Business Mathematics. Lecture Note #11 Chapter 6-(2)

Transcription

1 Business Mathematics Lecture Note #11 Chapter 6-(2) 1

2 Applications of Differentiation Total functions 1) TR (Total Revenue) function TR = P Q where P = unit price, Q = quantity Note that P(price) is given by the demand function which is a function of Q. i.e. P=f(Q) 2) TC (Total Cost) function TC = FC + VC where FC = fixed cost, VC = variable cost and FC is a constant, and VC is a function of Q 2

3 Applications of Differentiation Marginal functions 1) MR (Marginal Revenue) function MR is the rate of change in TR per unit increase in Q MR = dtr dq 2) MC (Marginal Cost) function MC is the rate of change in TC per unit increase in Q MC = dtc = d(fc+vc) dq dq since FC is a constant = d(vc) dq = MVC dfc dq = 0 3

4 Applications of Differentiation Average functions 1) AR (Average Revenue) function AR is average revenue per unit output (Q) AR = TR Since TR = P Q, AR = TR = P Q = P Q Q Q where P is given by the demand function 2) AC (Average Cost) function AC = TC = FC+VC = FC + VC = AFC + AVC Q Q Q Q where AFC=average fixed cost, AVC=average variable cost 4

5 TR and MR Worked Example 6.6: MR function Given the demand function P = 6-0.5Q, find the value of MR for Q = 1, 2, 3, 4, 5, 6, 7 TR = P Q = (6-0.5Q)Q = 6Q 0.5Q 2 MR = dtr dq = 6 Q Q TR = 6Q-0.5Q MR = 6 Q

6 TC and MC Worked Example 6.8: MC function (a) Given the total cost function TC = Q, (i) Derive an equation for MC. Does MC vary with output Q? (ii) Show that derivative of TC and that of VC with respect to output Q are the same. (i) MC = d(tc) dq = d(10+4q) dq = 4 (ii) VC = 4Q, so MVC = d(vc) dq Therefore, MC = MVC = 4 = d(4q) dq = 4 6

7 MR & AR for a Perfectly Competitive Firm and a Monopolist Worked Example 6.9: MR and AR (a) Given a perfectly competitive firm s demand function P = 20, find MR and AR functions. (b) A monopolist is faced with a linear demand function P = 50-2Q, find MR and AR functions. 7

8 MR & AR for a Perfectly Competitive Firm and a Monopolist Solutions for Worked Example 6.9 (a) For a perfectly competitive firm; P = 20 TR = P Q = 20Q So MR = d(tr) dq = d(20q) dq = 20 and AR = TR Q = = 20Q Q = 20 AR function is always equal to the demand function P. For a perfectly competitive firm, MR function is also equal to the demand function. Therefore, AR = P = MR for a perfectly competitive firm. 8

9 MR & AR for a Perfectly Competitive Firm and a Monopolist Solutions for Worked Example 6.9 (b) For a monopolist; P = 50-2Q TR = P Q = (50-2Q)Q = 50Q - 2Q 2 MR = d(tr) = d(50q 2Q2 ) dq dq = 50 4Q AR = TR 50Q 2Q2 = = = 50 2Q Q Q AR function is always equal to the demand function P. But for a monopolist, the slope of the MR function is twice that of AR. For a monopolist, AR = P MR 9

10 Derive MC from AC Worked Example 6.10 Find MC functions for the following given AC funtions (a) AC = 2Q Q (b) AC = 3Q 2-4Q Q And calculate the value of MC when Q = 50 10

11 Derive MC from AC Worked Example 6.10 Find MC functions for the following given AC funtions (a) AC = 2Q Q AC = TC Q is given TC = AC Q = (2Q Q )Q = 2Q2 + 5Q + 30 MC = d(tc) = d(2q2 + 5Q + 30 ) dq dq = 4Q + 5 So, when Q=50, MC = =

12 Derive MC from AC Worked Example 6.10 Find MC functions for the following given AC funtions (b) AC = 3Q 2-4Q Q is given AC = TC Q TC = AC Q = (3Q2-4Q Q )Q = 3Q 3-4Q 2 + 6Q MC = d(tc) = d(3q3 4Q 2 + 6Q ) = 9Q 2-8Q + 6 dq dq So, when Q=50, MC = 9(50) 2-8(50) + 6 = 22,106 12

13 Production Function and MP L and APL Firms transform inputs(or factors of production) into units of output. Inputs(or production factors): L = labor K = physical capital (buildings, machines) R = raw materials T = technology(including information technology) S = land E = enterprise Outputs: products and services 13

14 Production Function and MP L and APL Production function is the function that illustrates the relationship between inputs and output. Q = f(l, K, R, T, S, E) where Q = quantity of output Production function states that the level (or quantity) of output (Q)is dependent on the quantities of inputs used in the production process. 14

15 Production Function and MP L and APL In the short run, the inputs K, R, T, S and E can be assumed to be fixed so the level of output, Q, becomes the function of labor only. i.e. Q = f(l) MP L (Marginal Product of Labor) is the rate of change in Q(production quantity or total output) with respect to labor(l). MP L = dq dl APL (Average Product of Labor) is the average output per unit of labor. So it is total output(q) divided by the quantity of labor used (L). APL = Q L 15

16 Worked Example 6.11 Production Function and MP L and APL Given the short-run production function Q = 15L 2-0.5L 3 (a) Deduce the equation for MP L. Calculate and comment on the marginal product of labor when 10 units of labor are used. (b) Derive the equation for APL. Calculate and comment on the average product of labor for the first 10 units of labor used. 16

17 Worked Example 6.11 Production Function and MP L and APL (a) Q = 15L 2-0.5L 3 MP L = dq = 30L 1.5L2 dl When L = 10, MP L = 30(10) 1.5(10) 2 = 150 At the point where 10 units of labor are used, the production is increasing at the rate of 150 units of output per additional unit of labor used. 17

18 Worked Example 6.11 Production Function and MP L and APL (b) Q = 15L 2-0.5L 3 APL = Q = 15L2 0.5L 3 = 15L 0.5L 2 L L When L = 10, APL = 15(10) 0.5(10) 2 = 100 Each of the first 10 units of labor used produced on average 100 units of output. i.e. The average productivity per unit of labor is 100 units of output for each of the first 10 units of labor employed. So, the total output of the first 10 units of labor is APL L = = 1000 units of output 18

19 MPC, MPS, APC, APS Income(Y) is defined to be the sum of consumption(c ) and savings(s): Y = C + S MPC(marginal propensity to consume) is defined as the change in consumption per unit change in income MPC = dc dy MPS(marginal propensity to save) is defined as the change in savings per unit change in income MPS = ds dy 19

20 MPC, MPS, APC, APS It is obvious that dy dy = 1, so dy = d(c+s) dy dy = dc dy + ds dy = MPC + MPS = 1 MPC + MPS = 1!!! 20

21 Suppose C = C 0 + by MPC, MPS, APC, APS (i.e. consumption is a linear function of income) Then, MPC = dc = d(c 0+bY) = b dy dy MPS = ds = d(y C 0 by) = 1 - b S = Y C = Y - C dy dy 0 - by APC = C Y = (C 0+bY) Y APS = S Y = (Y C 0 by) Y = C 0 Y + b = 1 - C 0 Y - b Note that MPC + MPS =1 and APC + APS = 1, MPC < APC and MPS > APS 21