Shadow price

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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.

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 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.

[edit] Illustration #1

Suppose a consumer faces prices \,\! p_1,p_2 and is endowed with income \,\!m, then the consumers problem is: \max \{\,\!u(x_1,x_2)\mbox{ } :\mbox{ } p_1x_1+p_2x_2=m\}. Forming the Langrangian auxilliary 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 it's saddle point we obtain \,\! x^*_1\mbox{, }x^*_2\mbox{, }\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 Langrange 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.

[edit] 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):= \arg\max \{\,\!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 that \,\! \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_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

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

This again gives the obvious interpretation, on extra dollar of optimal expenditure will lead to \,\! \lambda^* units of optimal utility.

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