Wrapping set

The envelope theorem (also called envelope theorem , envelope theorem or envelope theorem ) is a fundamental theorem of the calculus of variations that is often used in microeconomics . It describes how the optimal value of the objective function of a parameterized optimization problem behaves when the parameters are changed.

A distinction is usually made between two versions of the envelope theorem: one for optimization problems without and one for those with constraints, the first version being a special case of the second.

presentation

Optimization problem without constraints

(Envelope theorem for optimization problems without constraints :) Be a continuously differentiable function and a scalar - in short . The problem is given ${\displaystyle f(\mathbf {x} ;q)}$ ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}$ ${\displaystyle q}$ ${\displaystyle f(\mathbf {x} ;q)=f(x_{1},\ldots ,x_{n};q)}$

${\displaystyle \max _{\mathbf {x} }f(\mathbf {x} ,q)}$

that has a solution that is continuously differentiable. Then is given by what is known as the optimal value function of (that is, the original function is evaluated at the point - here only dependent on - where it assumes its maximum). The wrapping theorem then says: ${\displaystyle \mathbf {x} ^{*}(q)=(x_{1}^{*},\ldots ,x_{n}^{*})}$ ${\displaystyle v(q)\equiv f(\mathbf {x} ^{*}(q);q)}$ ${\displaystyle f}$ ${\displaystyle q}$

${\displaystyle {\frac {\mathrm {d} v(q)}{\mathrm {d} q}}=\left.{\frac {\partial f(\mathbf {x} ;q)}{\partial q}}\right|_{\mathbf {x} =\mathbf {x} ^{*}(q)}}$

It turns out that when calculating the first-order effect, a variation of has no influence on the change of . ${\displaystyle q}$ ${\displaystyle v(q)=f(\mathbf {x} (q),q)}$ ${\displaystyle \mathbf {x} }$

Extension: The sentence applies analogously to several parameters. It then holds for the maximization problemwith(,) andfor any(): ${\displaystyle \max _{\mathbf {x} }f(\mathbf {x} ,\mathbf {q} )}$ ${\displaystyle f(\mathbf {x} ;\mathbf {q} )=f(x_{1},\ldots ,x_{n};q_{1},\ldots ,q_{k})}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle q\in \mathbb {R} ^{k}}$ ${\displaystyle v(\mathbf {q} )\equiv f(x_{1}^{*}(\mathbf {q} ),\ldots ,x_{n}^{*}(\mathbf {q} );\mathbf {q} )}$ ${\displaystyle j}$ ${\displaystyle 1\leq j\leq k}$

${\displaystyle {\frac {\partial v(\mathbf {q} )}{\partial q_{j}}}=\left.{\frac {\partial f(\mathbf {x} ;\mathbf {q} )}{\partial q_{j}}}\right|_{\mathbf {x} =\mathbf {x} ^{*}(\mathbf {q} )}}$

Optimization problem with constraints

(Generalized Envelope theorem for optimization problems with constraints :) Be a continuously differentiable function and a scalar - in short . The problem is given ${\displaystyle f(\mathbf {x} ;q)}$ ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle q}$ ${\displaystyle f(\mathbf {x} ;q)=f(x_{1},\ldots ,x_{n};q)}$

${\displaystyle \max _{\mathbf {x} }f(\mathbf {x} ,q)\quad }$ under the constraints ${\displaystyle (g_{1}(\mathbf {x} ;q)=b_{1},\ldots ,g_{m}(\mathbf {x} ;q)=b_{m})}$

that has a solution that is continuously differentiable. It is the corresponding Lagrange function . The long-range multipliers are also continuously differentiable. In addition, the Jacobi matrix has the rank . ${\displaystyle \mathbf {x} ^{*}(q)=(x_{1}^{*},\ldots ,x_{n}^{*})}$ ${\displaystyle {\mathcal {L}}(\mathbf {x} ,\mathbf {\lambda } ;q)\equiv f(\mathbf {x} ;q)-\sum _{i=1}^{m}\lambda _{i}g_{i}(\mathbf {x} ,q)}$ ${\displaystyle \lambda _{1}(q),\ldots ,\lambda _{m}(q)}$ ${\displaystyle D\mathbf {g} (\mathbf {x} ^{*})}$ ${\displaystyle m}$

Then is an optimal value function of and says the enveloping theorem: ${\displaystyle v(q)\equiv f(\mathbf {x} ^{*}(q);q)}$ ${\displaystyle f}$

${\displaystyle {\frac {\mathrm {d} v(q)}{\mathrm {d} q}}=\left.{\frac {\partial {\mathcal {L}}(\mathbf {x} ;q)}{\partial q}}\right|_{\mathbf {x} =\mathbf {x} ^{*}(q)}=\left.{\frac {\partial f(\mathbf {x} ;q)}{\partial q}}\right|_{\mathbf {x} =\mathbf {x} ^{*}(q)}-\sum _{i=1}^{m}\lambda _{i}\left.{\frac {\partial g_{i}(\mathbf {x} ,q)}{\partial q}}\right|_{\mathbf {x} =\mathbf {x} ^{*}(q)}}$

Extension: The sentence can also be used in cases with several parameters. With analogous definitions the following applies for any(): ${\displaystyle j}$ ${\displaystyle 1\leq j\leq k}$

${\displaystyle {\frac {\partial v(\mathbf {q} )}{\partial q_{j}}}=\left.{\frac {\partial {\mathcal {L}}(\mathbf {x} ;\mathbf {q} )}{\partial q_{j}}}\right|_{\mathbf {x} =\mathbf {x} ^{*}(\mathbf {q} )}=\left.{\frac {\partial f(\mathbf {x} ;\mathbf {q} )}{\partial q_{j}}}\right|_{\mathbf {x} =\mathbf {x} ^{*}(\mathbf {q} )}-\sum _{i=1}^{m}\lambda _{i}\left.{\frac {\partial g_{i}(\mathbf {x} ,\mathbf {q} )}{\partial q_{j}}}\right|_{\mathbf {x} =\mathbf {x} ^{*}(\mathbf {q} )}}$

Remarks

${\displaystyle v(\mathbf {q} )}$is the envelope of the family of curves , hence the name of the sentence. ${\displaystyle \{f(\mathbf {x} ,\mathbf {q} ):\mathbf {x} \in \mathbb {R} ^{n}\}}$

Example without constraints

The following problem is given as an example:

${\displaystyle \max _{x}f(x,\mathbf {q} ),\quad \mathbf {q} \in \mathbb {R} ^{2}}$with . ${\displaystyle f(x,\mathbf {q} )=-5x^{2}+10xq_{1}+30xq_{2}+12q_{1}}$

Is the first order condition of the maximization problem

${\displaystyle {\frac {\partial f(x,\mathbf {q} )}{\partial x}}=-10x+10q_{1}+30q_{2}{\overset {!}{=}}0}$.

Placing them condition to, the following for the "optimal" : . If this is put back into the original function, the optimum value function is obtained . It is now of interest how this feedforward function changes when it changes. This should first be shown with the wrapping sentence and then “directly” for illustration. With the wrapping sentence it immediately follows: ${\displaystyle x}$ ${\displaystyle x^{*}(\mathbf {q} )=(10q_{1}+30q_{2})/10=q_{1}+3q_{2}}$ ${\displaystyle v(\mathbf {q} )=f(x^{*}(\mathbf {q} ),\mathbf {q} )}$ ${\displaystyle \mathbf {q} }$

${\displaystyle {\frac {\partial v(\mathbf {q} )}{\partial q_{1}}}=\left.{\frac {\partial f(x,\mathbf {q} )}{\partial q_{1}}}\right|_{(x^{*}(\mathbf {q} ),\mathbf {q} )}=10(q_{1}+3q_{2})+12}$

and

${\displaystyle {\frac {\partial v(\mathbf {q} )}{\partial q_{2}}}=\left.{\frac {\partial f(x,\mathbf {q} )}{\partial q_{2}}}\right|_{(x^{*}(\mathbf {q} ),\mathbf {q} )}=30(q_{1}+3q_{2})}$

The same result could have been calculated “directly”. To do this, however, the optimal value function must be calculated explicitly:

${\displaystyle {\begin{aligned}v(\mathbf {q} )&=-5(q_{1}+3q_{2})^{2}+10(q_{1}+3q_{2})q_{1}+30(q_{1}+3q_{2})q_{2}+12q_{1}\\&=-5(q_{1}^{2}+6q_{1}q_{2}+9q_{2}^{2})+10q_{1}^{2}+30q_{2}q_{1}+30q_{1}q_{2}+90q_{2}^{2}+12q_{1}\\&=5q_{1}^{2}+12q_{1}+30q_{1}q_{2}+45q_{2}^{2}\end{aligned}}}$

And so too

${\displaystyle {\begin{aligned}{\frac {\partial v(\mathbf {q} )}{\partial q_{1}}}&=10q_{1}+12+30q_{2}=10(q_{1}+3q_{2})+12\\{\frac {\partial v(\mathbf {q} )}{\partial q_{2}}}&=10q_{2}+30q_{1}+90q_{2}=30(q_{1}+3q_{2})\end{aligned}}}$

application

One application can be found in microeconomics . There you can use the wrapping theorem both in the theory of enterprises and in the theory of households.

In the field of theory of enterprises referred to the volume of production, depending on the input , it follows by sets as the price vector for output and Inputgut, and as producers gain , Hotelling's lemma . However, it is also possible to use the envelope theorem to minimize costs. This works analogously to Shephard's lemma . ${\displaystyle y(x)}$ ${\displaystyle x}$ ${\displaystyle \mathbf {q} =(p,w)}$ ${\displaystyle f}$ ${\displaystyle f(x,p,w)=py(x)-wx}$

In household theory, the envelope theorem is used in connection with indirect utility functions . It is easy to use Roy's identity to analyze what happens if there is a change in income or price. For this, the indirect utility function is partially derived according to income and price.

See also

• Hicks's demand function
• Roy's identity
• Shephard's lemma

Web links

• Consumer Theory and the Envelope Theorem (on the relationship between the Envelope Theorem and other microeconomic concepts of household theory) (PDF, 0.1 MB)

literature

• Andreu Mas-Colell, Michael Whinston, Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-195-07340-1 . [For the wrapping set, pp. 964–966.]
• Carl P. Simon, Lawrence Blume: Mathematics for Economists. WW Norton, New York and London 1994, ISBN 0-393-95733-0 . [For the wrapping set, pp. 453–457.]
• Thorsten Pampel: Mathematics for Economists. , Springer-Verlag 2009, ISBN 3-642-04489-1 , Chapter 15.3: The cover set

Individual evidence

1. ↑ See Simon / Blume 1994, pp. 453 f.
2. ↑ See Simon / Blume 1994, pp. 455 f .; Mas-Colell / Whinston / Green 1995, pp. 965 f.