Ch04. Maximum Theorem, Implicit Function Theorem and Envelope - PowerPoint PPT Presentation

Ch04. Maximum Theorem, Implicit Function Theorem and Envelope Theorem Ping Yu Faculty of Business and Economics The University of Hong Kong Ping Yu (HKU) MIFE 1 / 27

The Maximum Theorem 1 The Implicit Function Theorem 2 The Envelope Theorem 3 Ping Yu (HKU) MIFE 2 / 27

Overview of This Chapter The maximum theorem: the continuity of the optimizer and optimum with respect to (w.r.t.) a group of parameters. The implicit function theorem: the differentiablity of the optimizer w.r.t. a group of parameters. The envelope theorem: the differentiablity of the optimum w.r.t. a group of parameters. In the previous two chapters, we are " moving along a curve", that is, f and G are given (or the parameter value is given) and we are searching over x along f to find the optimal x � . In this chapter, we are " shifting a curve", i.e., we change the parameter values to shift f and G , and check how the optimizer and optimum respond to such shifting. Ping Yu (HKU) MIFE 2 / 27

The Maximum Theorem The Maximum Theorem Ping Yu (HKU) MIFE 3 / 27

The Maximum Theorem History of the Maximum Theorem Claude J. Berge (1926-2002), French Ping Yu (HKU) MIFE 4 / 27

The Maximum Theorem The Problem Problem: We want to know how x � (or the maximized function) depends on the exogenous parameters a , rather than what x � is for a particular a . - For example, in consumer’s problem max x 1 , x 2 u ( x 1 , x 2 ) s.t. p 1 x 1 + p 2 x 2 = y , how ( x � 1 , x � 2 ) or ( u ( x � 1 , x � 2 )) depends on ( p 1 , p 2 , y ) rather than what ( x � 1 , x � 2 ) is when p 1 = 2, p 2 = 7, and y = 25, i.e., we are interested in what the demand function is. Mathematically, x 1 , ��� , x n f ( x 1 , ��� , x n , a 1 , ��� , a k ) max s.t. g 1 ( x 1 , ��� , x n , a 1 , ��� , a k ) = c 1 , . . . g m ( x 1 , ��� , x n , a 1 , ��� , a k ) = c m . When is the "demand" function continuous? Ping Yu (HKU) MIFE 5 / 27

The Maximum Theorem Continuous Correspondence Define the maximized value of f when the parameters are ( a 1 , ��� , a k ) as v ( a 1 , ::: , a k ) = f ( x � 1 ( a 1 , ::: , a k ) , ::: , x � n ( a 1 , ::: , a k ) , a 1 , ::: , a k ) . Using the feasible set, the problem can be restated as x 1 , ��� , x n f ( x 1 , ��� , x n , a 1 , ��� , a k ) max s.t. ( x 1 , ��� , x n ) 2 G ( a 1 , ��� , a k ) , where G ( a 1 , ��� , a k ) � f ( x 1 , ��� , x n ) j g j ( x 1 , ��� , x n , a 1 , ��� , a k ) = c j , 8 j g is a set valued function or a correspondence. Two sets of vectors A and B are within ε of each other if for any vector x in one set there is a vector x 0 in the other set such that x 0 2 B ε ( x ) . The correspondence G is continuous at ( a 1 , ::: , a k ) if 8 ε > 0, 9 δ > 0 such that if ( a 0 1 , ��� , a 0 k ) is within δ of ( a 1 , ��� , a k ) then G ( a 0 1 , ��� , a 0 k ) is within ε of G ( a 1 , ��� , a k ) . [Figure here] - If G is a function, then the continuity of G as a correspondence is equivalent to the continuity as a function. - The continuity of the functions g j does not necessarily imply the continuity of the feasible set. (Exercise) Ping Yu (HKU) MIFE 6 / 27

The Maximum Theorem Figure: Continuous and Discontinuous Correspondence Ping Yu (HKU) MIFE 7 / 27

The Maximum Theorem The Maximum Theorem Theorem Suppose that f ( x 1 , ��� , x n , a 1 , ��� , a k ) is continuous (in ( x 1 , ��� , x n , a 1 , ��� , a k ) ), that G ( a 1 , ��� , a k ) is a continuous correspondence, and that for any ( a 1 , ��� , a k ) the set G ( a 1 , ��� , a k ) is compact. Then (i) v ( a 1 , ��� , a k ) is continuous, and (ii) if ( x � 1 ( a 1 , ��� , a k ) , ��� , x � n ( a 1 , ��� , a k )) are (single valued) functions then they are also continuous. Compactness of G ( a 1 , ��� , a k ) and continuity of f w.r.t. x guarantees the existence of x � for any a by the Weierstrass Theorem. Uniqueness of x � is guaranteed by the uniqueness theorem in Chapter 3. Ping Yu (HKU) MIFE 8 / 27

The Implicit Function Theorem The Implicit Function Theorem Ping Yu (HKU) MIFE 9 / 27

The Implicit Function Theorem History of the IFT Augustin-Louis Cauchy (1789-1857), French Ping Yu (HKU) MIFE 10 / 27

The Implicit Function Theorem The Problem Problem: How to solve x � 2 R n from n FOCs and how sensitive of x � is to the parameter? Suppose that we have n endogenous variables x 1 , ::: , x n , m exogenous variables or parameters, b 1 , ::: , b m , and n equations or equilibrium conditions f 1 ( x 1 , ��� , x n , b 1 , ��� , b m ) = 0 , f 2 ( x 1 , ��� , x n , b 1 , ��� , b m ) = 0 , . . . f n ( x 1 , ��� , x n , b 1 , ��� , b m ) = 0 , or, using vector notation, f ( x , b ) = 0 , where f : R n + m ! R n , x 2 R n , b 2 R m , and 0 2 R n . When can we solve this system to obtain functions giving each x i as a function of b 1 , ::: , b m ? Ping Yu (HKU) MIFE 11 / 27

The Implicit Function Theorem The Case of Linear Functions Suppose that our equations are a 11 x 1 + ��� + a 1 n x n + c 11 b 1 + c 1 m b m = 0 a 21 x 1 + ��� + a 2 n x n + c 21 b 1 + c 2 m b m = 0 . . . a n 1 x 1 + ��� + a nn x n + c n 1 b 1 + c nm b m = 0 We can write this, in matrix notation, as � x � [ A j C ] = 0 or Ax + Cb = 0 , b where A is an n � n matrix, C is an n � m matrix, x is an n � 1 (column) vector, and b is an m � 1 vector. As long as A can be inverted or is of full rank, x = � A � 1 Cb or ∂ x ∂ b 0 = � A � 1 C . Ping Yu (HKU) MIFE 12 / 27

The Implicit Function Theorem The General Nonlinear Case If there are some values ( x , b ) for which f ( x , b ) = 0 , then a parallel result holds in the neighborhood of ( x , b ) if we linearize f in that neighborhood. Theorem Suppose that f : R n + m ! R n is a C 1 function on an open set A � R n + m and that ( x , b ) in A is such that f ( x , b ) = 0 . Suppose also that 0 1 ∂ f 1 ( x , b ) ∂ f 1 ( x , b ) ��� B ∂ x 1 ∂ x n C ∂ f ( x , b ) B . . C ... . . = B C . . ∂ x 0 @ A ∂ f n ( x , b ) ∂ f n ( x , b ) ��� ∂ x 1 ∂ x n is of full rank . Then there are open sets A 1 � R n and A 2 � R m with x 2 A 1 , b 2 A 2 and A 1 � A 2 � A such that for each b in A 2 there is exactly one g ( b ) in A 1 such that f ( g ( b ) , b ) = 0 . Moreover, g : A 2 ! A 1 is a C 1 function and � ∂ g ( b ) � � ∂ f ( g ( b ) , b ) � � 1 � ∂ f ( g ( b ) , b ) � = � . ∂ b 0 ∂ x 0 ∂ b 0 n � m n � n n � m Ping Yu (HKU) MIFE 13 / 27

The Implicit Function Theorem Explanations of the IFT Intuition: For ( x , b ) in a neighborhood of ( x , b ) such that f ( x , b ) = 0 with x = g ( b ) , we have ∂ f ( g ( b ) , b ) ∂ g ( b ) + ∂ f ( g ( b ) , b ) = 0 n � m , ∂ x 0 ∂ b 0 ∂ b 0 so � ∂ f ( g ( b ) , b ) � � 1 � ∂ f ( g ( b ) , b ) � ∂ g ( b ) � � A � 1 C in the linear case . = � ∂ b 0 ∂ x 0 ∂ b 0 This is a local result, rather than a global result. If b is not close to b we may not be able to solve the system, and that for a particular value of b there may be many values of x that solve the system, but there is only one close to x . [Figure here] For all values of b close to ¯ b we can find a unique value of x close to ¯ x such that f ( x , b ) = 0. However, (1) for each value of b there are other values of x far away x that also satisfy f ( x , b ) = 0, and (2) there are values of b , such as ˜ from ¯ b for which there are no values of x that satisfy f ( x , b ) = 0. x , ¯ That ∂ f ( ¯ b ) is of full rank means j g 0 ( ¯ x ) j 6 = 0 in this example. ∂ x Ping Yu (HKU) MIFE 14 / 27

The Implicit Function Theorem Figure: Intuition for the IFT: f ( x , b ) = g ( x ) � b , x , b 2 R Ping Yu (HKU) MIFE 15 / 27

The Implicit Function Theorem Comparative Statics The IFT does not provide conditions to guarantee the existence of ( x , b ) such that f ( x , b ) = 0 ; rather, it provides conditions such that if such an ( x , b ) exists, then we can also uniquely solve f ( x , b ) = 0 in its neighborhood. So the most important application of the IFT is to obtain ∂ g ( b ) rather than ∂ b 0 guarantee the existence or uniqueness of the solution. Example The seller of a product pays a proportional tax at a flat rate θ 2 ( 0 , 1 ) . Hence, the effective price received by the seller is ( 1 � θ ) P , where P is the market price for the good. Market supply and demand are given by the differentiable functions Q d D ( P ) , with D 0 ( � ) < 0 = Q s S (( 1 � θ ) P ) , with S 0 ( � ) > 0 = and equilibrium requires market clearing, that is, Q s = Q d . Analyze, graphically and analytically, the effects of a decrease in the tax rate on the quantity transacted and the equilibrium price. Ping Yu (HKU) MIFE 16 / 27