- Information
- AI Chat
Real+Valued+Functions+of+n+Variables
Microeconomic Theory I (ECON 210A)
University of California Irvine
Students also viewed
Related documents
- Study Guide of development transitions and adulthood (Child Development (psci 111D))
- Study Guide of Prenatal Development (Child Development (psci 111D))
- Learning Objectives for child development
- The Divine comedy summary (COM LIT 60A)
- World Literature Vocabulary (COM LIT 60A)
- Learning Objective (COM LIT 60A)
Preview text
Real-Valued Functions of n Real Variables
Cole Williams
October 4, 2015
Letf:X⊂Rn→Rbe a real-valued function ofnreal variables x= [x 1 , ..., xn]. The operator “·” represents thedot product, ifa= [a 1 , ..., an] andb= [b 1 , ..., bn] thena·b=a 1 b 1 +a 2 b 2 +...+anbn. We shall specify‖ · ‖to be theEuclidean norm,‖a−b‖= [(a 1 −b 1 ) 2 + (a 2 −b 2 ) 2 +...+ (an−bn) 2 ] 1 / 2.
Definition 1. Define thepartial derivativeoff(x) with respect toxii= 1, .. atx 0 = [x 1 , 0 , x 2 , 0 , ..., xi, 0 , ..., xn, 0 ]∈Xis given by
∂f ∂xi(x 0 ) = limh→ 0
f(x 1 , 0 , x 2 , 0 , ..., xi, 0 +h, ..., xn, 0 )−f(x 1 , 0 , x 2 , 0 , ..., xi, 0 , ..., xn, 0 ) h ∈R. (1)
Definition 2. Thegradientoffatx 0 is the 1×nvector of partial derivatives
∇f(x 0 ) =
[∂f ∂x 1 (x 0 ),
∂f ∂x 2 (x 0 ), ...,
∂f ∂xn(x 0 )
]
(2)
Definition 3. f is said to be differentiable at x 0 iff there exists a vector
f′(x 0 )∈Rncalled thederivativewith
xlim→x 0 f(x)−f(x 0 )−f
′(x 0 )·(x−x 0 ) ‖x−x 0 ‖ = 0. (3)
Proposition 4. Iffis differentiable atx 0 , then
f′(x 0 ) =∇f(x 0 ). (4)
Definition 5. Theincremental changeoffatx 0 , denoted by∇fis
∆f=f(x 0 +h)−f(x 0 ). (5)
Definition 6. Thetotal differentialoff, denoteddf(x 0 ), is
df(x 0 ) =∂x∂f 1 (x 0 )dx 1 +∂x∂f 2 (x 0 )dx 2 +...+∂x∂fn(x 0 )dxn. (6)
Letdx= [dx 1 , ..., dxn] be an 1×nvector of infinitesimal changes to our variables. We can express the total differential in vector notation asdf=∇f·dx. From (3) and (4), forh≈ 0 we can approximate the incremental change in offby the total differential, ∆f≈df. (7)
Definition 7. Thetotal derivativeof f with respect toxican be found by dividing both sides of (6) bydxi
df dxi=
∂f ∂x 1
dx 1 dxi+
∂f ∂x 2
dx 2 dxi+...+
∂f ∂xn
dxn dxi. (8)
Sketch of Proof.
Divide our nvariables into two collectionsz= {x 1 , ..., xn− 1 }andy= xn with f(z 0 , y 0 ) =c and without loss of generality ∂f∂y(z 0 , y 0 )>0. We want to show that for small changes inznearz 0 , say toz 1 that there is only one choice ofy, sayy 1 such thatf(z 1 , y 1 ) =c. Hence, could define a functiony=g(z) such that f(z, g(z)) =c.
Notice that asfhas continuous derivatives ∂f∂y(z, y)>0 for (z, y) near (z 0 , y 0 ).
Suppose we changezfromz 0 toz 1 while keepingyfixed aty 0. If this change increasesthe value of the functionf(z 1 , y 0 )> f(z 0 , y 0 ) =c then we mustdecrease y to somey 1 < y 0 so that f(z 1 , y 1 ) =c, furthermore sincef is locally strictly monotonic iny, there is only one suchy 1 that will satisfy this equality.
Similar arguments can be made for the cases thatf(z 1 , y 0 )< f(z 0 , y 0 ) andf(z 1 , y 0 ) = f(z 0 , y 0 ). Hence there exists angsatisfying (10). (A complete proof would likewise need to show thatgis continuously differentiable.)
Having established (10) we can write
f(x 1 , ..., xn− 1 , g(x 1 , ..− 1 )) =c.
Taking the partial derivate of both sides with respect to somexii= 1, ..., n− 1 and utilizing the chain rule∂x∂fi+∂x∂fn∂x∂gi= 0 for∂x∂giyields (11).
Example 1 - Convexity of the Isoquant
Letf(x 1 , x 2 ) be a real-valued production function exhibiting
Positive marginal product in both inputs∂x∂fi>0,i= 1, 2
Diminishing returns to both inputs∂∂x 2 i 2 f<0,i= 1, 2
Weak complementarity of inputs∂x∂j 2 ∂xfi≥0,i= 1,2,j 6 =i.
Define anisoquantto be the graph consisting of the various combinations of inputs which would result in the production of a fixed output ̄y∈R+,L( ̄y) ={(x 1 , x 2 )∈ R 2 :f(x 1 , x 2 ) = ̄y}.
Prove that the graph of each isoquant can be represented by a decreasing, convex functionx 1 =g(x 2 )
Sketch of Proof.
Select any ( ̄x 1 , ̄x 2 ) ∈ L( ̄y). Having assumed f to be C 1 , exhibiting positive marginal product in both inputs, andf(x 1 , x 2 ) = ̄ythe conditions for the implicit function theorem. Hence by (10), there exists aC 1 functionx 1 =g(x 2 |( ̄x 1 ,x ̄ 2 )) in the intersection of a neighborhood of ( ̄x 1 , ̄x 2 ) andL( ̄y). As this is true for every choice of ( ̄x 1 ,x ̄ 2 ), then as eachg(x 2 |( ̄x 1 , ̄x 2 )) isC 1 there exists a function x 1 =g(x 2 ) for all ordered pairs inL( ̄y) which is likewiseC 1.
From (11) we have∂x∂g 2 =−∂f /∂x∂f /∂x 21 which is negative as a result of positive marginal products, henceg(x 2 ) is decreasing.
Applying the quotient rule, we find the second derivative ofgwith respect tox 2 to be
∂ 2 g ∂x 22 =−
[( ∂ 2 f ∂x 1 ∂x 2 ∂x∂g 2 +∂ 2 f ∂x 22
)∂f ∂x 1 −
(∂ 2 f ∂x 21 ∂x∂g 2 + ∂ 2 f ∂x 2 ∂x 1
)∂f ∂x 2
]
[∂f ∂x 1
] 2 > 0.
where the inequality holds as a result ofg(x 2 ) being decreasing and assumptions (1)-(3). Hence g(x 2 ) is convex. As the choice of ̄y∈ R+was arbitrary, these
Real+Valued+Functions+of+n+Variables
Course: Microeconomic Theory I (ECON 210A)
University: University of California Irvine
Students also viewed
Related documents
- Study Guide of development transitions and adulthood (Child Development (psci 111D))
- Study Guide of Prenatal Development (Child Development (psci 111D))
- Learning Objectives for child development
- The Divine comedy summary (COM LIT 60A)
- World Literature Vocabulary (COM LIT 60A)
- Learning Objective (COM LIT 60A)