Essential Microeconomics -- CHAPTER -: SHADOW PRICES An intuitive approach: profit maimizing firm with a fied supply of an input Shadow prices 5 Concave maimization problem 7 Constraint qualifications 8 Kuhn-Tucker conditions 4 Quasi-concave maimization problem 5 Eample: Consumer Choice 6
Essential Microeconomics -- Maimizing profit with a fied supply of an input Ma{ f ( ) b g( ) 0, 0} We shall interpret this mathematical problem as the decision problem of a profit maimizing firm This firm can produce any vector of outputs which satisfies the non-negativity constraint, 0and a resource constraint g ( ) b A simple eample is a linear constraint Each unit of requires a units of resource b Then n a = a b =
Essential Microeconomics -3- Suppose that solves the optimization problem If the firm increases, the direct effect on profit is f However, the increase in also utilizes additional resources so that there must be offsetting changes in other commodities We introduce a shadow price λ 0 to reflect the opportunity cost of using the additional resources The etra resource use is g Multiplying this by the shadow price of the resource gives the opportunity cost of increasing The net gain to increasing is therefore f g ( ) λ ( )
Essential Microeconomics -4- If the optimum for commodity,, is strictly positive, this marginal net gain must be zero That is f g > 0 ( ) λ ( ) 0 = If is zero, the marginal net gain to increasing cannot be positive Hence f g = 0 ( ) - λ ( ) 0 Summarizing f g ( ) - λ ( ) 0, with equality if > 0 Since must be feasible b g ( ) 0 Moreover, we have defined λ to be the opportunity cost of additional resource use Then if not all the resource is used, λ must be zero Summarizing, b g ( ) 0, with equality if λ > 0
Essential Microeconomics -5- There is a convenient way to remember these conditions First write the i-th constraint in the form hi ( ) 0, i =,, m In vector notation h ( ) 0 Thus in our eample we write the constraint as h( ) b g ( ) 0 = Then introduce a vector of Lagrange multipliers or shadow prices λ and define the Lagrangian L (, λ) = f( ) + λ h ( ) The first order conditions are then all restrictions on the partial derivatives of L (, λ) f h (i) = + λ 0, i with equality if > 0, =,, n L (ii) hi ( ) 0, λ = i with equality if λ i > 0, i =,, m Eercise: Solve the following problem Ma{ U ( ) = ln + ln( + 3) p+ p + p33 60} (i) if p = (,, 6) (ii) p = (,,) (iii) p = (,,4)
Essential Microeconomics -6- Restatement of FOC (i) L f h 0 and L = + λ = 0 i, =,, n (ii) = hi( ) 0 and λi = 0 λ λ i i, i =,, m In vector notation (i) L f h 0 and L = + λ = 0 (ii) = h ( ) 0 and λ = 0 λ λ, i =,, m These are called the complementary slackness conditions Note: (ii) implies that L (, λ ) = f( )
Essential Microeconomics -7- Special case: Concave constrained optimization problem Proposition Sufficient Conditions for a Maimum For the problem Ma{ f ( ) X} where X= { 0, h ( ) 0}, suppose that the concave functions f, h,, h m are differentiable at If (i) L f h 0 and L = + λ = 0 and (ii) = h ( ) 0 and λ = 0 λ λ, i =,, m then solves Ma{ f ( ) X} where X= { 0, h ( ) 0} Proof: The sum of concave functions is concave so L (, λ) = f( ) + λ h ( ) is a concave function of Hence L(, λ) L (, λ) + (, λ) ( ) Appealing to (i), the second term on the right hand side is negative Appealing to (ii) L (, λ ) = f( ) Therefore L (, λ) = f( ) + λ h( ) f( ) But λ h ( ) 0 for any feasible Therefore f( ) f( ) QED
Essential Microeconomics -8- Non concave problem: Constraint Qualifications Define X to be the set of feasible vectors, that is X = { 0, h( ) 0, i =,, m} i The constraint qualifications holds at X if hi (i) for each constraint that is binding at the associated gradient vector ( ) 0 (ii) X, the set of non-negative vectors satisfying the linearized binding constraints has a non-empty interior L hi Note: The linear function hi ( ) = hi( ) + ( ) ( ) has the same value and gradient as the function hi ( ) at Thus the linear approimation at on the boundary of the constrained set satisfying the constraint ( ) 0 i hi ( ) ( ) 0 hi h is hi ( ) + ( ) ( ) 0 Since hi ( ) = 0 the linear constraint is
Essential Microeconomics -9- Eample : Constraint qualification holds Ma{ f ( ) = ln 0, h( ) = 0} As is readily confirmed, the maimizing value of is = (,) The feasible set and contour set for f through = (,) are depicted The Lagrangian is L (, λ) = ln + ln + λ( ) The first order conditions are therefore (i) L = λ 0, with equality if > 0, =, Fig -: Constrained maimum L (ii) 0, λ = with equality if λ > 0 As is readily checked, the FOC are all satisfied at (, λ ) = (,,)
Essential Microeconomics -0- Eample : Constraint qualification does not hold 3 { ( ) ln ln 0, ( ) ( ) 0} Ma f = + h = Since the feasible set and maimand are eactly the same as in eample, the solution is again = (,) The Lagrangian is (, ) ln ln ( ) 3 L λ = + + λ Differentiating by, L = = 3 λ( ) at = (,)
Essential Microeconomics -- Thus the first order condition does not hold at the maimum Our intuitive argument breaks down h because, the gradient of the constraint function, is zero at the maimum Thus the partial derivatives of h no longer reflect the opportunity cost of the scarce resource More formally, at the maimum, the linear approimation of the constraint is L hi hi ( ) = hi( ) + ( ) ( ) Then, as long as the constraint is binding ( hi ( ) = 0) and the gradient vector is not zero, the linearized constraint is h i ( ) ( ) 0, i =,, m
Essential Microeconomics -- There is a second (though unlikely) situation in which the linear approimations fail Consider the following problem Ma{ f ( ) = + h( ) = ( ) 0, 0} 3 The feasible set is the shaded region in the figure below Also depicted is the contour set for f through = (,0) Fig -a: Original problem From the figure it is clear that f takes on its maimum at = (,0) However, L ( ) = 3 λ( ) = Thus again the first order conditions do not hold at the maimum
Essential Microeconomics -3- This time the problem occurs because the feasible set, after taking a linear approimation of the constraint function, looks nothing like the original feasible set h At, the gradient vector ( ) = (0, ) Thus the linear approimation of the constraint h ( ) 0 through is h h h ( ) ( ) = ( )( ) + ( ) = 0 Linearized feasible set Fig -b: Linearized problem Since must be non-negative, the only feasible value of is = 0 In Figure -b the linearized feasible set is therefore the horizontal ais Then the solution to the linearized problem is not the solution to the original problem
Essential Microeconomics -4- As long as the constraint qualifications hold, the intuitively derived conditions are indeed necessary conditions This is summarized below Proposition: Necessary Conditions for a Constrained Maimum Suppose solves Ma{ f ( ) X} If the constraint qualifications hold at then there eists a vector of shadow prices λ 0 such that and (, λ) 0, =,, n with equality if > 0 (, λ) 0, i =,, m with equality if λi > 0 λ FOC (Kuhn-Tucker conditions) Since Kuhn and Tucker were the first to publish a complete set of constraint qualifications, the first order conditions (FOC) are often called the Kuhn-Tucker Conditions
Essential Microeconomics -5- Proposition: Sufficient Conditions for a Constrained Maimum Suppose that and are quasi-concave and the feasible set has a non-empty interior If the Kuhn-Tucker conditions hold at, and for each binding constraint,, then solves Intuition: Under these assumptions one can draw a hyperplane through which separates X and the upper contour set C f f U 0 ( ) = { ( ) ( )}
Essential Microeconomics -6- Eample: Consumer Choice Consider the following consumer choice problem: Ma{ U ( ) = ln( + 3 ) + ln p I} 0 (a) Solve if p = (,,4) 3 (b) Show that there is a unique maimizer * p p ( pi, ) unless = 3 Note that z = ln y is an increasing concave function and y = + 3 is linear and hence concave Therefore ln( + 3 ) is concave Also ln is concave Therefore U( ) is 3 concave Then the Kuhn-Tucker conditions are sufficient conditions for a maimum L = U( ) + λ( I p ) (a) FOC = λ 0 + 3 with equality if * > 0 3 = λ 0 + 3 with equality if * > 0 = 4λ 0 3 3 with equality if * 3 > 0
Essential Microeconomics -7- Try * >> 0 + 3 λ = 0, 3 + 3 λ = 0, 3 = 4λ From the first two equations λ = + 3 and λ = 3 + 3 But this is impossible Then try > 0 = * * = λ = 0, = 4λ = 0 3 3 Eliminating λ it follows that = 43 From the budget constraint + + 4 = 8 = I Then 3 3 = I and * 3 8 = I * Since the problem is concave, the necessary conditions are sufficient
Essential Microeconomics -8- (b) Once the consumer has chosen * 3, the consumer has commodities The consumer s problem is now to solve y= I p to spend on the other two * 3 3 Ma{ U (, ) p + p y}, That is, * 3 Ma{ln( + 3 ) + ln p + p y} * 3 If If 3 >, the consumer strictly prefers commodity over commodity p p 3 <, the consumer strictly prefers commodity over commodity p p Remark: This kind of utility function seems plausible for two products with similar characteristics such as two brands of beer Brand two is preferred but if the price premium is too high the consumer chooses brand