Introduction to second order approximation. SGZ Macro Week 3, Day 4 Lecture 1

Transcription

1 Introduction to second order approximation 1

2 Outline A. Basic concepts in static models 1. What are first and second order approximations? 2. What are implications for accuracy of welfare calculations (Kim and Kim, puzzle and response)? 3. What are implications for behavior? first and second order comparative statics decision-making under uncertainty B. Approximation of RE models 1. What general RE model to approximate? 2. What concepts? 3. What tools? 4. What results? 2

3 A. Basic concepts in Static Models Follow same strategy as with theory of taxation (positive and normative) by starting with static examples to show issues. Why? Background to dynamics Especially important because best general methods is not yet established for dynamic models How? Use approach (value function and decision rules) motivated by dynamic models 3

4 A. First and Second Order Approximations Tool: Taylor series approximation Consider application to indirect utility/value function, v(a) 1 va ( ) va ( ) + va *[ a ( a)] + vaa *[ a ( a)] 2 2 4

5 Behind value function General point: decision rules determine value function va ( ) = uca ( ( ), la ( )) implies that va ( ) = uca ( ( ), la ( )) v ( a) = u ( a)* c ( a) + u ( a)* l ( a) a c a l a v ( a) = u ( a)*[ c ( a)] + 2* u ( a)*[ l ( a) c ( a)] + u ( a)*[ c ( a)] 2 2 aa cc a cl a a ll l Tool: chain rule + u ( a) *[ c ( a)] + u ( a) *[ l ( a)] c aa l aa 5

6 A2: Welfare implications KK puzzle and resolution Literature: compute first order approximations to policy functions (open economy models, with complete and incomplete international financial markets) Simulate utility using these solutions and average to get expected utility Average utility for all countries higher with incomplete markets than with complete markets (pareto optimum): can t be the case! What literature did: omitted 2 nd order decision rule terms from calculations above. KK work stresses: accurate 2 nd order approximation of welfare requires 2 nd order approximation of decision rules 6

7 A: Implications for behavior 3. Second order comparative statics 4. Behavior under uncertainty 7

8 Comparative statics is based on implicit functions u ((),()) c a l a = λ() a c u ((),()) c a l a = a* λ() a l ca ( ) = a*[1 la ( )] 8

9 A3: Calculating 2 nd order approximation to behavior Based on conditions we ve just seen Requires level, first and second derivatives of decision rules with respect to exogenous variable(s), in this case a. 9

10 First order approximation (related to standard comparative statics) Need unknown first derivatives of decision rules. Tool is implicit function theorem Must solve 3 equations in 3 unknowns These depend on second derivatives of utility (since FOCs depend on 1 st derivatives) and on point of approximation (level of decision rules) More generally, p*m equations in p*m unknowns if there are p endogenous variables and m exogenous variables 10

11 Equations to be solved u * c + u * l = λ cc a cl a a u * c + u * l = + a* λ λ lc a ll a a c = [1 l] a* l a a 11

12 Second order approximation Evaluation of behavior and of welfare requires second derivatives of decision rules Tool is again implicit function theorem Unknowns are second derivatives of c,l,λ so again three equations in three unknowns. Generally, if p endogenous and m exogenous, then have p*m*m equations in p*m*m unknowns (or, using symmetry restrictions, can reduce to p*m*(m+1)/2. 12

13 u * c + u * l cc aa cl aa + u *( c ) + 2* u * c * l + u *( l ) = λ 2 2 ccc a ccl a a cll a aa u * c + u * l lc aa ll aa + u *( c ) + 2* u * c * l + u *( l ) 2 2 lcc a lcl a a lll a = 2* λ + a* λ a aa c = 2* l a* l aa a aa 13

14 Note Tedious: motivates use of symbolic mathematical packages to do the necessary differentiations Second order derivatives of decision rules depend depend on levels and first derivatives of decision rules. Practical implication: (1) solve for level of decisions (2) solve for first derivatives of decision rules (3) solve for second derivatives of decision rules 14

15 what we get Second order approximation to decision rules 2 nd order aspects of behavior 2 nd order approximation of welfare 15

16 A4: decision-making under uncertainty Want example that is close to model we ve just studied. Suppose wage is stochastic, but labor income must be decided on prior to seeing wage. For simplicity: focus on case of additively separable objective (not too many derivatives to calculate) and on case in which decision rules are exactly linear. 16

17 Model c = a * n v = max E{ u( c )} + h( n) n FOC:0=E{ a * u ( c )} + h ( n) c n 17

18 Question and approach How does behavior change as the individual faces more risk? a = a+ φε Increasing risk approach: perturbations in stretching parameter of 1 st and 2 nd order form E ε = 0 18

19 Decision rules and approximations n( φ) c= a * n( φ) = ( a+ φε )* n( φ) 19

20 Certain setting (φ=0) c,n such that { a* u ( c)} + h ( n) = 0 c c = a* n n 20

21 First order approximation 0= E{ a * uc( c )} + hn( n) φ = E{ ε* u ( c ) + a * u ( c )*[ ε* n+ a * n ]} c + h ( n)* n nn cc = a u c + h n φ 2 { * cc ( ) nn}* φ φ n = φ 0 21

22 Interpretation Decision-maker is locally risk neutral with respect to labor supply choice, given that a small risk is introduced from an initial position of certainty. 22

23 Second order approximation 2 2 0= E{ a * u 2 c( c )} + h ( ) 2 n n ( φ) ( φ) = E{ ε* u ( ) * ( )*[ * c c + a ucc c ε n+ a* nφ ]} ( φ) + h n n + h n n 2 nnn( )*( φ) nn( )*( φφ) 2 2 = E{( ε) * u ( c ) + a* u ( c )*[ ε* n+ a* n ] } cc ccc + E{ a * u ( c )*[2* ε * n + a * n ]} cc + h n n + h n 2 nnn( )*( φ ) nn( )*( φ φφ n φφ ) φ 2 2 ucc c nφφ a ucc c hnn = E( ε ) *[ ( )] + [ * ( ) + ] at φ=0 23

24 Implication Level of labor supply depends on variance 1 n n+ n φ + n φ 2 φφ φ 2 = n+ 1 θ *var( a ) 2 24

25 Welfare approximation Above, we stressed that one motivation was for considering second order approximations was welfare. Current context: Welfare doesn t depend on uncertainty with a first-order approximation Welfare depends on variance with a secondorder approximation 25

26 First order approximation v( φ) = max { Eu( c) h( n)} cn, st.. c= ( a+ φε ) n v(0) = u( an) h( n) v ( φ) = E{[ εn+ an ] u ( c )]} h n φ φ c n φ 26

27 Second order approximation c= ( a+ φε ) n v ( φ) = E{[ εn+ an ] u ( c )]} h n φ φ c n φ 2 v () φ = E{[( εn + an ) u () c + ( ε + an ) u ()]} c φφ φ φφ c φ h n nn 2 φ = + E( ε ) φφ h n since n = 0,[ au h ] = 0, n = 0 φφ c 2 u n cc n φφ φφ cc 27

28 B. Dynamic model 1. What general RE model to approximate? 2. What concepts? 3. What tools? 4. What results? 28

29 Stochastic model from DP perspective vka (, ) = max { uc ( ) + β Evk ( ', a') a} c c+ k' = F( a, k) 29

30 Efficiency conditions c:0 = u ( cka (, )) λ( ka, ) k':0 = λ( ka, ) + β Eμ( k', a') a λ : c 0 = Fk (, a) k'( k, a) c( k, a) ET : μ( ka, ) = λ( ka, )* F ( k, a) k 30

31 General form 0 = u ( cka (, )) λ( ka, ) c 0 = μ( ka, ) = λ( ka, )* F ( k, a) k β Eμ( k', a') a = λ( k, a) k '( k, a) ϕ( ka, ) = F( k, a) ck (, a) 31

32 Nature of general form 4 equations restricting 4 decision rules (highlighted in red) Decision rules are implicit functions of endogenous and exogenous state variables. Some compound structure necessitates chain rule: future marginal value depends on future capital which depends on current capital and productivity. 32

33 Nature of General Form Some equations are not dynamic (1) and (2) Some equations are state equations: normalized to have k with unit coefficient and as only future variable Some equations have future marginal values entering in an additively separable manner and a linear manner. 33