University of Nottingham

School of Economics University of Nottingham Pre-sessional Mathematics Masters (MSc) Dr Maria Montero Dr Alex Possajennikov Topic 3 Multivariate Calculus Pre-sessional Mathematics Topic 3

Multivariate Calculus Functions of n variables y = f(x1, … ,xn) F: Rn R Example: f(x,y) = x2 - y2 In economics: Utility function u(x,y) Production function y(K,L) = KL1- Limits limx  a f(x) = b for any  > 0, there exists  > 0 such that |f(x) - b| <  for any x such that ||x - a|| <  x = (x1, … ,xn), a = (a1, … ,an) Continuity f(x) is continuous at x0 if limx  x0 f(x) = f(x0) f(x) is continuous on D if it is continuous at every point in D Pre-sessional Mathematics Topic 3

Multivariate Calculus Differentiation xi0 + xi xj0 Consider function f(x1, … ,xn ). Fix x0 = (x10, …,xn0). The partial derivative of f with respect to xi f/xi gives the slope of the tangent line to the graph of the function in the hyperplane parallel to xi axis and z In economics: Marginal product of labour: Y/L Marginal utility: u/xi Partial derivatives are functions If all partial derivatives of f(x)exist and continuous, then f(x) is continuously differentiable (C1) Pre-sessional Mathematics Topic 3

Multivariate Calculus Rules of differentiation The same as for ordinary differentiation The chain rule f(x1, … ,xn ) and xi(t1, … ,tm ) Total derivative (gradient) Approximation f(x0 + x)  f(x0) + Df(x0)·x Total differential Pre-sessional Mathematics Topic 3

Theorem If f  C2, then Multivariate Calculus Second order derivatives f/xi(x1, … ,xn ) is a function of n variables Hessian cross-partials Example: f(x) = 4x12x2 - 2x22 - x1 - 1 Pre-sessional Mathematics Topic 3

D y y x x D Multivariate Calculus Concavity and convexity Convex set D is a convex set if for any x,y  D, (1-)x+y  D, for any   [0,1] Convex set Non-convex set f(x) is convex on convexD if f((1 - )x1 + x2)  (1 - )f(x1) + f(x2) f(x) is concave on convexD if f((1 - )x1 + x2)  (1 - )f(x1) + f(x2) Pre-sessional Mathematics Topic 3

Multivariate Calculus Theoremf  C2. f is concave on convex D when D2f is negative semidefinite on D f is convex on convex D when D2f is positive semidefinite on D Example f(x,y) = x1/3y1/3on D = {x > 0, y > 0} Pre-sessional Mathematics Topic 3

TheoremF(x1,…,xn,y) = 0. F  C1. F(x0,y0) = 0. F/y(x0,y0)  0 Then there exists y = f(x1,…,xn) such that F(x1,…,xn, f(x1,…,xn)) = 0 and Multivariate Calculus Implicit function y = f(x1,…,xn)explicit function F(x1,…,xn,y) = 0implicit function It may be difficult to solve F(x1,…,xn,y) = 0 to get y = f(x1,…,xn) Example: y3 - xy + ln x = 0 Suppose we are interested in y/xi Pre-sessional Mathematics Topic 3

Multivariate Calculus n = 2 F(x,y) = 0 Slopes of the level curves of the graph Example: utility function u(x,y) = x2/3y1/3 Slope of indifference curve at x = 1, y = 1 Pre-sessional Mathematics Topic 3

Multivariate Calculus Optimisation x0 = (x10, …, xn0) is the global maximum of f(x) if f(x0)  f(x) for all x in D x0 = (x10, …, xn0) is the local maximum of f(x) if f(x0)  f(x) for all x in small neighbourhood of x0 Theoremf  C1. If x0 is an interiorlocalmaximum or minimum, then f/xi(x0) = 0 for all i. Theorem f  C2. f/xi(x0) = 0 for all i. If D2f(x0) is negative definite, then x0 is a local maximum. If D2f(x0) is positive definite, then x0 is a local minimum. If D2f(x0) is indefinite, then x0 is neither local maximum nor local minimum Pre-sessional Mathematics Topic 3

Multivariate Calculus Theorem f  C2. f/xi(x0) = 0 for all i. If f is concave (D2f is negative semidefinite) on D, then x0 is a global maximum. If f is convex (D2f is positive semidefinite) on D, then x0 is a global minimum. General approach to maximisation of C1 function without constraints 1. Find all critical points of f(x) by solving f/xi(x) = 0for all i. 2. Find values of f(x) at these points 3. “Find” values of f(x) at the boundary of D 4. Choose the highest value from 2 and 3 There may be no maximum Pre-sessional Mathematics Topic 3

Multivariate Calculus Example: f(x1,x2) = 4x12x2 - 2x22 - x1 - 1 D = {x1[0,1],x2 [0,1]} Pre-sessional Mathematics Topic 3

Multivariate Calculus Constrained optimisation maxf(x1,…,xn) s.t.gi(x1,…,xn)  0 i=1,…,m hj(x1,…,xn) = 0 j=1,…,k Examples max u(x1,x2) s.t. p1x1 + p2x2 I utility maximisation subject to budget constraint profit maximisation subject to production possibilities welfare maximisation subject to individuals’ reaction Pre-sessional Mathematics Topic 3

Multivariate Calculus Substitution method maxf(x1,x2) s.t.h(x1,x2) = 0 Solveh(x1,x2) = 0for x2 = H(x1) Substitutex2 = H(x1) to get the problem max f(x1,H(x1)) When there are inequality constraintsg(x1,x2)  0, sometimes argument can be made that at maximumg(x1,x2) = 0 maxf(x,y)= x1/2y1/2 s.t.g(x,y)= x + y - 4  0 Example: f is increasing in x and y At maximum, g(x,y)= 0 g is increasing in x and y Pre-sessional Mathematics Topic 3

Multivariate Calculus Lagrangean method maxf(x1,…,xn) s.t.gi(x1,…,xn)  0 i=1,…,m hj(x1,…,xn) = 0 j=1,…,k Lagrangean ‘Maximise’ the Lagrangean i, j are Lagrangean multipliers Idea: represent common tangent hyperplane of the set defined by the constraints and the level curves of function f At maximum there is no way to move along constraints without decreasing the value of f Interpretation of the multipliers: shadow price of the constraint Pre-sessional Mathematics Topic 3

Jacobian matrix of the set of constraints Multivariate Calculus Equality constraints maxf(x1,…,xn) s.t.hj(x1,…,xn) = 0 j=1,…,k Lagrangean Assumek  n Rows of Dh as vectors hi / x Dh has rankm Pre-sessional Mathematics Topic 3

Multivariate Calculus Theoremf ,hj C1. If x0 is a local maximum or minimum and if rank Dh(x0) = k then there exist j0 such that Example maxf(x,y) = x2 - y s.t.x2 + y2= 1 Pre-sessional Mathematics Topic 3

Multivariate Calculus Inequality constraints maxf(x1,…,xn) s.t.gi(x1,…,xn)  0 i=1,…,m Binding constraintgi(x0) = 0 Non-binding constraintgi(x0) > 0 Lagrangean Theoremf ,hj C1. x0 is a local maximum. g1(x0) = 0,…, gk(x0) = 0 and gk+1(x0) > 0,…, gm(x0) > 0. rank Dg1,..,k(x0) = k. Then there exist i0 such that: gi(x0)  0for all i i0·gi(x0) = 0for all i Complementary slackness condition i0 0for all i [At minimum i0 0] Pre-sessional Mathematics Topic 3

Multivariate Calculus Example maxf(x,y) = x2 - y s.t.x2 + y2 1 Pre-sessional Mathematics Topic 3

Multivariate Calculus Mixed constraints maxf(x1,…,xn) s.t.gi(x1,…,xn)  0 i=1,…,m hj(x1,…,xn) = 0 j=1,…,k Lagrangean Theoremf ,hj C1. x0 is a local maximum. [Technical condition on rank of a Jacobean]. Then there exist i0,j0 such that: hj(x0) = 0for all j gi(x0)  0for all i i0·gi(x0) = 0for all i i0 0for all i Pre-sessional Mathematics Topic 3

Multivariate Calculus Example maxf(x,y) = x2 - y s.t.x2 + y2= 1 x + y  0 Pre-sessional Mathematics Topic 3

Multivariate Calculus Sufficient conditions for local maximum to be global maximum L is concave Can be written using concavity of f and h General approach to maximisation of C1 function with constraints 1. Form the Lagrangean 2. Find all critical points by solving the appopriate system of equations and inequalities 3. Find values of f(x) at these points 4. Choose the highest value from Step 3 (boundary of D is taken care of by some of the equations in the system) Pre-sessional Mathematics Topic 3

Multivariate Calculus Comparative statics f(x ; a) = 0 F1(x1 , x2 ; a) = 0 F2(x1 , x2 ; a) = 0 Pre-sessional Mathematics Topic 3

Multivariate Calculus The envelope theorem max f(x1 , x2 ; a) = 0 s.t. h (x1 , x2 ; a) = 0 Lagrangean FOC Derivative dL / da Pre-sessional Mathematics Topic 3

Multivariate Calculus The envelope theorem max f(x1 , x2 ; a) = 0 s.t. h (x1 , x2 ; a) = 0 Letx10, x20be a solution FOC Value function Derivative dV / da Derivative dh / da Pre-sessional Mathematics Topic 3

Multivariate Calculus The envelope theorem maxf(x1,…,xn; a) s.t.hj(x1,…,xn;a) = 0 j=1,…,k Suppose x0(a) is the solution [Technical conditions on smoothness of the solution] V V(·;a) V(·;2) Example: max u(x1,x2) s.t. p1x1 + p2x2= I V(·;1) a,x Pre-sessional Mathematics Topic 3

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