Mathematics for Economics and Business Jean Soper chapter two Equations in Economics

1. Mathematics for Economics and BusinessJean Soper chapter twoEquations in Economics

2. Equations in Economics – Objectives 1 • Understand how equations are used in economics • Rewrite and solve equations • Substitute expressions • Solve simple linear demand and supply equations to find market equilibrium • Carry out Cost–Volume–Profit analysis • Identify the slope and intercept of a line • Plot the budget constraint to obtain the budget line • Appreciate there is a constant rate of substitution as you move along a line

3. Equations in Economics – Objectives 2 • Solve quadratic equations • Find the profit maximizing output and also the supply function for a perfectly competitive firm • Solve simultaneous equations • Discover equilibrium values for related markets • Model growth using exponential functions • Use logarithms for transformations and to solve certain types of equations • Plot and solve equations in Excel

4. Rewriting and Solving Equations • Equation: two expressions separated by an equals sign such that what is on the left of the equals sign has the same value as what is on the right • Transposition: rearranging an equation so that it can be solved, always keeping what is on the left of the equals sign equal to what is on the right

5. When Rewriting Equations • Add to or subtract from both sides • Multiply or divide through the whole of each side (but don’t divide by 0) • Square or take the square root of each side • Use as many stages as you wish • Take care to get all the signs correct

6. Solution in Terms of Other Variables • Not all equations have numerical solutions • Sometimes when you solve an equation for x you obtain an expression containing other variables • Use the same rules to transpose the equation • In the solution x will not occur on the right-hand side and will be on its own on the left-hand side • Inverse function: expresses x as a function of y instead of y as a function of x

7. Substitution • Substitution: to write one expression in place of another • Always substitute the whole of the new expression and combine it with the other terms in exactly the same way that the expression it replaces was combined with them • It is often helpful to put the expression you are substituting in brackets to ensure this

8. Demand and Supply • We plot supply and demand with P on the vertical axis • Before plotting a supply or demand function, write it so that P is on the left, Q is on the right

9. Market Equilibrium • Market equilibrium occurs when the quantity supplied equals the quantity demanded of a good • The supply and demand curves cross at the equilibrium price and quantity • You can read off approximate equilibrium values from the graph • Solving algebraically for the point where the demand and supply equations are equal gives exact values

10. Changes in Demand or Supply • Changes in factors other than price alter the position(s) of the demand and/or supply curves Effects of a per unit tax • To obtain the new supply equation when a per unit tax, t, is imposed substitute P – t for P in the supply equation

11. Cost–Volume–Profit (CVP) Analysis • Two simplifying assumptions are made: namely that price and average variable costs are both fixed  = P.Q – (FC + VC) = P.Q – FC – VC • Multiplying both sides of the expression for AVC by Q we obtain AVC.Q = VC and substituting this  = P.Q – FC – AVC.Q

12. Special Assumptions of CVP Analysis • P is fixed • AVC is fixed •  is a function of Q but P, FC, and AVC are not • We can write the inverse function expressing Q as a function of  • Adding FC to both sides gives  + FC = P.Q– AVC.Q • Interchanging the sides we obtain P.Q– AVC.Q =  + FC

13. Solving for Desired Sales Level • Q is a factor of both terms on the left so we may write • Q(P – AVC) =  + FC • Dividing through by (P – AVC) gives • Q = ( + FC)/(P – AVC) • If the firm’s accountant can estimate FC, P and AVC, substituting these together with the target level of profit, , gives the desired sales level

14. Linear Equations • Slope of a line: distance up divided by distance moved to the right between any two points on the line • Coefficient: a value that is multiplied by a variable • Intercept: the value at which a function cuts the y axis

15. Representing a Line as y = mx + b • The constant term, b, gives the y intercept • The slope of the line is m, the coefficient of x • Slope = y/x = (distance up)/(distance to right) • Lines with positive slope go up from left to right • Lines with negative slope go down from left to right • Parameter: a value that is constant for a specific function but that changes to give other functions of the same type; m and b are parameters

21. Budget Line • If two goods x and y are boughtthe budget line equation is x.Px + y.Py = M • To plot the line, rewrite as y = M/Py – (Px/Py)x • Slope = – Px/Pythe negative of the ratio of the prices of the goods • Intercept = M/Pythe constant term in the equation

22. The Parameters of a Budget Line • Changing Px rotates the line about the point where it cuts the y axis • If Py alters, both the slope and the y intercept change • the line rotates about the point where it cuts the x axis • An increase or decrease in income M alters the intercept but does not change the slope • the line shifts outwards or inwards  

23. Constant Substitution Along a Line • The rate at which y is substituted by x is constant along a downward sloping line, but not along a curve

24. Diminishing Marginal Rate of Substitution Along an Indifference Curve • Indifference curve: connects points representing different combinations of two goods that generate equal levels of utility for the consumer • Diminishing marginal rate of substitution: as a consumer acquires more of good x in exchange for good y, the rate at which he substitutes x for y diminishes because he becomes less willing to give up y for a small additional amount of x

25. Quadratic Equations • A quadratic equation takes the form ax2 + bx + c = 0 • You can solve it graphically • or sometimes by factorizing it • or by using the formula where a is the coefficient of x2, b is the coefficient of x and c is the constant term

26. Intersection of MC with MR or AVC • Quadratic equations arise in economics where a quadratic function, say marginal cost, cuts another quadratic function, say average variable cost, or cuts a linear function, say marginal revenue • Equate the two functional expressions • Subtract the right-hand side from both sides so that the value on the right becomes zero • Collect terms • Solve the quadratic equation

27. Simultaneous Equations • Simultaneous equations can usually (but not always) be solved if number of equations = number of unknowns

28. Solving Simultaneous Equations • Solution methods for two simultaneous equations include • Finding where functions cross on a graph • Eliminating a variable by substitution • Eliminating a variable by subtracting (or adding) equations • Once you know the value of one variable, substitute it in the other equation

29. Simultaneous Equilibrium in Related Markets • Demand in each market depends both on the price of the good itself and on the price of the related good • To solve the model use the equilibrium condition for each market demand = supply • This gives two equations (one from each market) in two unknowns which we then solve

30. Exponential Functions • Exponential function: has the form ax where the base, a, is a positive constant and is not equal to 1 • The exponential function most used in economics is y = ex • The independent variable is in the power and the base is the mathematical constant e = 2.71828… • Use your calculator or computer to evaluate ex

31. Logarithmic Functions • Logarithm: the power to which you must raise the base to obtain the number whose logarithm it is • Common logarithms denoted log or log10are to base 10 • Natural logarithms denoted ln or loge are to base e and are more useful in analytical work • Equal differences between logarithms correspond to equal proportional changes in the original variables

32. Working with Logarithms • log (xy) = log (x) + log (y) • log (x/y) = log (x) – log (y) • log (xn) = n log (x) • ln (ex) = x • The reverse process to taking the natural logarithm is to exponentiate

33. Solving a Quadratic Equation in Excel • You can enter formulae in Excel to calculate the two possible solutions to a quadratic equation • First calculate the discriminant b2 – 4ac • To make your formulae easier to understand, name cells a, b, const and discrim and use the names instead of cell references • By default, names are interpreted in Excel as absolute cell references • To name a cell, select it, type its name in the Name Box and press the Enter key

34. Plotting and Solving Equations in Excel • Excel includes many inbuilt functions that you can type in or access by clicking the Paste Function button • Those for exponentials and logarithms are =EXP() and =LN() where the cell reference for the value to which the function is to be applied goes inside the brackets

35. Solving Equations with Excel Solver • Excel includes a Solver tool designed to solve a set of equations or inequalities • Set out the data in a suitable format • Interact with the Solver dialogue box to find the solution • Excel does not solve the equations the same way as you do when working by hand • It uses an iterative method, trying out different possible values for the variables to see if they fit the specified requirements