代做ECON10005 - Mathematics for Economics代写Web开发

5. Multivariate optimisation

ECON10005 - Mathematics for Economics

Multivariate optimisation

Single-variable optimisation

Multivariate optimisation

Multivariate optimisation

Maximisation problem

Choice variables: (x1, · · · , xn) ∈ S

Target function: f : S −→ R

Similar to single-variable optimisation except:

the set of actions is a multivariate set, for instance [0, 1] × [0, 1]

the tangent function is a function of several variables

Stationary point

A stationary point should verify:

for all choice variables: (x1, · · · , xn)

Finding a maximum

1. We find the stationary points of f:

2. We evaluate f at the boundaries of S (finite or limit at infinity).

3. Among these candidate solutions, the action (x1, · · · , xn) providing the highest value is the maximum.

Concavity

If f is concave over S, a stationary point (if it exists!) will always be a maximum over S.

If f is convex over S, a stationary point (if it exists!) will always be a minimum over S.

Concavity with two variables

Concavity with one variable: f is concave over S if f ′′(x) ≤ 0 for all x ∈ S

Concavity/ convexity is more difficult to establish with many variables

Concavity with two variables

f is concave over S if

Convexity with one variable: f is convex over S if f ′′(x) ≥ 0 for all x ∈ S

Convexity with two variables

f is convex over S if

Why do we need condition 3?

Example

f(x, y) = x 2 + y 2 − 3xy

Why do we need condition 3?

f(x, y) = x 2 + y 2 − 3xy

Economic Example: Profit maximisation

Consider an entrepreneur (Tobacco Yoko) producing tobacco with inputs labour L and capital K:

We assume that markets are competitive: the entrepreneur takes the factor prices r = 1.2 and w = 0.6 as given.

The unit price of tobacco is P = 12