SGZ Macro Week 3, Lecture 2: Suboptimal Equilibria. SGZ 2008 Macro Week 3, Day 1 Lecture 2

Transcription

1 SGZ Macro Week 3, : Suboptimal Equilibria 1

2 Basic Points Effects of shocks can be magnified (damped) in suboptimal economies Multiple equilibria (stationary states, dynamic paths) in suboptimal economies Government policies i can improve welfare in these economies (or can be the source of suboptimality) 2

3 Outline A. Static example: production externality 1. Positive effect of one individual s effort on level and marginal product of another s effort 2. Optimal allocation 3. Decentralized allocation (Nash equilibrium) Output too small, if positive Output can be zero in equilibrium or much too small 4. Subsidy can raise output to efficient level. 3

4 Outline (cont d) B. Dynamic example: financing a given amount of government purchases with an output tax, whose rate is endogenously set to raise specified revenue 1. Formulation as a sequence problem (later: formulation as a dynamic program ) 2. Steady state outcomes: Accumulation too low Multiple equilibria can occur with endogenous taxes 3. Dynamic outcomes: multiple equilibria can arise with endogenous taxation 4

5 A. Static Example Consider an economy populated by many identical agents All agents have a preferences over goods and leisure which take the form χ ucl c l 1+ γ 1+ γ (, ) = log( ) (1 ) with γ > 0 5

6 A1: Externality Model: Individual id production function Output for individual depends positively on his own labor input (n) and the average amount of input (N) chosen by individuals in the economy. If there are J individuals, id then N J 1 N 1 = n j so that = J j= 1 n J i where n i is the effort of the agent under examination. We ll assume that δn/δn i = 0, which requires that there are an infinite number of individuals, but is approximately true for large numbers of individuals. 6

7 Production Function Cobb Douglas Marginal product if everyone works harder is higher than if just the individual under study works harder y = a n N < < 1 [ α α ] 0 α 1 y y N= α y y N= n= n n n n But note social production function is y=an if n=n) 7

8 Production Function (cont d) If no one else works, there is no point to my working y = a n N < < 1 [ α α ] 0 α 1 y = 0 N 0 n = 8

9 A2. Social production function and optimal allocations in economy Production function and time constraint c = y = a* n Lagrangian for optimal allocations n + l = L = u (,) c l + λ [ an c ] + ω[1 n l] 1 9

10 Social optimum FOCs for planner 1 c: 0 = uc ( c, l) λ = λ c l :0 = u ( c, l) ω = χ(1 l) γ ω n :0 l = λa ω plus constraints t Optimal consumption and work 1 1+γ 1 n= ( ) c= an χ 10

11 A3. Decentralized (Nash) Equilibrium i Each agent chooses n taking as given N Symmetric equilibrium: because all agents are same, it must be that equilibrium N=n Zero activity equilibrium (N=n=0) is transparent, so focus on positive activity equilibrium 11

12 Individual optimization Problem max ucl (, ) cln,, s.t. c= f( n, N) and n+ l = 1 12

13 Lagrangian and FOCs L = ucl (, ) + λ[ f( n, N ) c ] + ω [1 n l ] 1 c:0 = uc ( c, l) λ = λ c l:0 = u ( c, l) ω = χ(1 l) γ ω l y n:0 = λfn( n, N) ω = λα ω n plus constraints 13

14 Equilibrium (with n=n) Using above equations, can work out that 1 α 1 + γ n= ( ) n= N c=a*n χ Contrast to social optimum which sets α=1 in above, so that equilibrium n is inefficiently low (and consumption too because c=an) 14

15 Concepts Output and work is inefficiently low because people do not take into account the social benefit raising other individual s productivity of their own work effort. This is a direct result of the interaction of the externality and competitive behavior, treating ti aggregate conditions as uninfluenced by one s own actions. If all agents were to agree on a jointly optimal level of work effort, it would be higher. 15

16 Coordination failure Low level of activity equilibrium is sometimes called a coordination failure equilibrium, because individuals are unable to work together to bring about the higher eqbm. Zero is extreme case: think of an economy in which people must choose to work on one of two production methods (individual or joint), but do so knowing choices of others. Could have low level equilibrium just on relatively unproductive process 2. [Example is rigged so that n is higher h in process 2, but that t b has no effect on the level l of n, so that c can be made much lower on 2 than on 1. Hence, with higher work and lower consumption, 2 is evidently worse than 1]. α 1 α Process 1: y = a[ n1 N1 ] 0 < α < 1 Process 2: y = bn b<<a 2 16

17 A4. Corrective subsidy A government which can impose lump sum taxes can subsidize employment and Eliminate the zero equilibrium (by paying people to work even if others do not) Stimulate the equilibrium to an efficient level, via a subsidy such that (1+s)α=1. Question to think about: what if government must raise revenue via labor taxation? 17

18 B. Dynamic Example Study economy with a large number of agents concerned with maximizing j U = β u( c t ) j= 0 and sometimes uc ( ) = log( c) t t 18

19 Output taxation Private agents have budget constraint c + ( k (1 δ ) k ) = (1 τ ) f( k ) + T t t+ 1 t t t t where c is consumption k is capital f(k) ()soupu is output T is lump-sum transfer τ is tax rate 19

20 Agents take as given The sequence of tax rates The sequence of transfers But they recognize that t these may depend d on the actions of others e.g.: If others accumulate more k, they will pay higher taxes and my transfers will go up 20

21 Dynamic Lagrangian t L= β u( c t ) t= 0 t + βλ[(1 τ ) f ( k ) + T t= 0 t t t t c ( k (1 δ ) k )] t t + 1 t 21

22 FOCs c : 0 = u ( c ) λ t c t t k : 0 = λ + βλ [(1 τ ) f ( k ) + (1 δ)] t+ 1 t t+ 1 t+ 1 k t+ 1 λ : 0 = (1 τ ) f ( k ) + T c ( k (1 δ) k ) t t t t t t+ 1 t 22

23 Equilibrium conditions: words Private agent efficiency conditions Government budget constraint Consistency (k=k) Government rules for g, τ and T 23

24 Equilibrium Conditions: Equations c :0 = u ( c ) λ t c t t k :0 = λ + βλ [(1 τ ) f ( k ) + (1 δ)] t+ 1 t t+ 1 t+ 1 k t+ 1 λ : 0 = (1 τ ) f ( k ) + T c ( k (1 δ ) k ) t t t t t t+ 1 t T = τ f ( K ) g k t t t t t = K t plus appropriate rule (A,B,C) for g and τ 24

25 Equilibrium Conditions: Equations simplified 0 = u ( c ) λ c t t 0 = λ + βλ [(1 τ ) f ( k ) + (1 δ )] t t+ 1 t+ 1 k t+ 1 0 = f ( k ) c ( k (1 δ ) k ) g k t = t t t+ 1 t t = K t 25

26 What does government do with revenue? Case A (Redistribution/Endogenous Transfers): Refunds lump sum to individuals T t =τ t y t Case B (Endogenous purchases): Spends on a time-varying quantity of government purchases, depending on the exogenous tax rate and endogenous tax base: g t =τ t y t Case C: (Endogenous Tax Rates) Finances a given amount of goods, allow tax rate to vary endogenously with the tax base: τ t =g/y t 26

27 Implications for Stationary States Case A and B 1 = β [(1 τ ) f ( k ) + (1 δ )] k (1 τ ) f ( k ) = r + δ with β = 1/(1 + r ) k Case C 1 = β[(1 τ( k)) f ( k) + (1 δ)] with τ( k) = g/ f( k) k g (1 ) fk ( k ) = r +δ f( k) 27

28 CD example Case A and B: f ( k ) = k α f ( k ) = α k k α 1 α(1 τ ) (1 τ) fk( k) = ( r+ δ) kss = [ ] r + δ 1 1 α 28

29 Cases A and B Stationary capital stock in cases A and B r+δ f k (k) of return rate fraction of ss capital 29

30 Tax Rate and Stationary State capital 5 Output tax rate(τ) tax rate(τ) 0.8 margin nal product rev venue tax rate(τ) tax rate(τ) 30

31 Key points Higher tax rate yields lower capital; lower output; higher pre-tax marginal product Laffer curve: high tax rates can produce less revenue revenue maximizing tax rate 31

32 Stationary State in Case C rate of return Stationary capital stocks in case C r+δ (1-τ)f k (k) with τ(k)=g/f(k) fraction of ss capital rate tax fraction of ss capital 32

33 Key features Two stationary points, both of which produce the required revenue g One with high tax, low capital One with low tax, high capital Tax rates are sharply pydifferent across two stationary points From earlier figure, know that revenue is rising with taxes near low tax point and falling with taxes near high tax point 33

34 Topic C2: Analyzing dynamics Case A: Exogenous movements in tax rates highlighting substitution effects, because tax revenues rebated. Case B: Just like a productivity shift because revenues are used up : shift to output available for private consumption and investment c + g + i = f( k ) t t t t c + i = f( k ) g = (1 τ ) f( k ) t t t t t t 34

35 Analyzing Dynamics To consider the effects of endogenous government policy on local dynamics, let s use general policy rules that allow for rich dependence of taxes and purchases on capital, then we can specialize to A,B, C or cover other cases τt = τ( Kt ) gt = g( Kt) T = τ ( K ) f ( K ) g ( K ) t t t t 35

36 Recalling equilibrium conditions (note no shocks) 0 = u ( c ) λ c t t 0 = λ + βλ [(1 τ( k )) f ( k ) + (1 δ )] t t+ 1 t+ 1 k t+ 1 0 = f( k ) c ( k (1 δ ) k ) g( k ) k t = K t t t t+ 1 t t 36

37 Linearization (near stationary point) ct c λt λ 0 = σ *( ) c λ λt λ λt+ 1 λ kt+ 1 k 0 = ( ) + ( ) η( ) λ λ k kt k ct c kt+ 1 k 0 = l( ) φ ( ) ( ) k c k 37

38 Dynamic system λt+ 1 λ λt λ 1 η λ 1 0 λ = 0 1 kt+ 1 k φ/ σ χ k t k k k 38

39 Coefficients Standard With endogenous fiscal policy χ =f k +(1-δ)=1/β=(1+r)>1 ( ) χ=f k +(1-δ)- g k =(1+r) g k φ = c/k Same σ= -(c* u cc )/ u c Same η =- βkf kk >0 η = -βk(1-τ)f kk +βkf k τ k 39

40 Characteristic polynomial Write in convenient form to identify sum of roots [1+χ+ϕ ] and product (χ) of roots 2 0 = z [1 + + ] z+ = ( z 1)( z 2) with = / χ ϕ χ μ μ ϕ ησ φ Write in convenient form for graphical analysis ( z 1)( z χ) = ϕ z 40

41 41

42 Real root cases ϕ < ϕ > 0< χ < 1 χ > 1 0 A: 2 stable B: 2 unstable 0 C: 1 of each D: 1 of each 42

43 Complex root cases Stability is dictated entirely by χ since complex roots have common modulus equal to square root of χ Complex roots obtain if discriminant i i is negative, i.e., if [1+χ+ϕ] 2-4* χ<0. 43

44 What s the point? If we change values of χ and η in critical ways, then multiplicity obtains. Critical values are: Resources: χ<1 versus χ>1, which means that a higher capital stock reduces rather than increases future k, given the c level. Incentives: η <0 versus η >0, which means that return to investing rises with capital rather than fall. (η dictates sign of ϕ) 44

45 How does extra stable root change RE solutions? Recall that when we talked about the single variable case, we said that there could be multiple equilibria if the difference equation s root was less than one in absolute value y = ae y 1 + cx rootis ( az 1) = 0 or z = 1/ a t t t+ t 1 c yt+ 1 = ( ) Et yt+ 1 xt + ξt+ 1 with Et ξt+ 1 = 0 a a 45

46 Current setup Two variables. One initial condition. Two stable roots Implication can shock λ just as could shock y above, resetting initial conditions. 46

47 Sample dynamics (shock to λ) from region II 2 Consumption 0.6 Government purchases Tax Rate 4 Capital

48 Key features Note endogenous variation in tax rates due to fiscal rule Thought experiment on impact: if everybody else invests a lot, then tax rates will fall making it efficient for me to invest more 48

49 Key features Thought experiment subsequently: how will the attempts of others to increase or decrease on capital formation affect my incentives to invest. 49

50 Application to model C Low activity ss is unstable in prior model High activity it ss is stable (return to capital diminishes locally). Stability is in saddlepoint sense, as in neoclassical l growth model without distortions because product of roots (l) is still 1/β = (1+r)>1 1 50

51 Topic C. Recursive formulation of fdistorted deconomies Agents take as given the sequence of taxes and transfers above, now view as function of a state variable which is exogenous to them τ = τ ( K) T = T( K) K ' = H( K) 51

52 Dynamic program Bellman equation vkk (, ) = max { uc ( ) +βvk ( ', K')} ck, ' subject to k' = (1 τ ( K)) f( k) + (1 δ ) k+ T( K) c K ' = H ( K ) Decision rules: c = c ( k, K)and k' = h ( k, K) 52

53 Equilibrium T( K) = τ ( K) f( K) g( K) hkk (, ) = H ( K ) k= K = 53

54 An efficient way to construct h(k,k)=h(k) Take FOCs and use Envelope theorem Impose consistency conditions Equilibrium policies C(K), H(K) satisfy 0 = u ( C( K)) Λ( K) c 0 = Λ ( K) + βλ( K')[(1 τ( K') f ( K') + (1 δ)] 0 = f( K) (1 δ ) K C( K) g( K) K ' = H( K) k 54

55 Note restriction Made equilibrium just a function of natural state variable K Limit of finite horizon economies, each of which h takes such a function governing the future as given 55

56 Policy function viewpoint Think of K =H(K) as coming out of the problem above. Some old and new possibilities, suggested by this nonlinear perspective Old: Multiple steady states New: Multiple equilibria at point in time if backward bending policy function 56

57 Multiple steady-states (note stability) 57

58 Multiple equilibria at a point in time 58

59 Later picture Clearly rule out in standard concave (growth model problem) without distortions Similar pictures have arisen in recent work on optimizing, i i discretionary monetary policy authority in New Keynesian models like those to be discussed d next week. 59