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and Its Proof |
The Implicit Function Theorem (IFT) is a generalization of the result that
Note that without any loss of generality the constant C can be taken to be 0. If G*(x,y)=C then G(x,y)=G*(x,y)-C=0.
The IFT is a very important tool in economic analysis and so the conditions under which it holds must be carefully specified. The simplest way to do this is to give a formal, explicit proof of the theorem. First a proof of an artifically limited version of the IFT will be given and this will provide an understanding and a guide to the proof of the full version.
This means that within R z can be represented as a function of x and y; i.e., z=f(x,y).
Proof:
Without any loss of generality the set S can be taken to be a parallelpiped,
a box,centered on P because within the set S there will be at least one such box. Within any
such box there will be another box containing P such that ∂F/∂z
has the the same sign as at P.
Let the box be given as a triple (a,b,c) such that
Without any loss of generality the sign of ∂F/∂z at P can be taken to be positive.
This means that
Now consider any point (x_{1},y_{1}) such that
Because F(x_{0},y_{0},z_{0}+c) > 0 and F(x,y,z) is continuous, F(x_{1},y_{1},z_{0}+c) > 0. Likewise F(x_{1},y_{1},z_{0}-c) < 0. With x and y held fixed at x_{1} and y_{1}, G(z)=F(x_{1},y_{1},z) is a function such that G(z_{0}+c) > 0 and G(z_{0}-c) < 0. Therefore there is some z between z_{0}-c and z_{0}+c such that G(z)=0; i.e., F(x_{1},y_{1},z)=0. Moreover this value of z is unique. Since this holds for any (x,y) such
over this domain there a function z=f(x,y) such that F(x,y,z)=0.
Corollary 1: The function f(x,y) determined above is continuous.
Corollary 2: The partial derivatives of the function f(x,y) determined above are given by:
Proof:
The Mean Value Theorem says that for a function z=h(x) with a continuous derivative
This can be extended to a binary function w=G(x,z) with continuous partial derivatives so that
This is not the only generalization of the Mean Value Theorem but it is sufficient for the purpose here. Let G(x,z) be F(x,y,z) with y held fixed. Then
Now let Δz be the change in z for z on the surface F(x,y,z)=0. Thus ΔF=0 and hence
In the limit as Δx goes to zero this becomes
Likewise, by an analogous procedure,
The proof is essentially the same as for Theorem 1, but some unnecessary restrictions in the proof can be removed. Within the set S there will be a parallelpiped containing P and specified by its lower and upper corner points (X_{1}^{L},...,X_{n}^{L},z^{L}) and (X_{1}^{U}X_{1}^{U},z^{U}) such that:
and ∂F/∂z has the same sign as at P.
Then F(x_{1}(P),...,x_{1}(P),z^{U}) and F(x_{1}(P),...,x_{1}(P),z^{L}) will have opposite signs. For any point Q within the parallelpiped, ignoring the coordinate z,
will have the same signs as for P and are thus of opposite sign. For a fixed Q then G(z)=F(x_{1}(P),...,x_{1}(P),z) has opposite signs at z^{U} and z^{L} and therefore there is a z between these two levels such that G(z)=0. This value of z is a function of ((x_{1}(Q),...,x_{1}(Q)); i.e., z=f((x_{1}(Q),...,x_{1}(Q)).
Proof:
Consider F(x,y,u,v)=0 and the point P and apply Theorem 2 to get
v as a function of x,y and u; i.e., find v=h(x,y,u). Substitute h(x,y,u)
for v in G(x,y,u,v) to obtain H(x,y,u)=G(x,y,u,h(x,y,u)). H(x,y,u)=0 so
Theorem 1 applies and thus there exists f(x,y) such that u=f(x,y). A
substitution of this function for u in v=h(x,y,u) gives v=g(x,y)=h(x,y,f(x,y)).
Proof: The procedure is the same as in the proof of Theorem 3. Theorem 2 is applied to one of the F-functions to obtain one of the u-variables as a function of the x-variables and the other u-variables. This expression for the u-variable as function of the other variables is substituted into a second F-function and another u-variable is obtained as a function of the remaining variables until one u-variable is found as a function of only the x-variables. This u-variable is then substituted into the preceding expression for a u-variable. This process is continued until all of the u-variables are obtained as functions only of the x-variables.
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