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Mathematical knowledge
1 Functions of a single variable
- Function y = f (x)
- The first derivative of f with respect to x is
f 0 (x) = df dx
It gives, at each value x, the slope or instantaneous rate of change in f (x).
- The second derivative of f with respect to x is
f 00 (x) = d^2 f dx^2.
It gives the rate at which the slope of f changes. It is thus related to the curvature of the function f.
— For constants, α:
d dx α = 0 ,
— For sums:
d dx [f (x) ± g(x)] = f 0 (x) ± g^0 (x),
— Power rule:
d dx (αx
n) = nαxn− (^1) ,
— Product rule:
d dx [f (x)g(x)] = f 0 (x)g(x) + f (x)g^0 (x),
— Quotient rule:
d dx
[
f (x) g(x)
] =
f 0 (x)g(x) − f (x)g^0 (x) [g(x)]^2
— Chain rule:
d dx [f^ (g(x))]^ =^ f^
(^0) (g(x))g (^0) (x)
- Examples. Calculate the derivatives in each of the following cases:
- y = 5x−^2
- f (x) = 2x^2 + 3x + 4
- g(x) = 3x − 1
- (f (x)g(x))^0?
- ( f g^ ((xx)) )^0?
- f (g(x))^0?
- Concavity and first and second derivatives. If f is twice differentiable, the following statements 1 to 3 are equivalent: 1. f is concave. 2. f 00 (x) ≤ 0 ∀x. 3. If λ ∈ [0, 1], ∀x and ∀x^0 , f (λx + (1 − λ)x^0 ) ≥ λf (x) + (1 − λ)f (x^0 )
(For all x^0 : f (x) ≤ f (x^0 ) + f 0 (x^0 )(x − x^0 ) ∀x.)
- Necessary conditions for local interior optima in a single-variable case f (x) is twice continuously differentiable. It reaches a local interior 1. maximum at x∗^ ⇒ f 0 (x∗) = 0 (F OC) ⇒ f 00 (x∗) ≤ 0 (SOC)
- maximum at ex ⇒ f 0 (ex) = 0 (F OC) ⇒ f 00 (ex) ≥ 0 (SOC)
4 Constrained optimization
- Consider the following problem
( Max x 1 ,x 2 f (x 1 , x 2 ) subject to g(x 1 , x 2 ) = 0
- f (x 1 , x 2 ): objective function or maximand,
- x 1 and x 2 are choice variables,
- g(x 1 , x 2 ): constraint.
- Let solve this problem by substitution: — suppose we can rewrite the constraint g(x 1 , x 2 ) = 0 as x 2 = eg(x 1 ). — We can substitute this directly into the objective function: the 2 variable constrained maximization problem can be rephrased as the single variable problem with no constraints:
M ax x 1 f (x 1 , eg(x 1 ))
— x∗ 1 is defined by
∂f (x 1 , eg(x 1 )) ∂x 1
∂f (x 1 , eg(x 1 )) ∂x 2
dge(x 1 )) dx 1
— and x∗ 2 by
x∗ 2 = eg(x∗ 1 )
- Example: Cost minimizing firm (C = wL + rK) with Cobb-Douglas pro- duction functions
Min K,L wL + rK subject to Q = LαKβ
— The constraint Q = LαKβ^ becomes L = Q α^1 K−^ βα . — Unconstrained minimization
Min K wQ (^1) α K−^ βα
— FOC:
dC dK = −wQ (^1) α β α
K−^
αβ −^1
— SOC
d^2 C dK^2
β α
K−^
βα − 2 ≥ 0
— Solve FOC gives
K∗^ = [
w r
Q
α^1 β α
]
αα+β
— Substitute K∗^ in L = Q (^1) α K−^ αβ gives
L∗^ = [
r w
Q
β^1 α β
]
α+ββ
