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