Through Scarcity to Prosperity and Beyond: A Theory of the - - PowerPoint PPT Presentation

Through Scarcity to Prosperity and Beyond: A Theory

• f the Transition to Sustainable Growth

Pietro F. Peretto

Duke University

March 2013

Peretto (Duke University) Transition to sustainable growth March 2013 1 / 30

The literature: Two seemingly disconnected ideas

Resource economics emphasizes the role of exhaustible resources in generating diminishing returns to other inputs that worsen over time as resources run out. Two classic questions: Is level of consumption per capita sustainable? (Solow 1974) Is growth of consumption per capita sustainable? (Stiglitz 1974) Growth economics emphasizes the role of land, a non-exhaustible resource, in generating diminishing returns to labor that allow construction

• f Matlhusian equilibria (Galor-Weil 1996, 2000, Lucas 2002, Galor 2005,

2011). It then asks: How did interaction of endogenous technological change and fertility choice drive escape from Malthusian trap of low consumption per capita? transition to sustained growth of consumption per capita?

Peretto (Duke University) Transition to sustainable growth March 2013 2 / 30

Are these two ideas connected?

Resource economics studies the future behavior of economy under increasing scarcity. As we run out of natural inputs, and diminishing returns to man-made inputs reduce output per capita, how can we sustain

• ur current — and growing — standards of living?

Growth economics studies the past transition to sustained growth experienced by advanced economies. It emphasizes population-technology interactions but ignores exhaustible resources, perhaps because modeling escape from the Malthusian regime requires that sustained growth be feasible in the …rst place. So, the two ideas are two sides of the same coin and provide the foundation of a general theory of the interactions of population, resources and technology.

Peretto (Duke University) Transition to sustainable growth March 2013 3 / 30

This paper

Integrates fertility choice and exhaustible resource dynamics in model of endogenous innovation to develop a theory of the transition from resource-based to knowledge-based growth. Initial phase where agents build up the economy by exploiting exhaustible natural resources to support population growth. Intermediate phase where agents turn on Schumpeterian engine of innovation-led growth in response to market expansion. Terminal phase where growth becomes driven by knowledge accumulation and no longer requires growth of physical inputs. Last part is crucial: not only economics growth no longer requires growth

• f physical inputs, but also knowledge accumulation compensates for the

exhaustion of the natural resource. The paper thus proposes a theory of de-coupling.

Peretto (Duke University) Transition to sustainable growth March 2013 4 / 30

A Schumpeterian model with exhaustible resources

Final producers: Homogeneous good that is consumed, used to produce intermediate goods, or invested in R&D. (One-sector structure.) This good is the numeraire, so PY 1. Intermediate producers: Develop new goods and set up operations to serve market (variety innovation or entry) and, when already in operation, invest in R&D internal to …rm (quality innovation). Households: Consume, save and set optimal path of population growth and resource use. For simplicity, they play the role of the extraction sector. (Alternative assumptions are feasible, of course.)

Peretto (Duke University) Transition to sustainable growth March 2013 5 / 30

Representative …nal producer (i)

Technology: Y = N(σ1)(1θ)

Z N

X θ

i

• Z α

i Z 1αLγR1γ1θ di,

Z

Z N

1 N Zjdj. where: 0 < θ, γ < 1 standard parameters that map into factor shares; Z α

i Z 1α vertical technology index, with α 2 [0, 1) measure of private

returns to quality and 1 α measure of social returns to quality; N is horizontal technology index, with σ 2 [0, 1) measure of social returns to variety (love-of-variety e¤ect in production).

Peretto (Duke University) Transition to sustainable growth March 2013 6 / 30

Representative …nal producer (ii)

Demand for product i: Xi = Nσ1 θ Pi

• 1

1θ

Z α

i Z 1αLγR1γ.

Factor payments: N PX =

Z N

PiXidi = θY ; wL = γ (1 θ) Y ; pR = (1 γ) (1 θ) Y .

Peretto (Duke University) Transition to sustainable growth March 2013 7 / 30

Intermediate producers

Technologies: Costi = 1

• Xi + φZ α

i Z 1α

; ˙ Zi = Ii. Firm’s objective: Vi (0) =

Z ∞

e R t

0 r(s)ds

Xi (t) (Pi (t) 1) φZ α

i (t) Z 1α (t) Ii (t)

• dt.

In symmetric equilibrium: max

Pi,Ii Vi ) r = α

X 1

θ 1

• Z

φ ! rZ ; V max

i

= βY N ) r = X 1

θ 1

φZ I β Y

N

+ ˙ Y Y ˙ N N rN.

Peretto (Duke University) Transition to sustainable growth March 2013 8 / 30

Representative household

Chooses C (t), L (t), R (t) and b (t) to maximize U0 =

Z ∞

eρt

• log

C (t) M (t)Mη+1 (t)

• + f (b (t))
• dt,

ρ, η > 0, subject to: ˙ A = rA + wL + pR C ΨbM, M L 0, ψ > 0; ˙ M = M (b δ) , M0 > 0, δ > 0; S0

Z ∞

R (t) dt, R 0, S0 > 0, ˙ S = R. What’s new here? M evolves according to fertility choice b; preference for b increasing and (weakly) concave, i.e, f 0 > 0, f 00 0. S is stock of exhaustible resource that evolves according to extraction choice R (for simplicity, extraction cost is zero).

Peretto (Duke University) Transition to sustainable growth March 2013 9 / 30

Household behavior (i)

First-order conditions for control variables C, b, R: 1 = λAC; f 0 (b) + λMM = λAΨM; λAp = λS, where the λs denote the shadow values of A, M and S; for state variables A, M, S: r + ˙ λA λA = ρ;

η M + λA (w Ψb)

λM + ˙ λM λM + ˙ M M

• = ρ;

˙ λS λS = ρ. Plus usual transversality conditions.

Peretto (Duke University) Transition to sustainable growth March 2013 10 / 30

Household behavior (ii)

Let c C/Y and h λMM. Conditions for C and A yield Euler equation r = ρ + ˙ C C = ρ + ˙ c c + ˙ Y Y . To simplify, cost of reproduction in units of the …nal good (taken as given since depends on aggregate variables that household does not control): Ψ = ψ Y M .

Peretto (Duke University) Transition to sustainable growth March 2013 11 / 30

Household behavior (iii)

Conditions for C, b, A and M yield fertility rule f 0 (b) + h = ψ c and asset-pricing equation ˙ h = ρh η 1 c wM Y ψb

• .

Conditions for C, R, S and the Euler equation yield Hotelling rule p C = λS ) ˙ p p = ρ + ˙ C C = r.

Peretto (Duke University) Transition to sustainable growth March 2013 12 / 30

Equilibrium extraction (i)

Flow of resource supplied by household equals …nal sector demand pR = (1 γ) (1 θ) Y . Log-di¤erentiating and using Hotelling rule ˙ R R = ˙ Y Y ˙ p p = ˙ Y Y r = ˙ c c + ρ

• .

Integrating and de…ning average growth rate of extraction ‡ow between time 0 and time t as ε (t) 1

t

R t

• ˙

c(s) c(s) + ρ

• ds yields

R (t) = R0eε(t)t.

Peretto (Duke University) Transition to sustainable growth March 2013 13 / 30

Equilibrium extraction (ii)

Substituting into S0 =

Z ∞

R (t) dt yields R0 = Z ∞ eε(t)tdt 1 | {z } constant that depends

• n fundamentals

S0. Therefore, the extraction rule at time t is R (t) = eε(t)t R ∞

0 eε(t)tdt S0.

Peretto (Duke University) Transition to sustainable growth March 2013 14 / 30

Technology and resources: output per capita dynamics

In symmetric equilibrium aggregate output is Y = κNσZMγR1γ, κ θ

2θ 1θ

where NσZ is TFP. Output per capita is y Y M = κNσZ R M 1γ . Let n ˙ N/N, z ˙ Z/Z and g ˙ y/y. At time t, output per capita growth rate is: g = σn + z | {z }

TFP growth

(1 γ) (m + ˙ c/c + ρ) | {z }

growth drag

.

Peretto (Duke University) Transition to sustainable growth March 2013 15 / 30

Transition dynamics: di¤erential equation for …rm size

Final producer pays total compensation N PX = θY to intermediate producers who set P = 1/θ. Hence, NX = θ2Y . Let µ P 1 and x X/Z. Reduced-form production function yields x = θ2Y /NZ = θ2κMγR1γ/N1σ. Returns to innovation become: r = α (µx φ) ; r = 1 βθ2

• µ φ + z

x

• + ˙

x x + z. Note: …rm-level decisions depend on quality adjusted …rm size x, which follows di¤erential equation ˙ x x = ˙ Y Y n z = γm (1 γ) (˙ c/c + ρ) | {z }

market growth

• (1 σ) n

| {z }

market fragmentation

.

Peretto (Duke University) Transition to sustainable growth March 2013 16 / 30

Transition dynamics: key insight

Population growth net of resource exhaustion drives growth of market for intermediate goods. Adjusting for market fragmentation due to product proliferation yields dynamics of …rm size, the driver of agents’ investment decisions in quality and variety. Whether these decisions support positive output per capita growth depends on whether growth of TFP is larger than growth drag. This is the classic condition for sustainability derived by Stiglitz (1974, see also Brock and Taylor 2005), with the di¤erence that in this model TFP growth is endogenous and not necessarily positive.

Peretto (Duke University) Transition to sustainable growth March 2013 17 / 30

Transition dynamics: useful prelimiary results (i)

Lemma 1 There are two main regimes, one where entrants are inactive and one where they are active. In the latter the consumption ratio c and the fertility rate b jump to their respective steady-state values c, b and remain constant throughout the transition driven by the evolution of …rm size x. Moreover, the resource input R follows an exponential process with constant rate of exhaustion ρ, i.e., R (t) = ρS0eρt.

Peretto (Duke University) Transition to sustainable growth March 2013 18 / 30

Transition dynamics: useful prelimiary results (ii)

Lemma 2 Let xN denote the threshold of …rm size that triggers variety innovation and xZ the threshold of …rm size that triggers quality

• innovation. Assume ρβθ2αφ

µρβθ2 < ρ + γm, where m is the constant

(endogenous) growth rate of population in an equilibrium with active

• entrants. Then xN < xZ .

Lemma 3 Expenditure behavior of household is c + ψb = ( θ2 µ φ

x

• + 1 θ

φ < x xN ρβθ2 + 1 θ x > xN .

Peretto (Duke University) Transition to sustainable growth March 2013 19 / 30

Transition dynamics: the dynamical system

Dynamical system consists of fertility and expenditure rules derived from houshold’s …rst order conditions, three di¤erential equations in c, h, x, transversality condition on x, and initial value x0 = θ2κMγ

0 R1γ

/N1σ . As in Dasgupta and Heal (1974), R0 in numerator is a choice. So, given M0 and N0, x0 is set so to select path c (t) that sati…es S0 = R ∞

0 R (t) dt

(recall analysis of equilibrium extraction). Nothing crucial hinges on presence or speci…c functional form of f (b). I thus posit f (b) = 0 and reduce fertility rule to h = ψ/c, which allows me to eliminate h and work in terms of c and x only.

Peretto (Duke University) Transition to sustainable growth March 2013 20 / 30

Transition dynamics: the regime with no entry

After some algebra, the system reduces to: ˙ c c = η + 1 ψ c θ2 ψ

• µ φ

x

• (1 γ) (1 θ)

ψ ρ; ˙ x x = 1 + (1 γ) η ψ c + θ2 ψ

• µ φ

x

• +
• 1 γ + γ2 1 θ

ψ γδ. Key: feedback between consumption and …rm size, that, because of no innovation, is driven solely by evolution of population and resource stock.

Peretto (Duke University) Transition to sustainable growth March 2013 21 / 30

Transition dynamics: the regime with entry

In this case we have: ˙ c c = η + 1 ψ c ρβθ2 + (1 γ) (1 θ) ψ ρ; ˙ x x = γm (1 γ) ρ (1 σ) n (x) . Key: rate of entry n is a function of x; unstable di¤erential equations for c and h do not depend on x (because model has no scale e¤ect) and, consequently, c and h jump to their steady-state values c and h and determine b = b and m = b δ at all times (Lemma 1).

Peretto (Duke University) Transition to sustainable growth March 2013 22 / 30

The transition to sustained growth: phase diagram, unique equilibrium path

Peretto (Duke University) Transition to sustainable growth March 2013 23 / 30

The transition to sustained growth: innovation rates and law of motion of …rm size in entry region

When economy crosses threshold xN, rates of innovation are: n = 8 < :

1 βθ2

• µ φ

x

• ρ

x xZ

(1α)(µxφ)+γ(m+ρ)σρ βθ2xσ

ρ x > xZ ; z = 8 > < > : x xZ

(µxφ)

• α

σ βθ2x

• γ(m+ρ)+σρ

1

σ βθ2x

x > xZ . Law of motion of …rm size is ˙ x x = 8 < : γ (m + ρ) σρ (1 σ) µxφ

βθ2x

xN < x xZ γ (m + ρ) σρ (1 σ) (1α)(µxφ)+γ(m+ρ)σρ

βθ2xσ

x > xZ .

Peretto (Duke University) Transition to sustainable growth March 2013 24 / 30

The transition to sustained growth: phase diagram, non-unique equilibrium path or failure to launch?

Peretto (Duke University) Transition to sustainable growth March 2013 25 / 30

The transition to sustained growth: anlytical solution (i)

For xN < x < xZ economy follows linear di¤erential equation ˙ x = ¯ ν (¯ x x) . Integrating between time TN and time t yields x (t) = xNe ¯

ν(TN t) + ¯

x 1 e ¯

ν(TN t)

. If ¯ x > xZ , then there exists value TZ = TN + 1 ¯ ν log ¯ x xN ¯ x xZ

• such that x (TZ ) = xZ ,

which is date when economy turns on quality growth.

Peretto (Duke University) Transition to sustainable growth March 2013 26 / 30

The transition to sustained growth: anlytical solution (ii)

Note: TZ is watershed moment when de-coupling becomes feasible. For t > TZ economy follows nonlinear di¤erential equation and converges to x = 1 (1σ)(1α)φ

γ(m+ρ)+σρ

βθ2 (1σ)(1α)µ

γ(m+ρ)+σρ

. Time path of income per capita growth is g (t) =

• σn (t) (1 γ) (m + ρ)

0 t TZ σn (t) + z (t) (1 γ) (m + ρ) t > TZ . With this expression in hand, we can investigate one of the central questions of the paper.

Peretto (Duke University) Transition to sustainable growth March 2013 27 / 30

Is long-run growth sustainable? (i)

If economy fails to cross threshold xZ , it converges to semi-endogenous growth rate ¯ g = σγ (m + ρ) ρ 1 σ | {z }

¯ n

(1 γ) (m + ρ) . Sustainability condition is ¯ g > 0 i¤ σ > (1 γ) m + ρ m . Holds for small values of m + ρ. Important: given ρ, this condition holds for fast population growth. Implication: Given non-zero exhaustion rate, sustainable growth requires su¢ciently high fertility.

Peretto (Duke University) Transition to sustainable growth March 2013 28 / 30

Is long-run growth sustainable? (ii)

If economy crosses threshold xZ , it converges to (fully) endogenous growth rate g = α (µx φ) m ρ. Sustainability condition is g > 0 i¤ α @ 1 (1σ)(1α)φ

γ(m+ρ)+σρ

βθ2 (1σ)(1α)µ

γ(m+ρ)+σρ

φ 1 A > m + ρ, Also holds for low values of m + ρ. However: given ρ, this condition holds for slow population growth. Implication: Given non-zero exhaustion rate, sustainable growth requires su¢ciently low fertility.

Peretto (Duke University) Transition to sustainable growth March 2013 29 / 30

Final thoughts

Model makes rather loose predictions about population growth rate. In fact, it only says that it is constant. Not quite satisfactory. Can we develop model with tighter predictions, e.g., long-run population level has to be constant in world of limited resources? Caveat: It should not be driven by Malthusian-like mechanism that ties population size to the resource stock. Example: Peretto-Valente (2012). Model is also very stark in its treatement of resource supply to preserve tractability. Can we do better?

• Likely. To gain analytical insight, however, need to be quite clever as

things get easily quite complex since we are adding state variables and non-linearities.

Peretto (Duke University) Transition to sustainable growth March 2013 30 / 30