Shadow price
Encyclopedia
In constrained optimization in economics
, the shadow price is the instantaneous change per unit of the constraint in the objective value of the optimal solution of an optimization
problem obtained by relaxing the constraint
. In other words, it is the marginal utility
of relaxing the constraint, or, equivalently, 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. 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.
The value of the shadow price can provide decision-makers with powerful insights 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.
and is endowed with income 
, then the consumer's problem is:
. Forming the Lagrangian auxiliary function 
, taking first order conditions and solving for its saddle point we obtain 
which satisfy:
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
as above, then the change in maximal utility per dollar of additional income will be equal to 
since at the optimum the consumer gets the same amount of marginal utility per dollar from spending his additional income on either goods. In this case the shadow price concept does not carry much importance because the objective function (utility) and the constraint (income) are measured in different units.
,
then we have the identity
,
where
are the demand functions, i.e. 
Now define the optimal expenditure function
Assume differentiability and that
is the solution at 
, then we have from the multivariate chain rule:
Now we may conclude that
This again gives the obvious interpretation, one extra dollar of optimal expenditure will lead to
units of optimal utility.
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
.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, the shadow price is the instantaneous change per unit of the constraint in the objective value of the optimal solution of an optimization
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
problem obtained by relaxing the constraint
Constraint (mathematics)
In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...
. In other words, it is the marginal utility
Marginal utility
In economics, the marginal utility of a good or service is the utility gained from an increase in the consumption of that good or service...
of relaxing the constraint, or, equivalently, the marginal cost
Marginal cost
In economics and finance, marginal cost is the change in total cost that arises when the quantity produced changes by one unit. That is, it is the cost of producing one more unit of a good...
of strengthening the constraint.
In a business
Business
A business is an organization engaged in the trade of goods, services, or both to consumers. Businesses are predominant in capitalist economies, where most of them are privately owned and administered to earn profit to increase the wealth of their owners. Businesses may also be not-for-profit...
application, a shadow price is the maximum price that management is willing to pay for an extra unit of a given limited resource. 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
Optimization (mathematics)
In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
problem has a shadow price or dual variable.
The value of the shadow price can provide decision-makers with powerful insights 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.
Shadow Price of Foreign Exchange
Foreign exchange has emerged as a scarce resource for developing countries, as these nations face multiple problems of balance of payment. As a result, the demand price is greater than the official price of the foreign exchange. At this point, a cost-benefit analyst needs to decide whether the price of imports and exports is to be valuated at the nominal price or the adjusted price. Most economists believe that pricing should be adjusted upwards for both exports and imports, so that the adjusted price complies with the scarcity value, which is reflected by the demand price of foreign exchange.Illustration #1
Suppose a consumer faces prices and is endowed with income , then the consumer's problem is:. Forming the Lagrangian auxiliary function , taking first order conditions and solving for its saddle point we obtain which satisfy: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
as above, then the change in maximal utility per dollar of additional income will be equal to since at the optimum the consumer gets the same amount of marginal utility per dollar from spending his additional income on either goods. In this case the shadow price concept does not carry much importance because the objective function (utility) and the constraint (income) are measured in different units.Illustration #2
Holding prices fixed, if we define,then we have the identity
,where
are the demand functions, i.e. Now define the optimal expenditure function
Assume differentiability and that
is the solution at , then we have from the multivariate chain rule:Now we may conclude that
This again gives the obvious interpretation, one extra dollar of optimal expenditure will lead to
units of optimal utility.Control theory
In optimal controlOptimal control
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...
theory, the concept of shadow price is reformulated as costate equations, and one solves the problem by minimization of the associated Hamiltonian
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
via Pontryagin's minimum principle
Pontryagin's minimum principle
Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. It was formulated by the Russian mathematician Lev Semenovich...
.
Further reading
- Ravi Kanbur (1987). "shadow pricing ," The New Palgrave: A Dictionary of Economics, v. 4, pp. 316-17.
- Economics of Development and Planning(Theory and Practice)- S.K Mishra, V.K Puri