|
|
| Line 1: |
Line 1: |
| {{IEP assignment|course=Wikipedia:India_Education_Program/Courses/Fall_2011/Development Economics_Year_3_Group_A|university=Symbiosis School of Economics|term=2011 Q3}}
| | == 「フーボーは笑いながら言った == |
| 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 (mathematics)|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.<ref>[http://www.answers.com/topic/shadow-price Shadow Price: Definition and Much More from Answers.com<!-- Bot generated title -->]</ref> 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.
| | どの2宮殿それ [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタン バッグ]？<br><br>「トップグレードライフルの神である。あなたに今秦Yuは、魂の製油所を修復することです。「フーボーは笑いながら言った。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタン 東京] 「この神である主の血精錬を識別できないか？ '<br><br>秦Yuは彼の右手、銃の把握を首を縦に振った。<br>秦ゆう手のひらに沿った銃<br>と涼しい、暗い場所では、寒さの波を感じるように渡し、秦Yuは躊躇しなかった。指はすぐに垂れ血をドロップします。<br><br>は「オム...... '秦Yuはわずかに震えすぐに槍で一滴の血液を吸収する。再びかすか放出された光を赤ブラッド。その銃を突きつけも血走った上に、ますます明るい丸で囲んだ。<br><br>秦Yuqingの透明度は接続を感じる [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 靴]。この槍と魂の関係 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_11.php クリスチャンルブタン 靴 メンズ]！<br><br>「ガンズ名 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン スニーカー] '雪'トップグレード神が破壊されることはありませ難しいですが、身体を壊しています。2二次的影響をむさぼり。 '秦ゆう脳が突然ガンスリンガーの「雪」についての一連のメッセージを持っていた。<br><br>秦Yuは完全にこのロッドガンスリンガーの力を理解するショックの心によってではない攻撃を受けた場合 |
| | | 相关的主题文章： |
| 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 (mathematics)|optimization]] problem has a shadow price or [[Dual_problem|dual]] variable.
| | <ul> |
| | | |
| The value of the shadow price can provide decision-makers with insights into problems. For instance if a constraint limits the amount of labor available to you to 40 hours per week, the shadow price will tell you how much you should 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.
| | <li>[http://www.hddxdc.com/plus/feedback.php?aid=16 http://www.hddxdc.com/plus/feedback.php?aid=16]</li> |
| | | |
| == Shadow Price of Foreign Exchange ==
| | <li>[http://www.zhijie520.com/plus/feedback.php?aid=467 http://www.zhijie520.com/plus/feedback.php?aid=467]</li> |
| [[File:ForexShadowPrice.jpg|ForexShadowPrice|thumb|left|400px|400px|]]
| | |
| {{ - }}
| | <li>[http://www.osumi.or.jp/ryo/bbs/aska.cgi http://www.osumi.or.jp/ryo/bbs/aska.cgi]</li> |
| | |
| == Illustration #1 ==
| |
| Suppose a consumer faces prices <math>\,\! p_1,p_2</math> and is endowed with income <math>\,\!m</math>, then the consumer's problem is:
| |
| <math> | |
| \max \{\,\!u(x_1,x_2)\mbox{ } :\mbox{ } p_1x_1+p_2x_2=m\}</math>. Forming the Lagrangian auxiliary function <math>\,\! L(x_1,x_2,\lambda):= u(x_1,x_2)+\lambda(m-p_1x_1-p_2x_2)</math>, taking first order conditions and solving for its saddle point we obtain <math>\,\! x^*_1\mbox{, }x^*_2\mbox{, }\lambda^*</math> which satisfy:
| |
| :<math> \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}</math>
| |
| 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 <math>\,\! \lambda^*</math> as above, then the change in maximal utility per dollar of additional income will be equal to <math>\,\! \lambda^*</math> 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
| |
| :<math> U(p_1,p_2,m):= \max \{\,\!u(x_1,x_2)\mbox{ } :\mbox{ } p_1x_1+p_2x_2=m\}</math>,
| |
| then we have the identity
| |
| :<math>\,\! U(p_1,p_2,m)=u(x_1^*(p_1,p_2,m),x_2^*(p_1,p_2,m)) </math>,
| |
| where <math>\,\! x_1^*(\cdot,\cdot,\cdot),x_2^*(\cdot,\cdot,\cdot)</math> are the demand functions, i.e. <math> 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</math>
| |
| | | |
| Now define the optimal expenditure function
| | </ul> |
| :<math>\,\! E(p_1,p_2,m):=p_1x_1^*(p_1,p_2,m)+p_2x_2^*(p_1,p_2,m) </math>
| |
| Assume differentiability and that <math>\,\! \lambda^* </math> is the solution at <math>\,\! p_1,p_2,m</math>, then we have from the multivariate chain rule:
| |
| :<math>\,\! \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} </math>
| |
| Now we may conclude that
| |
| :<math>\,\! \lambda^* = \frac{\partial U/\partial m}{\partial E/\partial m} \approx \frac{\Delta \mbox{Optimal Utility }}{\Delta \mbox{Optimal Expenditure}}</math>
| |
| This again gives the obvious interpretation, one extra dollar of optimal expenditure will lead to <math>\,\! \lambda^*</math> units of optimal utility.
| |
| | |
| ==Control theory==
| |
| 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 mechanics|Hamiltonian]] via [[Pontryagin's minimum principle]].
| |
| | |
| ==See also==
| |
| * [[Market price]]
| |
| * [[Reduced cost]]
| |
| * [[Linear programming]]
| |
| | |
| == 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
| |
| | |
| == References ==
| |
| <references/> | |
| | |
| [[Category:Operations research]]
| |
| [[Category:Mathematical economics]]
| |