Chapter 3 Linear equations: further topics

1. Mathematics for Economics and Business By Taylor and Hawkins Chapter 3Linear equations: further topics

2. Chapter 3:Topics Covered • Solving linear equation with more than two unknowns • Market Equilibrium: Changes in supply and demand • Cost, volume and profit analysis • Macroeconomic Models

3. Solving linear equation with more than two unknowns • A set of simultaneous equations may have: • a unique solution – this occurs when there is just one set of values that satisfies all given equations. • no solution - this occurs when the two (or three lines) are parallel and hence the lines never intersect. • an infinite number of solutions – this occurs when the two or three equations are actually equations of the same line. • Simultaneous equations with more than two unknowns can still be solved using the substitution method. • An example illustrates the techniques best.

4. Solving linear equation with more than two unknowns • Example • x + 12y = 3z = 120 • 2x + y + 2x = 80 • 4x + 2y + 6z = 219 • Find the values of x, y and z that makes each of these equations true. • Step 1: Number each of the equations: • x + 12y + 3z = 120 (1) • 2x + y + 2z = 80 (2) • 4x + 2y + 6z = 219 (3)

5. Solving linear equation with more than two unknowns • Step 2: Choose two of the three equations and multiply one or both of the equations through by a constant that will result in the coefficient if one of the unknowns being the same in both equations. • Equation (1) x 2: • 2x + 24y + 6z = 240 (4) • Step 3: Subtract one equation from another inorder to eliminate one unknown. • Equation (4) – Equation (2): • 2x + 24y + 6z – (2x + y + 2z) = 240 – 80 • 23y + 4z = 160 (5)

6. Solving linear equation with more than two unknowns • Step 4: Using the same technique eliminate the same unknown from the third initial equation that you have not used yet. • Equation (2) x 2: • 4x + 2y + 4z = 160 (6) • Equation (3) – (6): • (4x + 2y + 6z) – (4x + 2y + 4z) = 219 – 160 • 2z = 59 → z = 59/2 • (note the intention here was to eliminate x so we could have combined the result with equation (5). However we have had the added bonus of eliminating y also and hence finding z).

7. Solving linear equation with more than two unknowns • Step 5 would be to use the two new equations in two unknowns to solve for one of the unknowns using the row operation method. • But we can skip Step 5 and move onto Step 6 which is the substitute the value of the first unknown back into either of the equations with just two unknowns to solve for the second unknown. • Substitute z = 59/2 into (5): • 23y + 4(59/2) = 160 → 23y + 118 = 160 → y = 42/23

8. Solving linear equation with more than two unknowns • Step 7: Substitute the values of the two unknowns into any of the initial equations to obtain the value of the third unknown. • Substitute in Equation (1): • x + 12(42/23) + 3(59/2) = 120 • x = 120 – 504/23 – 177/2 = 441/46 • Hence x = 441/46 (=9.59), y = 42/23 (=1.83) and z = 59/2 (=29.5). • You can check your results by using an online simultaneous equation solver such as the one at http://math.cowpi.com/systemsolver/3x3.html or use the techniques outlined in chapter 10.

9. Market Equilibrium:Changes in supply and demand • The quantities of a good demanded or supplied are assumed to be dependent on a number of factors all of which (except price) are assumed to remain constant when we draw supply and demand curves. • Changes in any of the factors other than price (such as income, prices of an alternative good, tastes) will cause a shift or pivot in the demand or supply curve. • Consider the following demand curve: • Qd = 96 – 4P + 0.1M + 0.3Pa • where P represents price, M represents consumers income and Pa represents the price of some alternative good.

10. Market Equilibrium:Changes in supply and demand • In order to find equilibrium we need to make some assumptions about the level of income and the price of the alternative good: • M = £2500 and Pa = £60 • We also need a supply curve: • QS = 6P • Equilibrium is where QD = QS. • 96 – 4P + 0.1(2500) + 0.3(60) = 6P • 96 + 250 + 18 = 10P • P = 364/10 = 36.4 and hence Q* = 6(36.4) = 218.4 • Consider this situation graphically.

11. Market Equilibrium:Changes in supply and demand

12. Market Equilibrium:Changes in supply and demand • Now consider a changes in tastes that increases demand for the good by 25%. The new demand curve is then: • 1.25 x Qd = 96 – 4P + 0.1M + 0.3Pa = 1.25 x (364 – 4P) • Qd’ = 455 – 5P • The new equilibrium would be: • 455 – 5P = 6P • 455 = 11P → P = 41.36 and so Q* = 6(41.36) = 248.16 • The chart shows the new equilibrium, note how the demand curve has pivoted outwards.

13. Market Equilibrium:Changes in supply and demand

14. Market Equilibrium:Changes in supply and demand • Now consider the situation in which instead of a percentage increase in demand, the consumers income is increased by 40%. • The new demand equation would be: • Qd’’ = 96 – 4P + 0.1M x 1.4 + 0.3Pa = 96 -4P + 0.1(2500)(1.4) + 0.3(60) • Qd’’ = 96 – 4P + 350 + 18 = 464 – 4P • The new equilibrium would be: • 464 – 4P = 6P • 464 = 10P → P = 46.4 and so Q* = 6(46.4) = 278.4 • The chart shows the new equilibrium, note how the demand curve has shifted outwards.

15. Market Equilibrium:Changes in supply and demand

16. Market Equilibrium:Changes in supply and demand • Information on quantity supplied is based on the prices that suppliers receive. • If a per unit tax is imposed, buyers will still pay a price of P per unit, but suppliers receive a price of only P-t. • Therefore a tax changes the supply equation and causes it to shift to the left. • Previously: • No tax: QS = 6P • With Tax: QS = 6(P – t) • If the tax is £2 then: • QS = 6(P – 2) = -12 + 6P

17. Market Equilibrium:Changes in supply and demand • Consider the impact on the equilibrium following the imposition of a per unit tax of £4.50. • Previously: • Qd = 96 – 4P + 0.1M + 0.3Pa • where M = £2500 and Pa = £60. • Qs = 6P • and now Qs =6(P - t) = 6(P – 4.5) = 6P – 27 • Equilibrium is where QD = QS. • 96 – 4P + 0.1(2500) + 0.3(60) = 6P – 27 • 10P = 250 + 18 + 96 + 27 = 391 • P = 39.1 and hence Q = 6(39.1 – 4.5) = 207.6 • See diagram and note how supply curve shifts.

18. Market Equilibrium:Changes in supply and demand

19. Cost, volume and profit analysis • Consider the following definitions: • Profit (P) = Total Revenue (TR) – Total Cost (TC) • TR = Price (P) x Quantity (Q) • Total Cost (TC) = Fixed Cost (FC) + Variable Cost (VC) • Average Variable Cost (AVC) = Variable Cost (VC)/Quantity (Q) • Variable Cost (VC)= Average Variable Cost (AVC) x Quantity (Q)

20. Cost, volume and profit analysis Example • A firm has FC= £555, AVC = £12 and is selling a good at a price of £17 per unit. • Find an expression for profit in terms of level of sales, Q. • What is the value of Q that will achieve a profit of £195? • At what sales level does the firm break even? • Plot the TR and TC functions to show the breakeven point.

21. Cost, volume and profit analysis • (a) Find an expression for profit in terms of Q. • P = TR - TC = P.Q – (FC + VC) • P = P.Q – (FC + AVC x Q) • P = 17Q – (555 + 12xQ) • P = 5Q – 555 • (b) What is the value of Q that will achieve a profit of £195? • P = 5Q – 555 = 195 → Q = (195+555)/5 = 150 • (c) At what sales level does the firm break even? • P = 5Q – 555 = 0 → Q = (0+555)/5 = 111 • (d) Plot the TR and TC functions to show the breakeven point. • TR = PQ, TC = FC + VC = 555 + 12Q • See diagram.

22. Cost, volume and profit analysis

23. Macroeconomic Models • Consider the national income identity: • Y = C + I + G + (X – M) • where: • Y = National Income • C = Consumption • I = Investment • G = Government Expenditure • X = Exports • M = Imports • This identity is best illustrated with an example

24. Macroeconomic Models • Example • C = 0.6Y + £120m • I = £45m • G = £634m • X = £160m • M = 0.45Y + £8m • Y = C + I + G + X – M • Y = 0.6Y + 120 + 45 + 634 + 160 – (0.45Y + 8) • Y = 0.15Y + 951 • Y(1-0.15Y) = 0.85Y = 951 → Y = £1118.82m. • This can also be illustrated graphically. • Note that the 45 degree lines shows all points were expenditure and income are equal.

25. Macroeconomic Models

26. Macroeconomic Models • Now let us consider some of the elements that male up this identity. • The consumption function was given as: • C = 0.6Y + £120m • This has two components: • £120 = autonomous consumption, that level of consumption that would occur even if income was zero. • 0.6Y = the part of consumption that is proportional to income. The 0.6 is referred to as the Marginal Propensity to Consume (MPC). • So if Income was £100 then £60 would be consumed and the remainder, £40, saved. Hence the Marginal Propensity to Save (MPS) is 0.4. • MPS + MPC = 1

27. Macroeconomic Models Example • Consider what happens to the last example if the MPC increases to 0.7. • Y = C + I + G + (X – M) • Y = 0.7Y + 120 + 45 + 634 + 160 – (0.45Y + 8) • Y = 0.25Y + 951 • Y(1-0.25Y) = 0.75Y = 951 → Y = £1268m. • This can also be illustrated graphically. Note how the National Income Identity has pivoted upwards.

28. Macroeconomic Models

29. Macroeconomic Models • (X-M) represents the trade identity. If (X-M) > 0 then the value of exports is larger than the value of imports and the country is said to have a trade surplus. • If (X-M) < 0 then the value of exports is less than the value of imports and the country is said to have a trade deficits. • In our example the Import function was presented in two parts (M = 0.45Y + £8m). • This is similar to the consumption function where we have a autonomous imports (£8m) and a level of imports which is proportional to income. This is the Marginal Propensity to Import (MPM).

30. Macroeconomic Models Example • Consider what happens to the last example if as a result of a Government “Buy British” campaign the MPM decreases to 0.35. • Y = C + I + G + (X – M) • Y = 0.6Y + 120 + 45 + 634 + 160 – (0.35Y + 8) • Y = 0.25Y + 951 • Y(1-0.25Y) = 0.75Y = 951 → Y = £1268m. • This is exactly the same outcome as the increase in the MPC.

31. Macroeconomic Models Example • Consider what happens to the last example if as a result of an increase in world income exports double to £320m. • Y = C + I + G + (X – M) • Y = 0.6Y + 120 + 45 + 634 + 320 – (0.45Y + 8) • Y = 0.15Y + 1121 • Y(1-0.15Y) = 0.85Y = 1121 → Y = £1318.82m. • This can also be illustrated graphically. Note how the National Income Identity has shifted outwards upwards.

32. Macroeconomic Models