm the same level of output -An indifference curve is a combination of goods X and Y which when consumed give the consumer the same level of utility oo