Lecture 5: Competitive Equilibrium in the Growth Model

Transcription

1 Lecture 5: Competitive Equilibrium in the Growth Model ECO 503: Macroeconomic Theory I Benjamin Moll Princeton University Fall /17

2 Competitive Eqm in the Growth Model Recall two issues we are interested in regarding resource allocation problems 1 efficient allocations 2 decentralized equilibrium allocations So far did (1). Now consider (2). Focus on particular decentralized equilibrium concept: competitive equilibrium see Lecture 1 benchmark notion of decentralized eqm, but not only one 2/17

3 Competitive Eqm in the Growth Model Static economies: one way to formulate CE Dynamic economies: three ways to formulate CE 1 Arrow-Debreu CE (ADCE) 2 Sequence of Markets CE (SOMCE) 3 Recursive CE (RCE) Outcomes same for all three. Just different representations. Begin with ADCE extension of static CE but defining commodities as pairs of goods time 3/17

4 Preliminaries Detail to consider in economy with capital: who owns capital? households who then rent it to firms? firms who own capital who are in turn owned by households? reality: see some of each Turns out this is of no substantive importance in this setting lecture: assume capital owned by HH and rented to firms homework: other extreme Also assume single stand-in firm homework: show that this is harmless We also go back to discrete-time formulation 4/17

5 ADCE Definition: An ADCE for the growth model are sequences {ct h,ht,k h t h,kt f,ht,p f t,w t,r t } s.t. 1 (HH max) Taking {p t,w t,r t } as given, {ct,h h t,k h t h } solves max β t u(c t ) s.t. {ct h,hh t,kh t } p t (ct h +kt+1 h (1 δ)kt h ) (R t k t +w t h t ) c h t 0, 0 h h t 1, k h t+1 0, k h 0 = k 0 2 (Firm max) Taking {p t,w t,r t } as given, {kt,h f t} f solves max (p t F(kt,h f t) w f t h f {kt f,hf t } t R t kt) f kt f 0, ht f 0. 3 (Market clearing) For each t: k h t = k f t, h h t = h f t, c h t +k h t+1 (1 δ)k h t = F(k f t,h f t) 5/17

6 Comments Single budget constraint for HH Prices take care of discounting implicitly Everything happens at t = 0 6/17

7 Simplifying An ADCE for the growth model are sequences {c t,h t,k t,p t,w t,r t } s.t. 1 (HH max) Taking {p t,w t,r t } as given, {c t,h t,k t } solves β t u(c t ) s.t. max {c t,h t,k t} p t (c t +k t+1 (1 δ)k t ) (R t k t +w t h t ) c t 0, 0 h t 1, k t+1 0, k 0 = k 0 2 (Firm max) Taking {p t,w t,r t } as given, {k t,h t } solves (p t F(k t,h t ) w t h t R t k t ) k t 0, h t 0. max {k t,h t} 3 (Market clearing) For each t: c t +k t+1 (1 δ)k t = F(k t,h t ) (k and h markets clear implicitly) 7/17

8 Characterizing ADCE First Welfare Theorem applies, so could simply use fact that ADCE allocation = planner s allocation But will later consider environments with various distortions (taxes, monopoly power, financial frictions) in which this fails want to know how to solve for ADCE even when First Welfare Theorem fails. Consider such method now. General idea: for each piece of ADCE: max problem necessary conditions + market clearing 8/17

9 Characterizing ADCE Necessary conditions for consumer problem (h t = 1 wlog) c t : β t u (c t ) = λp t, λ = multiplier on b.c. (1) k t+1 : λp t +λ[ p t+1 (1 δ) R t+1 ] = 0 (2) p t (c t +k t+1 (1 δ)k t ) TVC : (R t k t +w t ) (3) lim T βt u (c T )k T+1 = 0 (4) initial : k 0 = k 0 (5) Necessary conditions for firm problem Market clearing p t F k (k t,h t ) = R t (6) p t F h (k t,h t ) = w t (7) c t +k t+1 (1 δ)k t = F(k t,h t ) (8) 9/17

10 Characterizing ADCE: TVC? As before, can think of TVC (4) as coming from finite horizon problem max {c t,h t,k t} T β t u(c t ) s.t. T p t (c t +k t+1 (1 δ)k t ) T (R t k t +w t h t ), k t+1 0 Denote multipliers by λ,µ t, necessary conditions at t = T are β T u (c T ) = λp T λp T = µ T β T u (c T )k T+1 = 0 µ T k T+1 = 0 10/17

11 Characterizing ADCE Use (1) at t and t +1 From (2) From (6) u (c t ) βu (c t+1 ) = p t p t+1 (9) p t p t+1 = R t+1 p t+1 +1 δ u (c t ) βu (c t+1 ) = R t+1 p t+1 +1 δ R t = F k (k t,1) = f (k t ) p t u (c t ) βu (c t+1 ) = f (k t+1 )+1 δ (10) 11/17

12 Characterizing ADCE Recall u (c t ) βu (c t+1 ) = f (k t+1 )+1 δ (10) (10) + TVC (4) + initial condition (5) + market clearing (8) = same set of equations as for SP problem Hence: ADCE allocation is the same for the SP problem How get prices? can always normalize one price to unity: wlog set p 0 = 1 get R 0,w 0 from (6) and (7) at t = 0 get p 1 from (9) given c 0,c 1,p 0 get R 1,w 1 from (6) and (7) at t = 1 given p /17

13 Aside: Walras Law Q: Why didn t we use budget constraint (3)? A: because it is implied by the other equations, in particular firm s problem + market clearing (8) (3) Firm s problem and market clearing are (p t F(k t,h t ) w t h t R t k t ) = 0 (11) c t +k t+1 (1 δ)k t = F(k t,h t ) (12) Substituting (12) into (11) gives (3): [p t (c t +k t+1 (1 δ)k t ) w t h t R t k t ] = 0 This is Walras Law _law very general: all budget constraints resource constraint useful check when writing models: if Walras Law doesn t hold, you did something wrong (e.g. forgot term in mkt clearing) 13/17

14 Steady State ADCE Definition: A St.st. ADCE is a value of k and an ADCE for the economy with k 0 = k s.t. k t = k (and c t = c ) for all t. Clearly from (9) k same as in SP problem 1 β = f (k )+1 δ Question: what do you think prices look like in a st.st. ADCE? p t = constant? R t = constant? w t = constant? 14/17

15 Let s work it out Steady State ADCE Have u (c ) βu (c ) = p t p t+1 = β p t+1 p t normalizing p 0 = 1 p t = β t Also R t p t = F k (k,1), Summary: R t /p t,w t /p t constant R t,w t,p t decreasing at rate β w t p t = F h (k,1) So prices are not constant in st.st. ADCE prices implicitly reflect discounting of future values price of future output is lower return to future work is lower 15/17

16 Alternative Pricing Convention Denote factor prices relative to output price in each period Write budget constraint as p t (c t +k t+1 (1 δ)k t ) p t ( R t k t + w t h t ) This formulation is useful for thinking about real rates of return and interest rates in ADCE no explicit credit market in ADCE but can infer implicit real interest rate on one-period ahead borrowing and lending 16/17

17 Alternative Pricing Convention Denote real interest rate by r t Definition: 1+r t+1 = amount of consumption you can get tomorrow by giving up unit of consumption today giving up one unit today saves p t with this you buy p t /p t+1 tomorrow 1+r t+1 = p t p t+1 In steady state, 1+r t = 1/β Real rate of return on capital: from HH max. w.r.t. k t+1 p t = p t+1 Rt+1 +p t+1 (1 δ) buy 1 unit of k today, get p t+1 Rt+1 +p t+1 (1 δ) tomorrow must equal cost of doing so p t 1+r t+1 = R t+1 +1 δ R t = r t +δ Terminology: rental rate R t = user cost of capital r t +δ 17/17