Shadow price

Shadow price

Loosely, the shadow price is the change in the objective value of the optimal solution of an optimization problem obtained by relaxing the constraint by one unit. 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. [ [http://www.answers.com/topic/shadow-price Shadow Price: Definition and Much More from Answers.com ] ] For example, what is the price of keeping a production line operational for an additional hour if the production line is already operated at its maximum 40 hour limit? That price is the shadow price.

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 variable.

The value of the shadow price can provide decision makers powerful insight into problems. For instance if you have a constraint that limits the amount of labor available to 40 hours per week, the shadow price will tell you how much you would be willing to pay for an additional hour of labor. If your shadow price is $10 for the labor constraint, for instance, you should pay no more than $10 an hour for additional labor. Labor costs of less than $10/hour will increase the objective value; labor costs of more than $10/hour will decrease the objective value. Labor costs of exactly $10 will cause the objective function value to remain the same.

Illustration #1

Suppose a consumer faces prices ,! p_1,p_2 and is endowed with income ,!m, then the consumer's problem is:max {,!u(x_1,x_2)mbox{ } :mbox{ } p_1x_1+p_2x_2=m}. Forming the Lagrangian auxiliary function ,! L(x_1,x_2,lambda):= u(x_1,x_2)+lambda(m-p_1x_1-p_2x_2), taking first order conditions and solving for its saddle point we obtain ,! x^*_1mbox{, }x^*_2mbox{, }lambda^* which satisfy:: 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 ,! lambda^* as above, then the change in maximal utility per dollar of additional income will be equal to ,! lambda^* since at the optimum the consumer gets the same amount of marginal utility per dollar from spending his additional income on either good. In this case the shadow price concept does not carry much import-- the objective function (utility ) and the constraint (income) are measured in different units.

Illustration #2

Holding prices fixed, if we define : U(p_1,p_2,m):= max {,!u(x_1,x_2)mbox{ } :mbox{ } p_1x_1+p_2x_2=m},then we have the identity :,! U(p_1,p_2,m)=u(x_1^*(p_1,p_2,m),x_2^*(p_1,p_2,m)) ,where ,! x_1^*(cdot,cdot,cdot),x_2^*(cdot,cdot,cdot) are the demand functions, i.e. x_i^*(p_1,p_2,m):= argmax {,!u(x_1,x_2)mbox{ } :mbox{ } p_1x_1+p_2x_2=m} mbox{ for } i=1,2 Now define the optimal expenditure function :,! E(p_1,p_2,m):=p_1x_1^*(p_1,p_2,m)+p_2x_2^*(p_1,p_2,m) Assume differentiability and thaton ,! lambda^* is the solution at ,! p_1,p_2,m, then we have from the multivariate chain rule::,! 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_1frac{partial x_1^*}{partial m} + lambda^* p_2 frac{partial x_2^*}{partial m}=lambda^* left(p_1frac{partial x_1^*}{partial m} + p_2 frac{partial x_2^*}{partial m} ight) =lambda^* frac{partial E}{partial m} Now we may conclude that:,! lambda^* = frac{frac{partial U}{partial m{frac{partial E}{partial m approx frac{Delta mbox{Optimal Utility {Delta mbox{Optimal ExpenditureThis again gives the obvious interpretation, one extra dollar of optimal expenditure will lead to ,! lambda^* units of optimal utility.

ee also

* Market price

References

Further reading

* Ravi Kanbur (1987). "shadow pricing ," "", v. 4, pp. 316-17.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • shadow price — noun An estimated price for a product for which there is no market price • • • Main Entry: ↑shadow …   Useful english dictionary

  • shadow price — noun Economics the estimated price of something for which no market price exists …   English new terms dictionary

  • shadow price — The opportunity costs that arise in the solution to a linear programming model …   Accounting dictionary

  • Shadow Pricing — 1. The actual market value of one share of a money market fund. In this case, shadow pricing refers to securities that are accounted for based on amortized costs rather than a market valuation assignment. 2. The assignment of dollar values to non …   Investment dictionary

  • Shadow Hearts: Covenant — US cover Developer(s) Nautilus Publisher(s) …   Wikipedia

  • Shadow of the Colossus — Североамериканская обложка издания игры Разработчик Team Ico Издатель Sony Computer Entertainme …   Википедия

  • Shadow of the colossus — Разработчик Team Ico Издатель Дизайнеры Кэндзи Кайдо (продюсер, менеджер) Фумито Уэда (главный дизайнер, режиссер) Даты выпуска …   Википедия

  • Shadow system — is a term used in Information Services for any application relied upon for business processes that is not under the jurisdiction of a centralized Information Systems department. That is, the Information Systems department did not create it, was… …   Wikipedia

  • Shadow of the comet — Éditeur Infogrames Développeur Infogrames Date de sortie 1993 Genre aventure Mode de jeu Un joueur Pl …   Wikipédia en Français

  • Shadow of the Comet — Éditeur Infogrames Développeur Infogrames Date de sortie 1993 Genre aventure Mode de jeu Un joueur Plate form …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”