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, …, x_n)
-top left value = fⁿ ₁₁, top right = fⁿ_1n, bottom left = fⁿ_n1, bottom right = fⁿ _nn, with values in between showing when you partially differentiate with respect to the different x₁, x_n 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)