Lecture 6: Competitive Equilibrium in the Growth Model (II) ECO 503: Macroeconomic Theory I Benjamin Moll Princeton University Fall 204 /6
Plan of Lecture Sequence of markets CE 2 The growth model and the data 2/6
Arrow-Debreu CE period 0: markets for everything Sequence of Markets CE Sequence of Markets CE: particular markets at particular points in time Period 0 Period Period 2... market for period 0 capital,......... period 0 labor,......... period 0 output,......... period 0 labor,......... period ahead borrowing/leding......... Individ. formulates plan at t = 0, but executes it in real time in contrast, in ADCE everything happens in period 0 SOMCE features explicit borrowing & lending riskless one-period bond that pays real interest rate r t 3/6
Sequence of Market CE Definition: A SOMCE for the growth model are sequences {c t,h t,k t,a t,w t,r t,r t } s.t. (HH max) Taking {w t,r t,r t } as given, {c t,h t,k t,a t } solves β t u(c t ) s.t. max {c t,h t,k t,a t} c t +k t+ ( δ)k t +a t+ R t k t +w t h t +(+r t )a t c t 0, 0 h t, k t+ 0, k 0 = k 0, a 0 = 0 ( T ) lim a T+ 0 T +r t 2 (Firm max) Taking {w t,r t,r t } as given, {k t,h t } solves max k t,h t F(k t,h t ) w t h t R t k t 3 (Market clearing) For each t: k t 0, h t 0 t. c t +k t+ ( δ)k t = F(k t,h t ) ( ) a t+ = 0 ( ) 4/6
a t = HH bond holdings Comments a t > 0: HH saves, a t < 0: HH borrows period-t price of bond that pays off at t +: q t = /(+r t ) some people like to write c t +k t+ ( δ)k t +q t b t+ R t k t +w t h t +b t this is equivalent with b t = (+r t )a t and q t = /(+r t ) Interpretation of bond market clearing condition ( ) bonds are in zero net supply more generally, in economy with individuals i =,...,N N a i,t+ = 0 i= for every dollar borrowed, someone else saves a dollar here only one type, so a t+ = 0. Q: since a t = 0, why not eliminate? A: need to know eq. r t 5/6
( ) is a so-called no-ponzi condition Comments with period budget constraints only, individuals could choose time paths with a t no-ponzi condition ( ) rules out such time paths: a t cannot become too negative implies that sequence of budget constraints can be written as present-value (or time-zero) budget constraint return to this momentarily Could have written firm s problem as ( t ) max (F(k t,h t ) w t h t R t k t ) k t 0, h t 0 {k t,h t} +r s s=0 but this is a sequence of static problems so can split them up 6/6
Sequence BC + no-ponzi PVBC Result: If {c t,i t,h t } satisfy the sequence budget constraint c t +i t +a t+ = R t k t +w t h t +(+r t )a t and if the no-ponzi condition ( ) holds with equality, then {c t,i t,h t } satisfy the present value budget constraint ( t ) ( t ) (c t +i t ) = (R t k t +w t h t ) +r s +r s s=0 s=0 Proof: next slide 7/6
Proof Write period t budget constraint as a t+ = (R t k t +w t h t c t i t )+a t +r t +r t At t = 0,t =,... a = (R 0 k 0 +w 0 h 0 c 0 i 0 )+a 0 +r 0 +r 0 a 2 = (R k +w h c i ) +r 0 +r +r 0 +r + +r 0 (R 0 k 0 +w 0 h 0 c 0 i 0 )+a 0 By induction/repeated substitution ( T ) ( T t ) a T+ = (R t k t +w t h t i t c t ) +r t +r s s=0 Result follows from taking T and imposing ( ) 8/6
Why no-ponzi Condition? Expression also provides some intuition for no-ponzi condition ( T ) ( T t ) a T+ = (R t k t +w t h t i t c t ) +r t +r s s=0 Suppose for the moment this were a finite horizon economy would impose: die without debt, i.e. a T+ 0 in fact, HH s would always choose a T+ = 0 Right analogue for infinite horizon economy ( T ) lim a T+ 0 T +r t and HH s choose {a t } so that this holds with equality no-ponzi condition not needed for physical capital because natural constraint k t 0. 9/6
Characterizing SOMCE Necessary conditions for consumer problem (h t = wlog) c t : β t u (c t ) = λ t = multiplier on period t b.c. () k t+ : λ t = λ t+ (R t+ + δ) (2) a t+ : λ t = λ t+ (+r t+ ) (3) c t +k t+ ( δ)k t +a t+ = R t k t +w t h t +(+r t )a t (4) no-ponzi: lim T ( T TVC on k : TVC on a : +r t )a T+ 0 (5) lim T βt u (c T )k T+ = 0 (6) lim T βt u (c T )a T+ = 0 (7) initial : k 0 = k 0, a 0 = 0 (8) 0/6
Characterizing SOMCE Necessary conditions for firm problem F k (k t,h t ) = R t, F h (k t,h t ) = w t (9) Market clearing c t +k t+ ( δ)k t = F(k t,h t ), a t+ = 0 (0) /6
Characterizing SOMCE (), (3) and (5) β T u (c T ) = λ T = T +r t lim T βt u (c T )a T+ 0 No-Ponzi condition looks very similar to TVC on {a t } But no-ponzi and TVC are different conditions Kamihigashi (2008) A no-ponzi-game condition is a constraint that prevents overaccumulation of debt, while a typical transversality condition is an optimality condition that rules out overaccumulation of wealth. They place opposite restrictions, and should not be confused. 2/6
(2) and (3) Characterizing SOMCE +r t+ = R t+ + δ i.e. rate of return on bonds = rate of return on capital arbitrage condition if this holds, HH is indifferent between a and k (), (2) and (9) u (c t ) βu (c t+ ) = f (k t+ )+ δ () () + TVC (6) + initial condition (8) + market clearing (0) = same set of equations as for SP problem Hence: SOMCE allocation is same as social planner s allocation this is actually somewhat surprising, see next slide 3/6
Why is SOMCE allocation =SP s alloc.? Relative to ADCE, we closed down many markets Q: Why do we still get SP solution even though we closed down many markets? A: We only closed down markets that didn t matter In fact, ADCE and SOMCE are equivalent 4/6
Equivalence of SOMCE and ADCE Recall HH s problem in ADCE (last lecture): β t u(c t ) s.t. max {c t,h t,k t} p t (c t +k t+ ( δ)k t ) p t (R t k t +w t h t ) Have shown earlier: HH s problem in SOMCE is same with present-value budget constraint ( t ) ( t ) (c t +k t+ ( δ)k t ) = (R t k t +w t h t ) +r s +r s s=0 Clearly these are equivalent ADCE is SOMCE with p t = t s=0 +r s SOMCE is ADCE with +r t+ = p t /p t+ Firm s problems are also equivalent. s=0 5/6
Why is SOMCE allocation =SP s alloc.? riskless one-period bond is surprisingly powerful one period ahead borrowing and lending arbitrary period ahead borrowing and lending When is SOMCE allocation with one-period bonds SP s allocation? That is, when do the welfare theorems fail? risk (idiosyncratic or aggregate) welfare theorems may hold if sufficiently rich insurance markets financial frictions. Examples: interest rate = r t(a t) with r t 0. in more general environments: borrowing constraint a t 0 or collateral constraints (need to back debt with collateral) a t+ θk t+... 6/6