6) Derivatives, gradients and Hessian matrices
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1 30C00300 Mathematcal Methods for Economsts (6 cr) 6) Dervatves, gradents and Hessan matrces Smon & Blume chapters: 14, 15 Sldes by: Tmo Kuosmanen 1
2 Outlne Defnton of dervatve functon Dervatve notatons Partal dervatve Gradent vectors Total dfferental Hessan matrx Young s theorem Mcroeconomc content Margnal utlty Margnal product Margnal cost
3 Defnton: Let pont x s f the lmt exsts. f Dervatve functon : R R. The dervatve of functon f at df f ( x h) f ( x) f ( x) ( x) lm dx h0 h Note: the lmt does not always exst. If the lmt exsts for all x n doman R, functon f s dfferentable. Dfferentablty mples contnuty, but the converse s not true. Dfferentable functon must be smooth. 3
4 Tangent lne Consder a convex functon f : R R that s dfferentable. Then the followng nequalty holds f ( x h) f ( x) f ( x) h h R If f s concave, the nequalty s reversed as f ( x h) f ( x) f ( x) h h R Note: convexty and concavty of functons wll be dscussed n more detal n Lecture 7. 4
5 Subdervatve and subdfferental Consder a convex functon f : R Rthat s not dfferentable. The subdervatve of functon f at pont x s a real scalar d that satsfes f ( x h) f ( x) dh h R The subdfferental s of functon f at pont x s a closed set [a, b] where a b lm h0 lm h0 f ( x h) f ( x) h f ( x h) f ( x) h Note: f a = b, then f s dfferentable at pont x. 5
6 Partal dervatve n Defnton: Let f : R R. The partal dervatve of functon f wth respect to the th varable x s f f f ( x,..., x h,..., x ) f ( x,..., x,..., x ) 1 n 1 n ( x) ( x) lm x h0 h f the lmt exsts. 6
7 Margnal utlty n Consder utlty functon u : R R. Assume u s dfferentable and the commodtes are contnuously dvsble. Then the margnal utlty of the th commodty s the partal dervatve u u ( x) ( x) x 7
8 Margnal product k Consder producton functon f : R R. Assume f s dfferentable and the nputs are contnuously dvsble. Then the margnal product (MP) of the th nput s the partal dervatve f f ( x) ( x) MP( x) x 8
9 Consder producton functon elastcty of the th nput s Output elastcty f ( x) x x ( x) MP( x) x f ( x) f ( x) k : R R. The output Output elastcty ndcates the percentage change of output dvded by the percentage change of the th nput. f Theorem: Assume f s dfferentable. Then f exhbts constant returns to scale f and only f k ( x) 1 x R 1 k 9
10 Dfferentable functon f : R Gradent vector R has n partal dervatves. We can collect all partal dervatves and form a column vector f 1( x) f ( x) f ( x) n f ( ) n x Ths column vector s called the gradent or gradent vector. 10
11 Subgradent and subdfferental Consder a convex functon dfferentable. that s not The subgradent of functon f at pont x 0 s a real valued n- dmensonal vector d that satsfes f ( x h) f( x) d h h R f : R n R n The subdfferental s of functon f at pont x s a convex closed set that contans all subgradent vectors of f at pont x. 11
12 Tangent hyperplane and the total dfferental Assume a dfferentable functon We can approxmate the change of functon f near pont x by the total dfferental f ( x) f ( x) f ( x) df ( x) dx1 dx... dxn x x x n 1 1 f ( x) dx f ( x) x f n : R R. n The total dfferental can be used for lnear approxmaton of functon f near pont x by the tangent hyperplane f ( x h) f ( x) f ( x) h 1
13 Margnal rate of techncal substtton Consder producton functon k : R R. Suppose we ncrease nput by nfnatesmal amount, and decrease nput j such that the output remans constant. Applyng the total dfferental, we have f( x) f( x) dx f j( x) MPj( x) df ( x) dx dx j 0 x x dx f ( x) MP( x) j j Snce dx j s negatve, the margnal rate of techncal substtuton (MRTS) between nputs and j s defned as f MRTS j dx f j( x) MPj( x) dx f ( x) MP( x) j df ( x) 0 13
14 Substtuton elastcty Elastcty of substtuton measures the curvature of an soquant and thus, the substtutablty between nputs (.e., how easy t s to substtute one nput for another). Substtuton elastcty between nput and nput j s defned as j d ln( x / x ) d ln(( f ( x) / x ) / ( f ( x) / x )) d ln( x / x j ) d ln( x / x j ) d ln( MP( x)) / ( MP ( x)) d ln MRTS Note: the substtuton elastcty of the Cobb-Douglas producton functon s equal to 1 by constructon. j j j j 14
15 Second dervatve Defnton: Let f : R R. The second dervatve of functon f at pont x s d df d f f ( x) ( x) ( x) dx dx dx f ( x h) f ( x) f ( x) f ( x h) lm h h h0 h f ( x h) f ( x) f ( x h) lm h0 h f the lmt exsts. 15
16 Second partal dervatve Consder f n : R R. The second partal dervatve s denoted f f f j ( x) ( x) ( x) x j x x jx The formal defnton as a lmt s omtted: smply apply the standard dervatve rules to the partal dervatve. 16
17 The law of dmnshng margnal utlty The law of dmnshng margnal utlty s fundamental n economcs. The law states that as the consumpton of a commodty ncreases, keepng other commodtes at constant level, the margnal utlty decreases. Formally, u( x) u ( x) u( ) 0 x x R x x x 17
18 The law of dmnshng returns Smlarly n the context of producton, the law of dmnshng returns states that f the amount of any nput s ncreased, keepng other nputs fxed, at some pont, the margnal product decreases. Formally, MP x ( x) f ( x) x for large enough x. f ( x) 0 18
19 Hessan matrx n Dfferentable functon f : R Rhas n second partal dervatves at pont x. We can collect all second partal dervatves and form a matrx f 11( x) f 1( x) f 1n( x) f ( ) f ( ) f ( ) x x x 1 n f ( x) f n1( x) f n( x) f nn( x) Ths matrx s called the Hessan matrx. Note: We wll later use the Hessan n optmzaton for assessng whether a local optmum s a mnmum or maxmum. 19
20 Hessan matrx Note: The law of dmnshng margnal utlty and the law of dmnshng returns suggest the dagonal elements of the Hessan are negatve f 11( x) f 1( x) f 1n( x) f ( ) f ( ) f ( ) x x x 1 n f ( x) f ( x) f ( x) n1 n nn f ( x) 0
21 Young s theorem Theorem 14.5 Suppose that f(x) s twce contnuously dfferentable on an open regon J R n. Then, for all x J and for each par of ndces,j f( x) f( x) x x x x j j Note: Young s theorem mples the Hessan of any twce contnuously dfferentable functon s a symmetrc matrx. The order of dfferentaton does not matter. 1
22 Taylor polynomal Recall the lnear (frst-order) approxmaton of functon f near pont x by the tangent lne f ( x h) f ( x) f ( x) h or pont x by the tangent hyperplane f ( x h) f ( x) f ( x) h The second-order approxmaton can be obtaned as 1 f ( x h) f ( x) f ( x) h f ( x) h 1 f ( x h) f ( x) f ( x) h h f ( x) h
23 Taylor theorem Theorem 30.7 Let f: U R 1 be a k+1 tmes contnuously dfferentable functon defned on a connected nterval U n R 1. For any ponts x and x+h n U, there exsts a pont c* between x and x+h such that 1 f x h f x f x h f x h 1 1 f ( x) h f ( c* ) h k! ( k 1)! ( ) ( ) ( ) ( )... [ k ] k [ k 1] k 1 3
24 Translog functon Transcendental logarthmc functon (translog) s a wdely used flexble functonal form that can be nterpreted as the secondorder Taylor approxmaton k k k ln y lnx 0.5 ln x ln x j j 1 j 1 Note: although translog s hghly nonlnear n x varables, t s lnear n parameters α, β, γ. Hence, t can be estmated by lnear regresson. 4
25 Next week 1st Mdterm exam Next lecture 13 Aprl: Important propertes of functons: homogenety, homothetcty, convexty and quas-convexty Textbook: Smon & Blume, Ch. 0 and 1 5
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