# β

## SummaryEdit

A shadow price is commonly referred to as a monetary value assigned to currently unknowable or difficult-to-calculate costs. It is based on the willingness to pay principle - in the absence of market prices, the most accurate measure of the value of a good or service is what people are willing to give up in order to get it. Shadow pricing is often calculated on certain assumptions and premises. As a result, it is subjective and somewhat imprecise and inaccurate.[1] The origin of these costs is typically due to an externalization of costs or an unwillingness to recalculate a system to account for marginal production. For example, consider a firm that already has a factory full of equipment and staff. They might estimate the shadow price for a few more units of production as simply the cost of the overtime. In this manner, some goods and services have near zero shadow prices, for example information goods. Less formally, a shadow price can be thought of as the cost of decisions made at the margin without consideration for the total cost.

While shadow pricing may be imprecise and inaccurate, it is still frequently employed as a useful technique and is widely used in cost-benefit analyses. For instance, before taking on a project, businesses and governments may want to weigh the costs and benefits of the project to decide whether the project is worthwhile. While tangible costs and benefits such as the cost of labor are easy to quantify, intangible costs and benefits such as the number of hours saved is much more difficult to quantify. In this case, business owners and policymakers turn to shadow pricing to determine what these intangibles are. There are usually numerous tools that help those to determine what the monetary values of these intangibles are. Some of the most common are: contingent valuation, revealed preferences, and hedonic pricing.

In constrained optimization in economics, the shadow price is the change, per infinitesimal unit of the constraint, in the optimal value of the objective function of an optimization problem obtained by relaxing the constraint. If the objective function is utility, it is the marginal utility of relaxing the constraint. If the objective function is cost, it is the marginal cost of strengthening the constraint. In a business application, a shadow price is the maximum price that management is willing to pay for an extra unit of a given limited resource.[2] For example, if a production line is already operating at its maximum 40-hour limit, the shadow price would be the maximum price the manager would be willing to pay for operating it for an additional hour, based on the benefits he would get from this change.

More formally, the shadow price is the value of the Lagrange multiplier at the optimal solution, which means that it is the infinitesimal change in the objective function arising from an infinitesimal change in the constraint. This follows from the fact that at the optimal solution the gradient of the objective function is a linear combination of the constraint function gradients with the weights equal to the Lagrange multipliers. Each constraint in an optimization problem has a shadow price or dual variable.

## Shadow Price - Cost-Benefit AnalysisEdit

### Illustration #1Edit

Suppose a consumer with utility function ${\displaystyle u}$  faces prices ${\displaystyle \,\!p_{1},p_{2}}$  and is endowed with income ${\displaystyle \,\!m.}$  Then the consumer's problem is: ${\displaystyle \max\{\,\!u(x_{1},x_{2}){\mbox{ }}:{\mbox{ }}p_{1}x_{1}+p_{2}x_{2}=m\}}$ . Forming the Lagrangian auxiliary function ${\displaystyle \,\!L(x_{1},x_{2},\lambda ):=u(x_{1},x_{2})+\lambda (m-p_{1}x_{1}-p_{2}x_{2})}$ , taking first order conditions and solving for its saddle point we obtain ${\displaystyle \,\!x_{1}^{*}{\mbox{, }}x_{2}^{*}{\mbox{, }}\lambda ^{*}}$  which satisfy:

${\displaystyle \lambda ^{*}={\frac {\frac {\partial u(x_{1}^{*},x_{2}^{*})}{\partial x_{1}}}{p_{1}}}={\frac {\frac {\partial u(x_{1}^{*},x_{2}^{*})}{\partial x_{2}}}{p_{2}}}.}$

This gives us a clear interpretation of the Lagrange multiplier in the context of consumer maximization. If the consumer is given an extra dollar (the budget constraint is relaxed) at the optimal consumption level where the marginal utility per dollar for each good is equal to ${\displaystyle \,\!\lambda ^{*}}$  as above, then the change in maximal utility per dollar of additional income will be equal to ${\displaystyle \,\!\lambda ^{*}}$  since at the optimum the consumer gets the same amount of marginal utility per dollar from spending his additional income on either good.

### Illustration #2Edit

Holding prices fixed, if we define the indirect utility function as

${\displaystyle U(p_{1},p_{2},m)=\max\{\,\!u(x_{1},x_{2}){\mbox{ }}:{\mbox{ }}p_{1}x_{1}+p_{2}x_{2}=m\}}$ ,

then we have the identity

${\displaystyle \,\!U(p_{1},p_{2},m)=u(x_{1}^{*}(p_{1},p_{2},m),x_{2}^{*}(p_{1},p_{2},m))}$ ,

where ${\displaystyle \,\!x_{1}^{*}(\cdot ,\cdot ,\cdot ),x_{2}^{*}(\cdot ,\cdot ,\cdot )}$  are the demand functions, i.e. ${\displaystyle x_{i}^{*}(p_{1},p_{2},m)=\arg \max\{\,\!u(x_{1},x_{2}){\mbox{ }}:{\mbox{ }}p_{1}x_{1}+p_{2}x_{2}=m\}{\mbox{ for }}i=1,2.}$

Now define the optimal expenditure function

${\displaystyle \,\!E(p_{1},p_{2},m)=p_{1}x_{1}^{*}(p_{1},p_{2},m)+p_{2}x_{2}^{*}(p_{1},p_{2},m).}$

Assume differentiability and that ${\displaystyle \,\!\lambda ^{*}}$  is the solution at ${\displaystyle \,\!p_{1},p_{2},m}$ , then we have from the multivariate chain rule:

${\displaystyle \,\!{\frac {\partial U}{\partial m}}={\frac {\partial u}{\partial x_{1}}}{\frac {\partial x_{1}^{*}}{\partial m}}+{\frac {\partial u}{\partial x_{2}}}{\frac {\partial x_{2}^{*}}{\partial m}}=\lambda ^{*}p_{1}{\frac {\partial x_{1}^{*}}{\partial m}}+\lambda ^{*}p_{2}{\frac {\partial x_{2}^{*}}{\partial m}}=\lambda ^{*}\left(p_{1}{\frac {\partial x_{1}^{*}}{\partial m}}+p_{2}{\frac {\partial x_{2}^{*}}{\partial m}}\right)=\lambda ^{*}{\frac {\partial E}{\partial m}}.}$

Now we may conclude that

${\displaystyle \,\!\lambda ^{*}={\frac {\partial U/\partial m}{\partial E/\partial m}}\approx {\frac {\Delta {\mbox{Optimal Utility }}}{\Delta {\mbox{Optimal Expenditure}}}}.}$

This again gives the obvious interpretation, one extra dollar of optimal expenditure will lead to ${\displaystyle \,\!\lambda ^{*}}$  units of optimal utility.

## Control theoryEdit

In optimal control theory, the concept of shadow price is reformulated as costate equations, and one solves the problem by minimization of the associated Hamiltonian via Pontryagin's minimum principle.

## ReferencesEdit

1. ^ Staff, Investopedia (2003-11-26). "Shadow Pricing". Investopedia. Retrieved 2018-03-02.
3. ^ International encyclopedia of the social & behavioral sciences. Smelser, Neil J., Baltes, Paul B. (1st ed ed.). Amsterdam: Elsevier. 2001. p. 13272. ISBN 0080430767. OCLC 47869490.
4. ^ "Applications of the contingent valuation method in developing countries". www.fao.org. Retrieved 2018-03-02.
5. ^ "What is the contingent valuation method, CVM?" (PDF). 2017-10-31. Retrieved 2018-03-02.
6. ^ a b Jonathan,, Gruber,. Public finance and public policy (Fifth edition ed.). New York. p. 226. ISBN 9781464143335. OCLC 914290290.
7. ^ Diamond, Peter A; Hausman, Jerry A (1994). "Contingent Valuation: Is Some Number Better than No Number?". Journal of Economic Perspectives. 8 (4): 45–64. doi:10.1257/jep.8.4.45. ISSN 0895-3309.
8. ^ Staff, Investopedia (2010-06-28). "Revealed Preference". Investopedia. Retrieved 2018-03-03.
9. ^ "Demerits of the Revealed Preference Theory". economics-the-economy.knoji.com. Retrieved 2018-03-03.
10. ^ Staff, Investopedia (2003-11-20). "Hedonic Pricing". Investopedia. Retrieved 2018-03-03.
11. ^ "Hedonic Pricing Method". www.ecosystemvaluation.org. Retrieved 2018-03-03.
12. ^ "Hedonic Pricing - Transportation Benefit-Cost Analysis". bca.transportationeconomics.org. Retrieved 2018-03-03.
13. ^ "Carbon Shadow Pricing" from www.ClimateMoneyPolicy.com