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Linear Functions Application of Linear Equations 1. Keynesian Consumption Function Keynesian Consumption Function is a single mathematical function used to express consumer spending. The function is used to calculate the amount of total consumption in an economy. Keynesian consumption Function is also known as the Absolute Income Hypothesis as its only bases consumption on current income and ignores potential future income. The function can be written in variety ways: C a b(Y T ) C c c Yd 0 1 C=total consumption c =Autonomous consumption ( c 0 ) or the level of consumption that would still exist even 0 0 if income was $0. In estimation, this is always assumed to be positive. C1 =marginal propensity to consume( MPC)(which is a ratio of consumption changes to income changes) (0 C1 1) MPC measures the rate at which consumption is changing when income is changing. Geometrically, MPC is the slope of the consumption function. MPC is also assumed to be positive. Y d =disposable income (e.g. income after government intervention e.g. benefits, taxes, transfer payments or Y (G T ) ) Example on the study guide: If given: C 4141 0.78Y 1 C=consumption, Y=National Income To find (MPC) marginal propensity to consume MPC DC DY Dc C C 1 0 C originalconsumption 0 C consumption 1 Marginal Propensity to save: MPS 1 MPC If we increase income by N$1 for example then the new consumption will be: C 4141 0.78 Y 1 1 0 MPC=additional consumption from additional dollar of disposable income MPS=additional saving from an additional dollar of disposable income INCOME DETERMINATION MODELS Income determination models generally express the equilibrium level of income in a four-sector economy as: Y C I G (X Z) Y income C consumption I investment G government expenditure X exports Z imports 2 Example: Assume a simple two-sector economy where: Y C I, C C bY and I I 0 0 Assume further that: C 85, b 0.9 and I 55. 0 0 The equilibrium level of income can be calculated in terms of: The general parameter: Y C I and substituting C and I in the general parameter, Y C0 bY I0 we get the reduced form, e.g. Solving for Y: Y bY C I 0 0 Y 1 b C I 0 0 C I Y 0 0 1 b The term 1 is called the autonomous expenditure multiplier in economics’ 1 b It measures the multiple effect each dollar of autonomous spending has on the equilibrium level of income. Since b MPC in the income determination model, the multiplier= 1 1 MPC . IS-LM ANALYSIS: Seeks to find the level of income and the rate of interest at which both the commodity market and the money market will be in equilibrium. NB: We use the technique of solving equations simultaneously. EXAMPLE 1: The commodity market for a simple two – sector economy is in equilibrium when 3 Y C I. The money market is in equilibrium when the supply of money M s equals the demand for d which in turn is composed of the transaction – precautionary demand for money money M Mt and the speculative demand for money M z . Assume a two – sector economy where C 48 08Y , I 98 75i, M s 250, Mt 0.3Y and M z 52 150i . We can calculate the level of income Y and the rate of interest i that will concurrently bring equilibrium to the economy as follows: Commodity equilibrium IS exists when: Y CI Y 48 0.8Y 98 75i Y 0.8Y 146 75i 0.2Y 75i 146 0 Monetary equilibrium ( LM ) exists when M s Mt M z 250 0.3Y 52 150i 0.3Y 150i 198 0 Solving simultaneously: 0.4Y 150i 292 0.3Y 150i 198 0 6 Y 700 , i 0.08 75 The commodity and money market will be in simultaneous equilibrium when Y 700 and i 6 0.08 75 By substituting the available values in the given equations, one can obtain the values as follows: 4 C 48 0.8(700) 608 I 98 75(0.08) 92 M t 0.3(700) 210 M z 52 150(0.08) 40 C I 608 92 700 And Mt M z 210 40 250 M s Example 2: Find: (a) the level of income and the rate of interest that concurrently bring equilibrium to the economy and (b) estimate the level of consumption C, investment I, the transaction – precautionary demand for money M t , and the speculative demand for money M w ,when the money supply M s =300 and C 102 0.7Y , I 150 100i , Mt 0.25Y , M w 124 200i Work out: INCOME DETERMINATION MODELS EXAMPLE 1: (a) Find the reduced –form equation and (b) solve for the equilibrium level of income 1. directly and 2. with the reduced form equation, if given: Y C I G C C bY 0 I I G G 0 0 Where C 135, b 0.8, I 75, and G 30 0 0 0 5 From Y C I G (a) Substituting the given values and solving for Y in terms of the parameters and exogenous variables, C0 , I 0 and G0 Y C0 bY I 0 G0 Y bY C0 I 0 G0 Y (1 b) C0 I 0 G0 Y C0 I 0 G0 (1 b) This is the reduced form. (b) 1. Y C I G Y 135 0.8Y 75 30 Y 0.8Y 240 0.2Y 240 Ye 1200 b. (2) C0 I 0 G0 (1 b) 135 75 30 Y (1 0.8) 240 Y 1200 0.2 Y 4.1.6 Find the equilibrium level of income Ye , given Y C I G when C 125 0.8Y , I 45 and G 90 We know that the reduced form equation is: Y C0 I 0 G0 (1 b) 6 Given that C 125 0.8Y and we also know that C C0 bY Then 1 Ye C I G 0 1 b 0 0 1 Ye 125 45 90 1 0.8 Ye 1300 4.17 Find (a) the reduced form (b) the value of the equilibrium level of income Ye , and (c) The effect on the multiplier, given the following model in which not autonomous but a function of income. investment is Y CI C C bY I I aY 0 0 and C 65, I 70, b 0.6, and a 0.2 0 0 4.19 Find (a) the reduced form (b) the numerical value of Ye and (c) the effect on the multiplier if an income tax T with a proportional component t is incorporated into the model, given Y CI C C bY 0 d T T tY 0 And C 85, I 30, T 20, b 0.75 and t 0.2 0 0 0 7 Y Y T d I I 0 ECONOMIC APPLICATION OF GRAPHS AND EQUATIONS: ISOCOST LINES Represent the different combinations of two inputs or factors of production that can be purchased with a given sum of money. The general formula is: P K P L E, L k K Capital L Labor P CapitalPrice k P Labor Price L E=amount allotted(agreed to) to expenditures In ISOCOST analysis: The individual price and the expenditure are initially held constant. Only the different combinations of inputs are allowed to change. P K P LE L k P K EP L L k EP L L K P k E PL K L P P K k This is the familiar linear function to the form: y mx c where c P E E PL E in K L K L L P P P P P K K k K k P P k m L is the slope. 8 From the graph, the effects of a change in any one of the parameters are easily noticeable. An increase in the expenditure from E to E will increase the vertical intercept and cause the ISOCOST line to shift out to the right but parallel to the old line, Change in P will change the slope but not C. Change in P will change slope and C. L k Example 1: A person has N$120 to spend on two goods ( X , Y ) whose respective prices are N$3 and N$5. (a) Draw a budget line showing all the different combinations of the two goods can be bought with the given budget (B). SOLUTION: The general function for a budget line is: 9 that Px X PyY B If Px 3 and Py 5 and B 120 then 3 X 5Y 120 Tograph,weput instandardform 3 Y 24 X 5 coordinates : 0;24 and 40;0 What happens to the original budget line if: (b) the budget falls by 25%? Solution: 10 75% 120 90 3 X 5Y 90 3 y x 18 5 coordinates : 0;18 (30;0) Lowering the budget causes the budget line to shift parallel to the left. (c) If the price of X doubles? Px 3 2 6 X 5Y 120 6 Y X 24 5 coordinates : 0;24 20;0 11 Vertical intercept remains the same. Slope changes and becomes steeper. With a higher price of X, less X can be bought within given budget. (d) If the price of Y falls to 4? 3 X 4Y 120 3 y 30 X 4 coordinates: 0;30 (40;0) 12 With a change in Py both the vertical intercept and the slope change. Example 2: Either coal (C) or gas (G) can be used in the production of steel. The coast of coal is 100, the cost of gas is 500. Draw an isocost curve showing the different combinations of gas and coal that can be purchased. GRAPHS IN THE INCOME DETERMINATION MODEL MACRO-ECONOMIC MODEL Generally express the equilibrium level of income in a four-sector economy as: Y C I G ( X Z ), where Y income C consumption I investment G Government exp enditures X exp orts M or Z imports 3.4 NON-LINEAR FUNCTIONS- TYPES OF NON-LINEAR FUNCTIONS: 13 3.5 Application of Quadratic functions to economic theory Use of factorization, graphical method, quadratic formula, completion of squares: Cobb-Douglas production function COBB-DOUGLAS PRODUCTION FUNCTION Optimization of Cobb-Douglas production function: q AK L ( A 0; 0 , 1) q quantity of outputinphysicalunits K quantity of capital L quantity of labor (the output elasticity of capital) Elasticity is………. measures the % change in q for a 1% change in k while L is held constant. (the output elasticity of labor) is exactly parallel A= efficiency parameter reflecting the level of technology A strict Cobb – Douglas function, in which 1 , exhibit constant returns to scale. A generalized Cobb – Douglass function in which ( 1) exhibits increasing returns to scale if 1 and decreasing returns to scale if 1. 14 QUESTIONS FROM PAST EXAMS PAPERS 1. The market for a 6-pack of Aquafina bottled water (manufactured by Pepsi) is composed of the following demand and supply equations respectively: QD 120 40 P QS 60 50 P The calculated equilibrium price and quantity will be: 2. 3. A. P N $2; Q 40units C. P N $1.50; Q 60units B. P N $10; Q 18units D. P N $1.00; Q 50units Consider the relationship between total profit (π) and output (Q) is given by the function f (Q) . In this function, Q is described as the A. Dependent variable B. Independent variable C. Slope D. Intercept Suppose the production of flat screen televisions is represented by the following production function Q of production output is 0.4 AL K Q1 units. 0.4 . The manufacturer’s new level If all the inputs are doubled, the new level of output will be equal to: A. 4. Q1 20.4 Q B. Q1 20.8 Q C. Q1 0.8Q D. Q1 1.6Q The following is a national income function for a four-sector economy at the equilibrium level of income: Y C I G X Z It has the following equations to describe the above four-sector economy Y income, C consumption, I investment , Where: G government exp enditure, X Z net exports 15 For the IS sector C 900 0.80YD ; YD Y T ; T 250 0.25Y I 220 40i; G 1100 and ; X Z 800 where T taxes and YD disposable income For the LM sector M d 600 0.85i and M s 596 4i where M d =money demanded and M s =money supplied 4.1 Calculate the level of income Y and the rate of interest i that will concurrently bring equilibrium to the economy at equilibrium Y C I G X Z and M s M d Y 900 0.8 Y 250 0.25Y 220 40i 1100 800 Y 900 0.8Y 200 0.2Y 220 40i 1100 800 0.4Y 40i 1220 eqn1 Ms Md 600 0.85i 596 4i i 0.824742268 substitute i into eqn1 0.4Y 40(0.824742268) 1220 0.4Y 1187.010309 Y 2967.525775 i 0.824 82.4% and Y N $2967.53 4.2 At the equilibrium level of income and interest rate computed in 4.1, determine the new level of consumption, 16 Ce Ce 900 0.8 Y 250 0.25Y Ce 700 0.6Y Ce 700 0.6 2967.53 Ce N $2480.52 4.3 Evaluate the MPC at this new level of income Ce 900 0.8 Y 250 0.25Y Ce 700 0.6Y MPC 5. dC 0.6 dY A Cobb-Douglas production output function for Tangeni Holdings is given as 1 3 2 3 Q1 AK L 5.1 Determine the cost of production output Q1 if the cost of technology is N$55, cost of capital is N$105 and that of labour is 30% the cost of capital. 1 3 2 3 Q1 AK L 1 3 Q1 55(105) (31.5) 2 3 2588.0111241 cost of production N $2588.01 5.2 Show that the new level of production will be Q2 3.18Q1 If the cost of capital is doubled and the cost of labour is quadrupled 17 1 3 2 3 Q1 AK L now 1 3 Q2 A(2 K ) (4 L) 2 3 1 2 2 4 AK 3 L3 1 2 3.17480 AK 3 L3 Q2 3.175Q1 1 3 5.3 2 3 Determine the partial derivative of 1 3 Q1 keeping K constant, that is, determine Q1 L 2 3 Q1 AK L 1 Q1 2 13 3 A K L L 3 1 1 Q1 2 3 AK L 3 L 3 6. The demand for Levi’s blue jeans depends on the own price of the blue jeans, on consumer income, I and on the level of advertising, A, done by Levi Strauss & Co. The demand function is given as A where p is the price of the Levi’s blue jeans, I is consumer 2 income and A is the advertising budget of the company. The supply of the blue jeans by the company is given by the function Qd 100 p 3I Qs 40 2 p Find the equilibrium price and quantity for the blue jeans sold on the market if I $20; A $60 . 18 7. An economy has the following prevailing conditions: C 89 0.6Y ; I 120 150i ; M t 0.1Y ; M w 240 250i ; M s M t M w ; where M s 275 Note that C is the level of consumption, I the investment, Mt the transaction precautionary demand for money and Mw the speculative demand for money. Determine: 7.1 the level of income and the rate of interest that concurrently bring equilibrium to the economy Y CI M D MS Mt Mw and Y 89 0.6Y 120 150i 275 0.1Y 240 250i 0.4Y 150i 209....1 0.1Y 250i 35.........2 solve simult Y N $500 i 0.06 6% 7.2 the levels of C , I , M t , and M w when M s 275 , at the equilibrium level of income and interest rate computed in 7.1 8. The monthly revenue R obtained by selling a particular blend of tea is a function of the demand x in the market. It is observed that, as a function of price p per packet, the monthly revenue and monthly demand are given respectively as R 300 p 2 p 2 8.1 ; x 300 2 p Show that R 150 x (Hint: Express p x2 . 2 in terms of x and then substitute in the correct equation) 19 8.2 Copy and complete the following table for R 150 x x 100 120 140 150 160 x2 . 2 180 200 R x 8.3 At what value of is the revenue the highest? 8.4 At what price will the revenue stand at the highest level? 9. Voltron manufactures flat screen televisions and the weekly production cost of the televisions is given by: C 22 x 750 When x units are produced per week. The advertising costs, C A necessary to sell these x units per week is given by CA 18 x 0.1x3 750 , And the weekly revenue R resulting from the sale of these x units is given by R 340x 2.25x2 9.1 Determine the optimum production level, i.e. production level at which maximum profit occurs and hence determine the maximum weekly profit 9.2 Determine the level of output at which the advertising cost per unit is minimum, and thereafter calculate this minimum average cost. 10. Sarah makes customized dresses for a select clientele. In the relevant range of her production, she can produce dresses according to the production function Q 0.2M 0.67 L0.33 20 Where and Q her output per week is, L is the number of hours each week that she works M is the amount of material she uses per week. Sarah values her labour at N$3 per hour and pays N$6 per unit of materials. She is currently working 20 hours per week and using 50 units of material per week. 10.1 Determine how many dresses a week Sarah is producing. Q 0.2(50)0.67 (20)0.33 Q 7.39 7dresses / week 10.2 Calculate how much it costs Sarah to make one dress. TotalCost 3 20 6 50 N $360 / dress 10.3 Compute the MRTS for Sarah’s dressmaking business. MRTS MPL MPK 0.2 0.33 L0.67 M 0.67 0.2 0.67 L0.33 M 0.33 K 0.493 L MRTS 11. 12. Consider the relationship between total profit (π) and output (Q) is given by the function f ( ) . In this function, is described as the A. Dependent variable B. Independent variable C. Slope D. Intercept The market for a 6-pack of Aquafina bottled water (manufactured by Pepsi) is composed of the following demand and supply equations respectively: QD 120 40 P QS 60 50 P 21 The calculated equilibrium price and quantity will be: 13. A. P N $2; Q 40units C. P N $1.50; Q 60units P N $10; Q 18units D. P N $1.00; Q 50units The IS curve shows all combinations of income and A. B. C. D. 14. B. interest rate for which the goods market is in equilibrium interest rate for which the money market is in equilibrium price level for which the goods market is in equilibrium price level for which the money market is in equilibrium The following is a national income function for a four-sector economy at the equilibrium level of income: Y C I G X Z It has the following equations to describe the above four-sector economy Y income, C consumption, I investment , Where: G government exp enditure, X Z net exports For the IS sector C 950 0.85YD ; YD Y T ; T 200 0.20Y I 220 40i; G 1100 and ; X Z 800 where T taxes and YD disposable income For the LM sector M d 600 - 0.80i and M s 590 5i where Md = money demanded and Ms = money supplied 14.1 Calculate the level of income Y and the rate of interest i concurrently bring equilibrium to the economy 22 that will Y C I G X Z 0.32Y 40i 1300 Ms Md 600 0.8i 5905i i 1.724 Y N $3847.50 14.2 At the equilibrium level of income and interest rate computed in 14.1, determine the new level of consumption, Ce 14.3 Evaluate the MPC at this new level of income 15. An income tax T with a proportional component t is incorporated into the model, given Y CI C C bY 0 d T T tY 0 Y Y T d I I 0 And C 85, I 30, T 20, b 0.75 and t 0.2 0 0 0 Find 16. (a) the reduced form (b) the numerical value of Y and (c) the effect on the multiplier An economy has the following prevailing conditions: C 89 0.6Y ; I 120 150i ; M t 0.1Y ; M w 240 250i ; M s M t M w ; where M s 275 23 Note that C is the level of consumption, I the investment, Mt the transaction precautionary demand for money and Mw the speculative demand for money. 16.1 Determine: 16.1.1 The level of income and the rate of interest that concurrently bring equilibrium to the economy 16.1.2 The levels of C , I , M t , and M w when M s 275 , at the equilibrium level of income and interest rate computed in 2.1.1 FROM NOV 2014 1. Sarah makes customized dresses for a select client. In the relevant range of her production, she can produce dresses according to the production function Q 0.2M 0.67 L0.33 where and Q her output per week is, L is the number of hours each week that she works M is the amount of material she uses per week. Sarah values her labour at N$3 per hour and pays N$6 per unit of materials. She is currently working 20 hours per week and using 50 units of material per week. 1.1 Determine how many dresses a week Sarah is producing. 1.2 Calculate how much it costs Sarah to make one dress. 1.3 Compute the MRTS for Sarah’s dressmaking business. 2. A fishing company has a total revenue function given as R( x) 280 x 2000 And their total cost function is given as C ( x) 60 x 5600 24 Where x represents the number of units of fish sold 2.1 If there is no production of fish what amount of costs will the company incur? 2.2 If 1500 units were produced, what would be the company’s total revenue? 2.3 Calculate the total profit of this company from the production of 1500 units of fish. 3. The national income function for a four-sector economy at equilibrium level of income has the following equations to describe the above four-sector economy Y income, C consumption, I investment , Where: G government exp enditure, X Z net exports For the IS sector Where Y C I G X Z C 900 0.80YD ; YD Y 250 I 220 40i; G 1100 and ; X Z 800 where i interest rate; YD disposable income For the LM sector Where Ms Md M d 600 0.85i and M s 596 5.85i 3.1 Write down the national income function for this four-sector economy 3.2 Calculate the level of income Y and the rate of interest i concurrently bring equilibrium to the economy 3.3 What is the level of consumption (𝐶) at equilibrium? 25 that will