Topic 7 Part I Partial Differentiation Part II Marginal Functions Part II Partial Elasticity Part III Total Differentiation Part IV Returns to scale Jacques (4th Edition): 5.1-5.3 1
Functions of Several Variables More realistic in economics to assume an economic variable is a function of a number of different factors: Y f(x, Z) Demand may depend on the price of the good and the income level of the consumer Q d f(p,y) Output of a firm depends on inputs into the production process like capital and labour Q f(k,l) 2
Graphically Sketching functions of two variables Y f (X, Z) Sketch this function in 3-dimensional space or plot relationship between 2 variables for constant values of the third 3
For example I Y Ya+bX+cZ Consider a linear function form: y a+bx+cz a +cz 0 X For different values of z we can represent the relationship between x and y Y a+cz 1 a+cz 0 z 1 z0 x 0 x 1 X 4
For example II Consider a non-linear function form: YX α Z β 0<α< 1 & 0<β< 1 For different values of z we can represent the relationship between x and y Y z >1 z 1 x 0 x 1 X 5
Part I: Partial Differentiation (Differentiating functions of several variables) Recall, function of one variable y f(x): One first order derivative: dy dx One Second order derivative: d f' ( x) 2 dx y 2 f''( x) 6
Consider our function of two variables: Y f (x, z) a + bx + cz TWO First Order Partial Derivatives y x fx f b 1 Differentiate with respect to x, holding z constant y z fz f2 c Differentiate with respect to z, holding x constant 7
8 FOUR Second Order Partial Derivatives Second Own and Cross Partial Derivatives 0 11 2 2 f f x y xx 0 22 2 2 f f z y zz 0 12 2 f f z x y xz 0 21 2 f f x z y zx
Consider our function of two variables: Y f (X, Z) X α Z β First Partial Derivatives Y/ X f X αx α-1 Z β > 0 Y/ Z f Z βx α Z β-1 > 0 9
Since Y/ X f αx α-1 Z β X Y/ Z f βx α Z β-1 Z Second own partial 2 Y/ X 2 f XX (α-1)αx α-2 Z β < 0 2 Y/ Z 2 f ZZ (β-1) βx α Z β-2 < 0 Second cross partial 2 Y/ X Z f XZ βαx α-1 Z β-1 > 0 2 Y/ Z X f ZX βαx α-1 Z β-1 > 0 10
Example Jacques Y f (X, Z) X 2 +Z 3 f X 2X > 0 Positive relation between x and y f XX 2 > 0 but at an increasing rate with x f XZ 0, ( f zx ) and a constant rate with z impact of change in x on y is bigger at bigger values of x but the same for all values of z f Z 3Z 2 > 0 Positive relation between z and y f ZZ 6Z > 0 but at an increasing rate with z f ZX 0 ( f xz ) and a constant rate with x impact of change in z on y is bigger at bigger values of z, but the same for all values of x 11
Example Jacques Y f (X, Z) X 2 Z f X 2XZ >0 Positive relation between x and y f XX 2Z >0 but at an increasing rate with x f XZ 2X >0 and at an increasing rate with z impact of change in x on y is bigger at bigger values of x and bigger values of z f Z X 2 >0 Positive relation between z and y f ZZ 0 but at a constant rate with z f ZX 2X >0 and an increasing rate with x impact of change in z on y is the same for all values of z but is bigger at higher values of x 12
Production function example Y f(k L) K 1/3 L 2/3 First partial derivatives of input gives Marginal product of input MPLδY/δLY L ( 2 / 3 K 1/3 L 2/3-1 ) 2 / 3 Y/L >0 An increase in L holding other inputs constant will increase output Y MPKδY/δK Y K ( 1 / 3 K 1/3-1 L 2/3 ) 1 / 3 Y/K > 0 An increase in K holding other inputs constant will increase output Y 13
MPL δy/δl ( 2 / 3 K 1/3 L -1/3 ) 2 / 3 Y/L. MPK δy/δk ( 1 / 3 K -2/3 L 2/3 ) 1 / 3 Y/K. Second Own derivatives of input gives Marginal Returns of input (or the change in the marginal product of an input with the level of that input) δ 2 Y/δL 2-2 / 9 K 1/3 L -4/3 < 0 Diminishing marginal returns to L (the change in MPL with L shows that the MPL decreases at higher values of L) δ 2 Y/δL 2-2 / 9 K -5/3 L 2/3 < 0 Diminishing marginal returns to K (the change in the MPK with K shows that the MPK decreases at higher values of K) 14
Part II: Partial Elasticity e.g cobb-douglas production function Y f(k L) A K α L β Elasticity of Output with respect to L Y / Y Y L ε YL L / L L Y (βa K α L β-1 ). L/Y β Y/L. L/Y β Elasticity of Output with respect to K Y / Y Y K ε YK K / K K Y (αa K α-1 L β ). K/Y α Y/K. K/Y α 15
e.g. Y f(k L) K 1/3 L 2/3 Elasticity of Output with respect to L Y L Y ε YL L ( 2 / 3 K 1/3 L 2/3-1 ). L/Y 2 / 3 Y/L. L/Y 2 / 3 Elasticity of Output with respect to K Y K Y ε YK K ( 1 / 3 K 1/3-1 L 2/3 ). K/Y 1 / 3 Y/K. K/Y 1 / 3 16
e.g. demand function Q f( P, P S, Y) Q f(p,p S,Y) 100-2P+P S +0.1Y P 10, P S 12, Y 1000 Q192 Partial Own-Price Elasticity of Demand ε QP Q/ P. P/Q -2 * (10/192) - 0.10 Partial Cross-Price Elasticity of Demand ε QPS Q/ P S. P S /Q +1 * (12/192) 0.06 Partial Income Elasticity of Demand ε QI Q/ Y. Y/Q +0.1 * (1000/192) 0.52 17
Part III: Total Differential Total Differential: Y f (X) Y dy/dx. X If X 10 and dy/dx 2, Y 2. 10 20 Total Differential: Y f (X, Z) Y Y/ X. X + Y/ Z. Z or dy Y/ X. dx + Y/ Z. dz 18
Example: Y f (K, L) Y K 1/3 L 2/3 dy Y/ K. dk + Y/ L. dl or dy ( 1 / 3 K -2/3 L 2/3 ).dk + ( 2 / 3 K 1/3 L -1/3 ).dl Rewriting: dy ( 1 / 3 K 1/3 K -1 L 2/3 ).dk + ( 2 / 3 K 1/3 L 2/3 L -1 ).dl or dy 1 / 3. Y / K. dk + 2 / 3. Y / L. dl To find proportionate change in Y dy / Y 1 / 3. dk / K + 2 / 3. dl / L 19
Total Differential: Y f (X, Z) Y A f (K, L) Y A K α L β dy Y/ K.dK + Y/ L.dL + Y/ A.dA or dy α.ak α -1 L β dk + β.ak α L β-1.dl + K α L β. da dy α.y/k. dk + β.y/l. dl + Y/A. da Or for proportionate change in Y: dy/y α.dk/k + β.dl/l + da/a Or for proportionate change in A: da/a dy/y (α.dk/k+β.dl/l) 20
Part IV: Returns to Scale Returns to scale shows the change in Y due to a proportionate change in ALL factors of production. So if Y f(k, L) Constant Returns to Scale f(λk, λ L) λf(k, L) λy Increasing Returns to Scale f(λk, λ L) > λf(k, L) > λy Decreasing Returns to Scale f(λ K, λl ) < λf(k, L) < λy 21
Cobb-Douglas Production Function: Y AK α L β Quick way to check returns to scale in Cobb- Douglas production function Y AK α L β then if α + β 1 : CRS if α + β > 1 : IRS if α + β < 1 : DRS 22
Example: Y f(k, L) A K α L β Y * f(λk, λl) A (λk) α ( λl) β Y * A λ α K α λ β L β λ α+β AK α L β λ α+β Y α+β 1 Constant Returns to Scale α+β > 1 Increasing Returns to Scale α+β < 1 Decreasing Returns to Scale 23
Homogeneous of Degree r if: f(λx, λz ) λ r f(x, Z) λ r Y Homogenous function if by scaling all variables by λ, can write Y in terms of λ r Note superscripts! Note then, for cobb-douglas Y K α L β, the function is homogenous of degree α + β 24
Eulers Theorem X f X + Z f Z r f(x, Z) ry E.g. r 1, Constant Returns to Scale If Y f(k, L) A K α L (1-α) Does K f K + L f L ry Y? K (αy/k) + L((1-α)Y/L)? αy + (1-α)Y αy + Y -αy Y Thus, Eulers theorem shows (MPk * K ) + (MPL * L) Y in the case of homogenous production functions of degree 1 25
Example.. If Y K ½ L ½ then Y(λK, λl) (λk) ½ (λl) ½ λ ½ K ½ λ ½ L ½ λ 1 K ½ L ½ λy homogenous degree 1... constant returns to scale Eulers Theorem: show that K. δy / δk +L. δy / δl r.y Y (r1 as homog. degree 1) K. (½.K ½-1.L ½ ) + L.(½.K ½.L ½-1 ) K(½. Y/K) + L(½.Y/L) ½.Y + ½.Y Y 26
Summary: Function of Two Variables Partial Differentiation - Production Functions first derivatives (marginal product of K or L) and second derivatives (returns to K or L) Partial Elasticity Demand with respect to own price, price of another good, or income Total Differentials Returns to Scale Plenty of Self-Assessment Problems and Tutorial Questions on these things. 27