Growth
Neoclassical Growth Model: I
Mark Huggett2
2Georgetown
October, 2017
Growth
Growth Model: Introduction
I Neoclassical Growth Model is the workhorse model inmacroeconomics. It comes in two main varieties:infinitely-lived agent and overlapping generations.
I Some key features: (i) accounts for the NIPA aggregatesC, I,G, (ii) accounts for factor shares, (iii) accounts forlong-run growth, (iv) framework for analyzing business-cyclefluctuations (impluses and propagation mechanisms) and (v)one can address positive and normative questions within it.
I Generalization of Solow (1956) to endogenize savingsdecision.
Growth
Infinitely-lived agent Growth Model
1. Preferences:∑∞
t=0 βtu(ct)
2. Endowments: k0 > 0 and 1 unit of time each model period
3. Technology: F (kt, lt) C.R.S.
ct + kt+1 ≤ F (kt, lt) + kt(1− δ)
One can rewrite this also as follows:
ct + kt+1 − kt(1− δ) ≤ F (kt, lt)
Growth
Equilibrium
Def: A competitive equilibrium is {ct, kt+1, lt, wt, Rt}∞t=0 such that
1. {ct, kt+1, lt}∞t=0 solve P1.
2. wt = F2(kt, lt) and Rt = F1(kt, lt) holds ∀t3. ct + kt+1 = F (kt, lt) + kt(1− δ) holds ∀t
P1 max
∞∑t=0
βtu(ct) s.t.
ct + kt+1 ≤ wtlt +Rtkt + (1− δ)kt
ct, kt+1 ≥ 0, lt ∈ [0, 1], given k0
Growth
Equilibrium
Comments:
1. We assume a sequential market structure with two factorrental prices per period.
2. There is no direct requirement of profit maximization by afirm or firms. However, this is implicit in the second conditionof equilibrium. Macroeconomists often simply imposecompetitive pricing of factor inputs to shorten the analysis.There is no explicit ownership of shares of the firm and noallocation of profits of the firm in the definition. Since firmswill make zero profit in an equilibrium, abstracting fromownership and the allocation of profit shortens the analysis.
3. Sometimes we will impose different market structures: (i)time-0 AD markets and (ii) sequential markets but where theconsumer owns the firm and the firm owns the capital.
Growth
Planning Problem
Planning Problem
max
∞∑t=0
βtu(ct) s.t.
ct + kt+1 ≤ F (kt, lt) + kt(1− δ) and lt ∈ [0, 1], given k0
1. 1st Welfare Theorem: It will be understood that within thegrowth model comp. equil. allocations are Pareto efficient.Thus, competitive equilibrium allocations will solve theplanning problem above.
2. We will not prove the 1st Welfare Theorem. However, we willshow how to recast the equilibrium concept into “thelanguage of Debreu”.
Growth
Language of Debreu
Comments:
1. In our application the commodity space is infinite dimensional.Debreu’s (1959) analysis handles finite dimensional problems.Debreu (1954) shows how to prove the Welfare Theorems withfinite consumers but infinite dimensional commodity space.
2. To invoke standard proofs, one would need to restatecompetitive equilibria in the “language of Debreu”.
3. (Ui, Xi, ei, Yj) are the elements of an economy in thelanguage of Debreu. In discussing competitive equilibriumthere is also share ownership (i.e. θij is the fraction of firm jowned by consumer i).
Growth
Debreu: Static Warmup Example
1. One agent and one firm in our application:X = {(x1, x2) : x1, x2 ≥ 0}U(x) = u(x1) + v(x2)
e = (0, 1)
Y = {(y1,−y2) : 0 ≤ y1 ≤ F (y2), y2 ≥ 0}2. A competitive equilibrium is (x, y, p) such that (i)x ∈ argmax {U(x)|x ∈ X, px ≤ pe+ 1× py}, (ii)y ∈ argmax {py|y ∈ Y } and (iii) x = y + e.
Growth
Debreu: Dynamic Growth Model Example
1. One agent and one firm in our application:X = {x = {(x1t, x2t)}∞t=0 : (x1t, x2t) ∈ R+ × [0, 1], ∀t ≥ 0}U(x) =
∑∞t=0 β
tu(x1t, x2t)
e = {(e1t, e2t)}∞t=0 and (e1t, e2t) = (0, 1),∀t ≥ 0
Y = {y = {(zt,−lt)}∞t=0 : ∃{kt}∞t=0, k0 = k s.t. (1) holds}(1) zt + kt+1 ≤ F (kt, lt) + kt(1− δ), lt ≥ 0,∀t ≥ 0
2. Prices: px =∑∞
t=0(p1t, p2t) · (x1t, x2t)
3. A competitive equilibrium is (x, y, p) such that (i)x ∈ argmax {U(x)|x ∈ X, px ≤ pe+ 1× py}, (ii)y ∈ argmax {py|y ∈ Y } and (iii) x = y + e.
Growth
Debreu: Summary
1. We have considered two example economies. Both have oneagent and one firm.
2. A competitive equilibrium in both economies is (x, y, p) andthe three requirements of equilibria are the same in botheconomies.
3. What differs across the economies is that the elements(x, y, p) have very different dimensions across the twoexamples and that the notion of “price system” in example 2is perhaps very different from what you are used to thinkingabout.
4. The equilibrium concepts used in the remainder of these slidesand the notation employed will be quite different from theDebreu formalization. One reason for this is that we will beinterested in computing equilibria (at least an approximation).Debreu’s language is not usually the most natural way to doso.
Growth
Steady States of the Growth Model
A competitive equilibrium {ct, kt+1, lt, wt, Rt}∞t=0 is a steady-statecompetitive equilibrium providedct = c, kt = k, lt = l, wt = w, Rt = R for all t ≥ 0.
Analysis:
1. u′(ct) = βu′(ct+1)(1 +Rt+1 − δ) - FONC
2. u′(ct) = βu′(ct+1)(1 + F1(kt+1, lt+1)− δ) - FONC + equil
3. 1 = β(1 + F1(k, 1)− δ) - steady state
4. F1(k, 1) > 0 and F11(k, 1) < 0 imply that there exists aunique steady state k > 0
5. Golden Rule: kGR ≡ argmax F (k, 1)− δkImplication: k < kGR
Growth
Steady States
Generalize to u(c, l) rather than u(c)
1. u1(c, l) = βu1(c, l))(1 + F1(k, l)− δ) - FONC
2. u1(c, l)F2(k, l) = −u2(c, l) - FONC
3. The following restrictions characterize steady states:
1 = β(1 + F1(k, l)− δ)
u1(c, l)F2(k, l) = −u2(c, l)
c+ δk = F (k, l)
4. An extra assumption (e.g. u additively separable or u is GHHpreferences) would lead to a sharper characterization
Growth
Steady States
Theorem: Assume u(c, l) = u(c)− v(l), u′ > 0, u′′ < 0 andu′(0) =∞, limc→∞u
′(c) = 0 and v′ > 0, v′′ > 0 . Then there is aunique interior ss competitive equilibrium allocation (c, k, l).Proof: sketch
1. Rearrange ss restrictions.
(1) 1 = β(1 + F1(k/l, 1)− δ)
(2) u′(c)F2(k/l, 1) = v′(l)
(3) c+ δ(k/l)l = F (k/l, 1)l
2. Usual assumptions on F imply there exists a uniquesolution k/l > 0 to (1)
Growth
Steady States
Proof: sketch (continued)
1. Substitute (3) into (2) to get (*)
u′(c)F2(k/l, 1) = v′(l)
(∗) u′(l(F (k/l, 1)− δ(k/l))) F2(k/l, 1) = v′(l)
2. Given unique solution k/l to (1), LHS of (*) is st decreasingin l and RHS is st increasing in l.
3. LHS(l)−RHS(l) is a continuous and st monotonedecreasing fn of l. Intermediate-Value Thm delivers a solutionl to LHS(l)−RHS(l) = 0. It’s unique by st. monotonicity.
4. Conclusion: There are unique positive values (c, k, l) solving(1)-(3)
Growth
Steady States with Proportional Tax Rates
Perspective:
One use of the growth model is to provide insight into fiscal policy.While we know that the growth model implies that government taxand spending plans do not improve upon the market allocation, wewill still pursue an analysis of proportional taxes within this model.This establishes a useful benchmark and is an important point ofdeparture in the literature.
Growth
Steady States with Proportional Tax Rates
Def: A competitive equilibrium is {ct, kt+1, lt, wt, Rt, Tt}∞t=0 suchthat
1. {ct, kt+1, lt}∞t=0 solve P1.
2. wt = F2(kt, lt) and Rt = F1(kt, lt) holds ∀t3. ct + kt+1 = F (kt, lt) + kt(1− δ) holds ∀t4. Tt = τcct + τwwtlt + τk(Rt − δ)kt holds ∀t
P1 max
∞∑t=0
βtu(ct) s.t.
ct(1 + τc) + kt+1 ≤ wtlt(1− τw) + (Rt − δ)(1− τk)kt + kt + Tt
ct, kt+1 ≥ 0, lt ∈ [0, 1], given k0
Growth
Steady States with Proportional Tax Rates
Analysis: τk > 0 and τc, τw = 0
1. u′(ct) = βu′(ct+1)(1 + (Rt+1 − δ)(1− τk)) - FONC
2. 1/β = 1 + (F1(k, 1)− δ)(1− τk) - steady state
3. F1(k, 1) > 0 and F11(k, 1) < 0 imply that there exists aunique steady state k > 0 and k decreases as τk increases.
4. F (k, l) = Akαl1−α implies
k = ([1/β − 1
1− τk+ δ]× 1
αA)1/(α−1)
Growth
Steady States with Proportional Tax Rates
How large is the steady state output effect of capital incometaxation?
1. F (k, l) = Akαl1−α
2. k(τk) = ([1/β−11−τk + δ]× 1
αA)1/(α−1)
3. y(τk)y(0) = F (k(τk),1)
F (k(0),1)= ( k(τk)
k(0))α
4. y(τk)y(0) = (
[1/β−11−τk
+δ]
[1/β−1+δ])αα−1 > (1− τk)
α1−α for δ > 0
5. y(τk)y(0) > (1− τk)
α1−α
.= (.5).3/.7 = .74
6. When α = .3, then a 50 percent capital tax lowerssteady-state output by at most 25 percent.
Growth
Steady States with Proportional Tax Rates
Analysis: u(c, l) and τk, τc, τw > 0
1. u1(ct, lt)1
1+τc= βu1(ct+1, lt+1) (1+(Rt+1−δ)(1−τk))
1+τc- FONC
2. u1(ct, lt)wt(1−τw)
1+τc= −u2(ct, lt) - FONC
3. In steady state the following hold:
1/β = 1 + (F1(k, l)− δ)(1− τk)
u1(c, l)F2(k, l)(1− τw)
1 + τc= −u2(c, l)
c = F (k, l)− δk
4. First equation pins down the ratio k/l. l is not pinned downabsent further restrictions.
Growth
Application: Corporate Tax
Let’s analyze how the introduction of a proportional tax τ oncorporate income impacts the steady state properties of theone-sector growth model. To do so, we recast the definition ofequilibrium. Specifically, now the firm’s problem is dynamic and wemodel the value of the firm when the firm owns its capital.
Definition: A steady-state equilibrium is (c, l, b, s, k, d, w, r, p, T )such that
1. (ct, lt, bt, st) = (c, l, b, s),∀t solves P1.
2. (kt, lt) = (k, l),∀t solves P2
3. c+ δk = F (k, l) and s = 1 and b = 0
4. T = τ [F (k, l)− wl − δk]
5. d = (1− τ)[F (k, l)− wl − δk]
Growth
Application: Corporate Tax
P1 max
∞∑t=0
βtu(ct) s.t.
ct + st+1p+ bt+1 ≤ wlt + st(p+ d) + bt(1 + r) + T
ct, st+1, bt+1 ≥ 0, lt ∈ [0, 1], given (s0, b0) = (1, 0)
P2 max
∞∑t=0
(1
1 + r)tdt , given k0 = k
dt ≡ [F (kt, lt)−wlt− (kt+1− kt(1− δ))]− τ [F (kt, lt)−wlt− δkt]
Growth
Application: Corporate Tax
Necessary Conditions for interior optimization are for all t ≥ 0
1. u′(ct) = βu′(ct+1)(1 + r)
2. u′(ct)p = βu′(ct+1)(p+ d)
3. F2(kt, lt) = w
4. 1 = ( 11+r )[(1− τ)(F1(kt+1, lt+1)− δ) + 1]
Steady State:l = 1(1 + r) = 1/β via 1.k = F1(·, 1)−1([ 1
β − 1]/(1− τ) + δ) via 1 and 4w = F2(k, 1) via 3p = β
1−βd = β1−β (1− τ)[F (k, 1)− w − δk] via 2
p = β1−β (1− τ)[k(F1(k, 1)− δ)] = β
1−β ( 1β − 1)k = k
T = τ [F (k, 1)− w − δk]
Growth
Application: Corporate Tax
Theorem: In a steady state in the growth model with aproportional tax τ on corporate income,
1. capital k(τ) decreases as the tax rate τ increases.
2. wage w(τ) = F2(k(τ), 1) decreases as the tax rate τ increases.
3. interest rate r(τ) = 1/β − 1 is constant as the tax rate τincreases.
4. value p(τ) = k(τ) of the firm decreases as the tax rate τincreases.
Proof: see analysis on the previous slide.
[Could generalize this Theorem to handle a labor-leisure decision.]
Growth
Recursive Competitive Equilibrium
Motivation
It will be useful to develop the notion of a recursive competiitiveequilibrium and a recursive formulation of the planning problem forthe growth model. One reason for this is that to analyze thesemodels it will often be easier to computationally approximatefunctions rather than infinite sequences.
The recursive notions we employ in growth theory will be closelyconnected to the recursive methods employed in the analysis ofasset pricing models in exchange economies (e.g. Lucas (1978)).
Growth
Recursive Competitive Equilibrium
Def: A recursive competitive equilibrium is(c(x), g(x), w(K), R(K), G(K)) such that
1. optimization: (c(x), g(x)) solve BE.
2. factor prices: w(K) = F2(K, 1) and R(K) = 1 +F1(K, 1)− δ3. feasibility: c(K,K) + g(K,K) = F (K, 1) +K(1− δ)4. law of motion: G(K) = g(K,K)
State: x = (k,K)
(BE) v(k,K) = max u(c) + βv(k′,K ′) s.t.
c+ k′ ≤ w(K) +R(K)k and c, k′ ≥ 0 and K ′ = G(K)
Growth
Recursive Competitive Equilibrium
Standard big K little k trick
1. We employ the state variable x = (k,K). Little k is theagent’s capital. Big K is economy wide capital.
2. Big K determines factor prices and helps the agent forecastthe future economy wide capital. The agent knows factorprices via knowing the functions (w,R) and the level K.
3. We acknowledge that “in equilibrium” big K and little k areequal. Thus, for the purpose of computing equilibria thedistinction will not be important. However, for the purpose ofdefining equilibria and clarifying the analysis the distiction isuseful.
4. Absent the distinction, the agent would view future factorprices as being influenced by curent capital choices. Since wewant to capture competitive behavior, the distinction is useful.
Growth
Recursive Competitive Equilibrium
Recursive Planning Problem
1. v(k) = max u(c) + βv(k′) s.t. c+ k′ ≤ F (k, 1) + k(1− δ)2. Solution: Decision rule k′ = G(k)
3. 1st Welfare Thm. implies that Law of motion in competitiveequilibrium G(k) coincides with the optimal decision ruleG(k) solving the Planning Problem.
4. Implication: can analyze competitive equilibrium by solvingthe planning problem.
5. All the properties of the growth model reduce to properties ofG(k).
Growth
Recursive Competitive Equilibrium
Recursive Planning Problem
1. v(k) = max u(c) + βv(k′) s.t. c+ k′ ≤ F (k, 1) + k(1− δ)2. Properties: Decision rule G(k) is (i) continuous, (ii)
increasing, (iii) there is a unique positive k∗ such thatk∗ = G(k∗), and (vi) monotone convergence of capital to k∗
induced by recursively applying kt+1 = G(kt) from k0 > 0.
3. Properties (i)-(ii) follow standard lines from early in thesemester, but (iii)-(iv) need to be developed.
Growth
Recursive Competitive Equilibrium
Monotone Convergence: LS (1989, Ch 6)
v(k) = max u(c) + βv(k′) s.t. c+ k′ ≤ F (k, 1) + k(1− δ)
1. u′(F (k, 1) + k(1− δ)−G(k)) = βv′(G(k)) - nec condition
2. v′(k) = u′(F (k, 1) + k(1− δ)−G(k))(1 + F1(k, 1)− δ) - BS
3. v st. concave implies [v′(k)− v′(G(k))][k −G(k)] ≤ 0,= iffG(k) = k
4. [(1 + F1(k, 1)− δ)− 1β ][k −G(k)] ≤ 0
5. Let k∗ solve β(1 + F1(k∗, 1)− δ) = 1 . Consider k > 0.
6. Case 1: k < k∗ implies [+][k −G(k)] < 0. Thus, G(k) > k
7. Case 2: k > k∗ implies [−][k −G(k)] < 0. Thus, G(k) < k
Growth
Recursive Competitive Equilibrium
How fast does capital converge to steady state?
v(k) = max u(c) + βv(k′) s.t. c+ k′ ≤ F (k, 1) + k(1− δ)
1. kt+1 = G(kt).= G(k∗) +G′(k∗)(kt − k∗)
2. kt+1 − k∗.= G′(k∗)(kt − k∗)
3. 1− α ≡ kt+1−k∗kt−k∗ = G′(k∗)
4. Upshot: If G′(k∗) is close to 1 then convergence is slow.Fraction α of gap closed is then small.
5. Note: Solow model with Cobb-Douglas production andcapital’s share near .3 implies rapid convergence. Calculationsfor the optimal growth model suggest similar results.
Growth
Sunspot Equilibria and the Animal Spirits Hypothesis
Rational Sunspot Equilibria
Idea: Contemplate an economy for which sunspot realizations areindependent of shocks to preferences, endowments or technology.See if sunspot realizations impact allocations under thesecircumstances. We will do so within the growth model where theONLY shocks are sunspot shocks.
Notation:st ≡ (s0, ..., st) ∈ St = S × · · · × S history of realizations of stst ∈ S finite setP (st) probability of “sunspot” history
Growth
Sunspot Equilibria and the Animal Spirits Hypothesis
A competitive equilibrium is{ct(st), kt+1
(st), lt(st), wt
(st), Rt
(st)}∞
t=0such that:
1.{ct(st), kt+1
(st), lt(st)}∞
t=0solve P1.
2. wt(st)
= F2(kt(st−1
), lt(st)), Rt
(st)
=1 + F2(kt
(st−1
), lt(st))− δ
3. Feasibility: ∀t,∀stct(st)
+ kt+1
(st)≤ F
(kt(st−1
), lt(st))
+ (1− δ) kt(st−1
)and lt
(st)∈ [0, 1]
Decision ProblemP1: maxE
[∑∞t=1 β
t−1u(ct(st))]
subject toct(st)
+ kt+1
(st)≤ wt
(st)lt(st)
+Rt(st)kt
(st−1
),
lt(st)∈ [0, 1]∀t, ∀st
Growth
Sunspot Equilibria and the Animal Spirits Hypothesis
Theorem: Assume u(c) is st. increasing and st. concave and F isconcave and CRS. Then there is no comp equilibrium in whichc(st) 6= c(st) for some t ≥ 0 and two histories st and st ∈ St.Sketch of Proof:
1. SBWOC that there is an equilibrium w/ c(st) 6= c(st).2. Define ct = E
[ct(st)]
, kt+1 = E[kt+1
(st)]
, lt = E[lt(st)]
3. Claim:{ct, kt+1, lt
}∞t=0
satisfies Feasibility. First line belowtakes expectations of the feasibility condition. Second lineapplies Jensen’s inequality and uses l(st) = 1.E[ct
(st)
+ kt+1
(st)] ≤
E[F(kt(st−1
), lt(st))
+ (1− δ) kt(st−1
)]
ct + kt+1 ≤ E[F(kt(st−1
), lt(st))
] + (1− δ) kt ≤F (kt, lt) + (1− δ)kt
4. E[∑∞
t=1 βt−1u (ct)
]> E
[∑∞t=1 β
t−1u(ct(st))]
by Jensen’sInequality.
5. This contradicts 1st Welfare theorem.
Growth
Sunspot Equilibria and the Animal Spirits Hypothesis
Comments:
(1) While rational sunspot equilibria exist in some modeleconomies, they don’t matter in basic growth models.
(2) Even if they were to matter in growth models, it is not so easyto produce procyclical labor productivity, as observed in US data,with constant returns technology.
(3) Proof of Theorem used l(st) = 1. It remains to be seen if onecan extend the proof to handle endogenous labor-leisure decision.