Topic 3: Calculus of functions of many variables

Question 1

What are the representations of different types of functions (3)

Answer 1

-A univariate function is a line in 2D
-A bivariate function is a surface in 3D
-A multivariate function is a hypersurface in all definitions

Question 2

What is a level curve (2,1)

Answer 2

-A level curve c of function f(x1, x2) corresponds to all the pairs (x1, x2) satisfying f(x1, x2) = c
-A level curve is the set of combinations of x1 and x2 which lead to the same value of the function

-Downwards sloping level curves represent tradeoffs

Question 3

What is youngs theorem (2)

Answer 3

-If all the nth order derivatives are continuous, any of them that involve differentiating with respect to the same variables the same number of times are necessarily equal
-f’‘(1,2) = f’‘(2,1)

Question 4

What is a hessian matrix (3)

Answer 4

-A hessian matrix is a matrix to show the nth order partial derivatives of f(x1, …, xn)
-top left value = f^(n) 11, top right = f^(n)1n, bottom left = f^(n)n1, bottom right = f^(n) nn, with values in between showing when you partially differentiate with respect to the different x1, xn etc
-Youngs theorem should show that f’‘(ij) = f’‘(ji)

Question 5

How can you write the function for partial elasticities (3)

Answer 5

For a function f(x1, x2), the partial elasticities = :
-Elx1 f = (x1/f(x1, x2) x (∂f(x1, x2)/∂x1)
-Elasticity = variable/function x partially differentiate function with respect to variable
-Can also be written as ∂lnf/∂lnx1 (partially differentiate ln of function with respect to ln of x1)

Question 6

How to derive the 2 functions for partial elasticity from each other (5)

Answer 6

-∂lnf/∂lnx1
-(∂lnf/∂f)(∂f/∂x1)(∂x1/∂lnx1)
-(1/f)(∂f/∂x1)(∂lnx1/∂x1)^-1
-(1/f)(∂f/∂x1)(1/x1)^-1
-(x1/f)(∂f/∂x1)