Business Mathematics. Lecture Note #13 Chapter 7-(1)

Transcription

1 1 Business Mathematics Lecture Note #13 Chapter 7-(1)

2 Applications of Partial Differentiation 1. Differentials and Incremental Changes 2. Production functions: Cobb-Douglas production function, MP L, MP K, APL and APK, Isoquant 3. Returns to scale 4. Utility functions 5. Partial elasticities 6. Multipliers

3 Differentials and Incremental Changes Notations: dx = differential of x, dy = differential of y y = incremental(small) change in y due to x when y = f(x) Since the rate of change of y w.r.t. x is y = y y = y x x x x As x becomes infinitesimal, x dx and y dy So the differential of y may be written as dy = dy dx eq(1) dx where dy = differential of y dy dx = derivative of y w.r.t x dx = differential of x

4 Differentials and Incremental Changes Incremental change is a small change in the dependent variable (y), that results from a small changes in the independent variable(s). Formula for incremental change can be derived from eq(1) where the differentials dx and dy are replaced by x and y for small changes but not infinitesimally small changes in x and y y dy dx x This is called the small changes formula or incremental changes formula which gives the approximate change in y as a result of a small change in x.

5 Differentials and Incremental Changes Worked Example 7.5: Differentials and Incremental Changes for functions of one variable Supply function is given by the eq. P=Q 2 (a) Find the derivative of P. (b) Find the differential of P. (c) If Q is increased by 3%, find the approximate change in P.

6 Differentials and Incremental Changes Worked Example 7.5: Differentials and Incremental Changes for functions of one variable (a) Derivative of P is simply ordinary differentiation. Since P = Q 2, dp dq = 2Q (b) Differential of P: dp = dp dq = (2Q)dQ dq (c) The incremental change in P, i.e. P, when Q is increased by 3%: Since Q = Q P dp dq Q = (2Q)(0.03Q) = 0.06Q2 = 0.06P So, there is a 6% increase in P when Q is increased by 3%.

7 Differentials and Incremental Changes Differentials and Incremental Changes for functions of two variables If z = f(x, y), the total differential of z is written as dz = f x dx + f y dy And, for small changes in x and y, i.e. for x and y, the incremental change in z is written as z = f x x + f y y

8 Differentials and Incremental Changes Worked Example 7.6a: Differentials and Incremental Changes for functions of two variables Total revenue function is given by the eq. TR = 5W 2 A 3 where W = the payment on wage A = amount of advertising expenditure (a) Find the differential of TR. (b) Find the approximate % change in TR if i) only W is increased by 5% and no change in A. ii) W is increased by 5% and A is decreased by 2%.

9 Differentials and Incremental Changes Worked Example 7.6a: Differentials and Incremental Changes for functions of two variables (a) Find the differential of TR. dtr = TR TR dw + da = (5W2 A 3 ) dw + (5W2 A 3 ) da W A W A = 10WA 3 dw + 15W 2 A 2 da (b) Find the approximate % change in TR if i) only W is increased by 5% and no change in A. Since W = 5 W and A = TR = TR TR W + W TR = A = A 10WA3 W + 15W 2 A 2 A 5 W + 0 = 100 5W2 A = TR TR This means TR has increased by 10% = 10WA 3

10 Differentials and Incremental Changes Worked Example 7.6a: Differentials and Incremental Changes for functions of two variables (b) Find the approximate % change in TR if ii) W is increased by 5% and A is decreased by 2%. Since W = 5 2 W and A = A TR = TR TR W + A = W A 10WA3 W + 15W 2 A 2 A = 10WA 3 5 W W2 A 2 2 A 100 = 5W 2 A W2 A = TR = TR + TR TR = 4 TR This means TR has increased by 4% 100

11 Differentials and Incremental Changes Worked Example 7.6b: Incremental Changes for functions of two variables The following function is the Cobb-Douglas production function Q = 10L 0.7 K 0.3 where Q = quantity produced L = the number of units of labor used K = amount of capital invested Calculate the approximate % change in Q if L is increased by 5% and K decreased by 2%.

12 Differentials and Incremental Changes Worked Example 7.6b: Incremental Changes for functions of two variables Since Q = 10L 0.7 K 0.3, L = 5 2 L and K = K Q = Q L L + Q K K = 7L 0.3 K 0.3 = 10L 0.7 K L0.7 K = Q Q = 2.9 Q 100 = Q L L0.7 K K 100 Q = 2.9 Q This means Q has increased by 2.9% 100

13 Production Functions A general production function which describes output Q(production quantity) as a function of L(labor) and K(capital). Q = f(l, K) Cobb-Douglas production function which is widely used in economic analysis is Q = AL α K β where A, α and β are constants, and A > 0, 0 < α < 1, 0 < β < 1 L and K are positive variables A specific Cobb-Douglas production function would be Q = 50L 0.4 K 0.6

14 Production Functions: MP L and MP K Two Marginal functions of Q: MP L and MP K when production function is given as Q = AL α K β 1. Marginal Product of Labor (MP L ) and its deirvative MP L = Q L = Q L = AαLα 1 K β > 0 MP L = Q LL = MP L = 2 Q = Aα(α L L 2 1)Lα 2 K β = (α 1) αalα K β = α(α 1) Q < 0 L 2 L 2 2. Marginal Product of Capital (MP K ) and its derivative MP K = Q K = Q K = AβLα K β 1 > 0 MP K = Q KK = MP K = 2 Q = Aβ(β K K 2 1)Lα K β 2 = (β 1) βalα K β = β(β 1) Q < 0 K 2 K 2

15 Production Functions: MP L and MP K Two Marginal functions of Q: MP L and MP K when Q = AL α K β 1. Marginal Product of Labor (MP L ) and its deirvative MP L > 0 when K is held constant, Q increases as L increases MP L < 0 when K is held constant, Q increases as L increases but at a decreasing rate (because MP L < 0, MP L decreases as L increases though MP L is positive) Law of diminishing marginal returns to labor 2. Marginal Product of Capital (MP K ) and its derivative MP K > 0 when L is held constant, Q increases as K increases MP K < 0 when L is held constant, Q increases as K increases but at a decreasing rate (because MP K < 0, MP K decreases as K increases though MP K is positive) Law of diminishing marginal returns to capital

16 Production Functions Second mixed partial derivative of Q: Q LK = Q K L when Q = AL α K β Q LK = Q (= MP L K L K = MP K L ) = AαβLα 1 K β 1 > 0 This implies (i) MP L increases as K increases and (ii) MP K increases as L increases Therefore, (i) MP L increases as K increases, but decreases as L increases (ii) MP K increases as L increases but decreases as K increases.

17 Relationships between: MP L and APL, MP K and APK MP K and APK are defined similarly as MP L and APL were defined in chapter 6. i.e. MP K = Q K and APK = Q K When Q = AL α K β (i.e. Cobb-Douglas production function) APK = Q K = ALα K β 1 MP K = Q K = AβLα K β 1 = β APK So, MP K < APK (since 0 < β < 1) The same thing holds for labor(l) as you can see below: APL = Q = L ALα 1 K β MP L = Q = L AαLα 1 K β = α APL So, MP L < APL (since 0 < α < 1)

18 Production Conditions Normally, a producer would desire productivity to increase as the amount of each input increases, as indicated by positive marginal product(q =MP) functions. However, the rate of increase usually slows down as the amount of inputs become progressively larger, indicated by negative second derivatives(q =MP ). In practice, a firm produces output over a certain range of the production function outlined by the following production conditions. (1) Conditions for using labor (L): MP L = Q L = Q L > 0, MP L L = Q LL = 2 Q L 2 < 0 and MP L < APL (2) Conditions for using capital (K): MP K = Q K = Q > 0, MP K = Q K K KK = 2 Q < 0 and MP K 2 K < APK

19 Production Function, Isoquant and MRTS Let a production function be Q = f(l, K) The graph of this production function cannot be plotted on a two dimensional diagram because the function has two variables. Instead we can plot the isoquant as follows. To plot isoquant as a two dimensional graph, fix Q at Q 0 which is a constant. i.e. Q 0 = f(l, K) Then K can be expressed as a function of L. i.e. K=g(L). This isoquant will give all the combinations of L and K that will produce the same fixed value of Q which is Q 0. (refer to Figures 7.2 and 7.3 on the following pages) The slope dk dl can be derived directly from the equation of the isoquant. The this slope is called the marginal rate of technical substitutions(mrts).

20 Production Function, Isoquant and MRTS At any point on an isoquant, the value of the slope, dk dl, is a measure of the decrease in capital (K) for each unit increase in labor (L), i.e., the number of units of capital (K) which would be replaced by one unit of labor(l) increased, while still maintaining the same output, Q. ( Figure 7.5) In Figure 7.5 we see that dk < 0 and d2 K dl dl2 > 0 which exhibits a diminishing marginal rate of technical substitutions(mrts). Since the absolute value of dk ΔK dl ΔL is decreasing, the rate at which the amount of capital(k) decreases for each unit increase in labor (L)is getting smaller as L increases. This is referred to as diminishing MRTS.

21 The slope of an Isoquant: MRTS The slope of an isoquant, dk in terms of MP L and MP K dl, i.e. MRTS, can be expressed The total differential of Q is dq = Q L Along any isoquant, dq = 0, dq = Q L Q L dl = Q K dk Q L = Q K dk dl dl + Q K dk dl + Q K dk = 0 Q = Q L K dk dl so, MRTS = dk dl = Q L Q K = MP L MP K

22 The slope of an Isoquant: MRTS The slope of an isoquant, dk, i.e. MRTS, for a Cobb-Douglas dl production function Q=AL α K β From the previous page MRTS = dk dl = MP L MP K Now, MP L = Q L = AαLα 1 K β and MP K = Q K = AβLα K β 1 So, MRTS = MP L = AαLα 1 K β αk MP K AβL α Kβ 1 = βl

23 Graphs of the Isoquants Q=5LK, for Q=5, 10, 15, 18 i.e. K= 1 L, K=2 L, K=3 L, K=3.6 L

24 Graphs of the Isoquants Q=5LK, for Q=5, 10, 15, 18 i.e. K= 1 L, K=2 L, K=3 L, K=3.6 L

25 Graphs of the Isoquants

26 Returns to Scale If both inputs L and K in the Cobb-Douglas production function are changed by the same proportion (for example, each input is increased by 50% or each input is reduced by 20%, etc.), we can easily determine the proportionate change in output. For some constant, r, if L is replaced by rl and K is replaced by rk, then the proportionate change in Q may be determined from the equation of the production function. Given the Cobb-Douglas production function Q 1 = AL α K β If L rl and K rk, how would Q change? Q 2 = A(rL) α (rk) β = r α+β A L α K β = r α+β Q 1

27 Returns to Scale Q before the change: Q 1 = AL α K β Change in inputs: L rl and K rk, Q after the change: Q 2 = A(rL) α (rk) β = r α+β A L α K β = r α+β Q 1 If α + β < 1, then the proportionate change in output is less than the proportionate change in each input. This is described as decreasing returns to scale. For example, if r = 2 and α + β = 0.8, then Q 2 = r α+β Q 1 = Q 1 = 1.741Q 1 < rq 1 If α + β = 1, then the proportionate change in output is the same as the proportionate change in each input. This is described as constant returns to scale. For example, if r = 2, then Q 2 = r α+β Q 1 = 2 1 Q 1 = 2Q 1 = rq 1 If α + β > 1, then the proportionate change in output is greater than the proportionate change in each input. This is described as increasing returns to scale. For example, if r = 2, α + β = 1.5, then Q 2 = r α+β Q 1 = Q 1 = 2.828Q 1 > rq 학기 end

28 Returns to Scale In general, a Cobb-Douglas production function is said to be a homogeneous function of degree (or order) m, if f(rl, rk) = r m f(l, K) where m = α + β If both L and K change simultaneously by small but different proportions or amounts, then the total change in output can be determined by the small changes formula. Q = Q L L + Q K K

29 Utility Functions Utility may be described as a function of goods consumed. U = f(x, y) where U = utility, x = the number of units of good X consumed, y = the number of units of good Y consumed A widely used utility function in economic analysis is the Cobb-Douglas utility function which is expressed in general form as U = Ax α y β where A, α and β are constants, and 0 < α < 1, 0 < β < 1 A specific Cobb-Douglas utility function could be U = 10x 0.3 y 0.5

30 Utility Functions Marginal utilities are defined as partial derivatives of utility function U=f(x,y). Marginal utility w.r.t. good X: MU x = U = f(x,y) x x = U x Marginal utility w.r.t. good Y: MU y = U = f(x,y) y y = U y

31 Utility Functions Graphical representation of utility function is similar to that of production function. Utility function U=f(x, y) which is now a function of two variables can be represented by a series of two-dimensional graphs known as indifference curves. To plot an indifference curve, fix the utility at a constant, then y can be expressed as a function of x as was done for production function of two variables. An indifference curve gives all the combinations of x and y for which the utility value is the same. Recall that an isoquant for a production function. An isoquant is a set of all the combinations of L and K which give the same value of Q. Similarly an indifference curve is a set of all the combinations of x and y which give the same level of utility.

32 Utility Function Slope of an indifference curve: dy dx Slope of an indifference curve is called the marginal rate of substitution(mrs). MRS may be expressed in terms of MU x and MU y, the marginal utilities of X and Y. By using the equation for total differential and the fact that U is constant along a given indifference curve, so that du=0, we have U y du = U x dx + U y dy = 0 dy = U x dx U ydy = U x dx dy dx = U x U y = MU x MU y

33 Utility Function Slope of an indifference curve = dy dx = MRS = MU x MU y MRS or the slope of an indifference curve at any given point represents the number of units by which good Y decreases when good X increases by one unit, while maintaining the same level of utility. The rate at which the number of good Y decreases for each unit increase good X becomes smaller as x increases Diminishing MRS(marginal rate of substitution.

34 Production Function and Utility Function Utility function is similar to production function in many aspects Production function Q=f(L, K) Utility function U=f(x, y) Isoquant: points of same output Q Slope of an isoquant =MRTS = dk = MP L dl MP K Diminishing MRTS: decrease in K for each unit increase in L diminishes as L increases, while maintaining the same output Q. Indifference curve: points of equal utility U Slope of an indifference curve =MRS = dy = MU x dx MU y Diminishing MRS: decrease in y for each unit increase in x diminishes as x increases, while maintaining the same level of utility U.

35 Partial Elasticity of Demand When demand (Q) is a function of the price of the good (P) only, i.e. Q=f(P), the point price elasticity of demand was defined as ε d = dq P dp Q Now, if we have more general demand function of the form Q A = f(p A, Y, P B ) where Q A = quantity demanded of good A P A = price of good A Y = consumer income P B = price of another good B Note that now the demand is a function of 3 variables P A, Y and P B. We may examine the elasticity of demand with respect to each of these 3 variables.

36 Partial Elasticity of Demand [1] Price elasticity of demand P A ε d = Q A P A Q A ε d measures the % change in Q A w.r.t. P A, keeping Y and P B constant. [2] Income elasticity of demand Y ε Y = Q A Y Q A ε Y measures the % change in Q A w.r.t. Y, keeping P A and P B constant. [3] Cross-price elasticity of demand P B ε c = Q A P B Q A ε c measures the % change in Q A w.r.t. P B, keeping P A and Y constant.

37 Partial Elasticity of Demand Worked Example 7.11 Demand function of good A is given by Q A = 100 2P A + 0.2Y + 0.3P B Find the price, income and cross-price elasticities of demand at P A = 6, Y = 500, P B = 10. (1) Price elasticity of demand Since, P A = 6, Y = 500, P B = 10, Q A = = 191 and So ε d = Q A P A Q A P A = 2 P A Q A = =

38 Partial Elasticity of Demand Worked Example 7.11 Q A = 100 2P A + 0.2Y + 0.3P B and P A = 6, Y = 500, P B = 10. are given and so Q A = = 191 (2) Income elasticity of demand Since So ε d = Q A Y Q A Y = 0.2 Y = = Q A 191 (3) Cross-price elasticity of demand Since Q A P B = 0.3 So ε d = Q A P B P B Q A = =

39 Partial Elasticity of Labor Partial Labor(L) Elasticity of Production(Output Q), otherwise called the Partial Elasticity of Output(Q) w.r.t. Labor(L) is defined as the proportionate change in output (Q) resulting from a proportionate change in labor input(l) when capital (K) is held constant: ε QL = Q L L = Q Q L 1 Q L = MP L 1 APL Note that partial elasticity of output(q) w.r.t. labor(l) can be expressed as a ratio of MP L to APL. For the Cobb-Douglas production function Q = AL α K β, ε QL = MP L 1 APL = αa Lα 1 K β 1 Q L = αa L α K β 1 Q = αq 1 Q = α

40 Partial Elasticity of Capital Similarly, Partial Capital(K) Elasticity of Production(Output Q), otherwise called the Partial Elasticity of Output(Q) w.r.t. Capital(K) is defined as the proportionate change in output (Q) resulting from a proportionate change in capital input(k) when labor(l) is held constant: ε QK = Q K K = Q Q K 1 Q K = MP K 1 APK Note that partial elasticity of output(q) w.r.t. capital(k) can be expressed as a ratio of MP K to APK. For the Cobb-Douglas production function Q = AL α K β, ε QK = MP K 1 APK = βa Lα K β 1 1 Q K = βa L α K β 1 Q = βq 1 Q = β

41 Partial Elasticity of Labor & Capital Worked Example 7.12 Calculate the partial elasticities w.r.t. labor and capital for the production function Q = 10L 0.5 K 0.5 (1) Partial Elasticity of Q w.r.t. Labor ε QL = Q L L Q ={0.5(10)L 0.5 K 0.5 } L Q = 0.5(10)L0.5 K Q = 0.5Q 1 Q = 0.5 (2) Partial Elasticity of Q w.r.t. Capital ε QK = Q K K Q = {0.5(10)L0.5 K 0.5 } K Q = 0.5(10)L0.5 K Q = 0.5Q 1 Q = 0.5