Wealth breeds decline Reversals of leadership and consumption habits Lionel Artige, Carmen Camacho and David de la Croix IRES Universit´ e catholique de Louvain June 2003
Introduction • Neoclassical growth models: per capita incomes in identical regions converge in the long run. • Historical data: convergence, if any, is neither smooth nor monotonic (overtaking, decline, rebirth).
Literature 1. New economic geography: • Increasing returns to scale + low transportation costs ⇒ economic agglomera- tions. More accrued with physical capital mobility (Desmet (2002)). • Urban costs generate dispersion/agglomeration/re-dispersion (Ottaviano, Tabuchi and Thisse (2002)). • New technology adopted by the poor region generates leapfrogging if the rich region does not because of the loss of accumulated experience ⇒ leapfrogging, (Brezis, Krugman and Tsiddon (1993)).
2. Human capital literature: • Divergence forces: social stratification across districts, regional fund- ing of education. • Convergence forces (which predominate): – Benabou (1996) Housing rent as social stratification cost ⇒ decrease income growth. – Tamura (2001): Teacher’s quality main input of human capital production ⇒ human capital converges since teaching quality is higher in poor districts. – Galor, Moav and Vollrath (2002): a country can overtake a richer one if initial distribution of land is more egalitarian.
3. Institutional literature • Acemoglu, Johnson and Robinson (2002): rich countries around 1500, colonized by European powers, are now poor and vice versa . Extractive institutions were imposed in rich countries ⇒ reversal of fortune. In this model : • Leapfrogging does not rely on exogenous shocks. • Convergence not guaranteed despite of capital mobility. • Leadership is more than stocks of physical and human capital, consumption habits are introduced.
What we do: 1. Examples of alternating primacy 2. A model of reversal 3. Regional dynamics 4. The role of capital mobility 5. Case study: Belgium-the Netherlands (1500-2000)
Examples of alternating primacy • Roman Empire’s decline: physical and human resources where allocated to enter- tainment instead of investment in knowledge and infrastructure. • Amalfi, largest city-state in Italy in the X century (60-80,000 people). Now, 7,000 inhabitants. Neighboring Naples now has 1 mil. people. • Belgium - the Netherlands 1 1.4 0.9 1.3 0.8 1.2 0.7 1.1 0.6 Belgium 0.5 1 Netherlands 0.4 0.9 0.3 0.8 Belgium 0.2 Netherlands 0.7 0.1 0.6 0 1000 1200 1400 1600 1800 2000 1000 1200 1400 1600 1800 2000
• Vienna-Prague 1 0.9 Vienna Praha 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1000 1200 1400 1600 1800 2000 • Sicily-Naples 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Sicily Naples 0.1 0 800 1000 1200 1400 1600 1800 2000
Model • OLG model with physical capital and knowledge. • 2 regions A,B. • Population does not grow, normalized to one. • Generations live two periods: – Period 1: Households work, consume and invest in physical capital and to accu- mulate knowledge. – Period 2: they consume their savings. • Firms: produce a single good with a c.r.s. technology. • Physical capital is perfectly mobile across regions. Labor is immobile.
• The model economy in a nutshell: – 2 regions, physical capital is perfectly mobile across them, – both produce the same good, with identical technology, – individuals have the same preferences, – regions have different levels of knowledge and consumption habits. • Dynamics: 1. Consumption habits are built from previous generation. 2. If young generation has high living standards ⇒ ↓ investment in knowledge. 3. That region loses leadership.
Preferences ln( ci,t − γai,t ) + β ln( di,t +1) + λ ln( ei,t ) i = A, B, where • c i,t : youth-age consumption, • a i,t : consumption habits, • d i,t : old-age consumption, • e i,t : spending on knowledge, • γ ∈ (0 , 1): influence of habits on preferences, • λ > 0: taste for spending on education, • β > 0: discount factor.
We assume that: • Depreciation rate ( forgetting rate ) of consumption habits is so high that old persons are not affected by them. Empirically : old households put less weight on comparisons to com- pare their welfare (Clark, Oswald and Warr (1996)). • λ ln ( e i,t ): ” joy of giving ”. Providing education to their children make parents happy.
Stock of habits ai,t = ci,t − 1 Technology Y i,t = A i,t K α i,t N 1 − α , i,t where A i,t = A h µ i,t , and • h i,t : knowledge, h i,t +1 = ψe i,t , ψ is a technological parameter (w.l.g. ψ = 1). • µ : elasticity of total factor productivity to knowledge. We assume that α + µ < 1.
Optimal behaviors Max c,d,e ln( c i,t − γa i,t ) + β ln( d i,t +1 ) + λ ln( e i,t ) subject to c i,t + s i,t + e i,t = ω i,t , d i,t +1 = R i,t +1 s i,t . First order conditions yield: 1 c i,t = 1 + δ ( ω i,t + γδa i,t ) , λ e i,t = 1 + δ ( ω i,t − γa i,t ) , s i,t = β λe i,t , where δ = λ + β . • Firm level: max profits. MP equal factor prices.
Equilibrium Perfect mobility of physical capital: R A,t = R B,t = R t Total stock of capital K t +1 = K A,t +1 + K B,t +1 = N A,t s A,t + N B,t s B,t .
Given initial conditions { h i, 0 , a i, 0 , k i, 0 } i = A,B satisfying µ µ k A, 0 + k B, 0 = β 1 − α 1 − α h A, 0 k B, 0 = h B, 0 k A, 0 , and λ ( h A, 0 + h B, 0 ) a competitive equilibrium is characterized by a path { h i,t , a i,t , k i,t } i = A,B,t> 0 , such that λ (1 − α ) Ah µ i,t − 1 k α � � h i,t = i,t − 1 − γa i,t − 1 , 1 + δ 1 (1 − α ) Ah µ i,t − 1 k α � � a i,t = i,t − 1 + γδa i,t − 1 , 1 + δ k A,t + k B,t = β λ ( h A,t + h B,t ) , µ µ 1 − α 1 − α h A,t k B,t = h B,t k A,t .
Regional dynamics and consumption habits • Steady State: 1 � A (1 − α )(1 − γ ) λ 1 − α β α � 1 − µ − α h A = ¯ ¯ h B = ¯ h = , 1 + ( β + λ )(1 − γ ) k = β k A = ¯ ¯ k B = ¯ ¯ h, λ ¯ h a A = ¯ ¯ a B = ¯ a = λ (1 − γ ) . • Dynamics: described by a system of four difference equations.
Proposition 1 [Hopf bifurcation] There exists a value γ 1 ∈ (0 , 1) such that at γ = γ 1 the steady state (1) is non- hyperbolic, the eigenvalues of the Jacobian of the linearized system have moduli less than unity with the exception of a conjugate pair of complex eigenvalues of modulus 1, { ℓ γ , ¯ ℓ γ } . This pair of eigenvalues also satisfies ℓ 3 γ � = 0 , ℓ 4 γ � = 0 and ∂ℓ γ /∂γ > 0 at γ = γ 1 . γ 1 is given by the following expression: (1 + δ ) 2 (1 + α + µ ) 2 − 4 δ (1 + δ )( α + µ ) � γ 1 = (1 + δ )(1 + α + µ ) − . 2 δ ( α + µ )
0 1 γ γ 1 γ 3 Depending on the value of γ , the dynamic system can show three differ- ent behaviors: • ∃ ¯ γ s.t. for γ < ¯ γ ⇒ Monotonic convergence. • For γ ∈ (¯ γ, γ 1 ) ⇒ Oscillatory convergence. • For γ > γ 1 ⇒ the system does not converge to the steady state.
Numerical example • Parameter values: α = 1 / 3, β = 1 / 2, λ = 1 / 2, µ = 1 / 2 and A = 10. Then γ 1 = 0 . 6379. We choose γ = 0 . 62. • Initial conditions: – a A, 0 = a B, 0 = ¯ a ; – k A, 0 , k B, 0 set to match their long run values. • Four different cases: 1. Alternating primacy; 2. Irreversible decline; 3. Synchronized waves; 4. Monotonic convergence;
1. h A, 0 = ¯ h ∗ 1 . 751 and h B, 0 = ¯ h/ 1 . 751; 3 5 7 9 11 13 15 17 19 21 23 25 27 29 2. h A, 0 = ¯ h ∗ 1 . 752 and h B, 0 = ¯ h/ 1 . 752; 3 5 7 9 11 13 15 17 19 21 23 25 27 29
3. h A, 0 = ¯ h ∗ 0 . 8 and h B, 0 = ¯ h ∗ 1 . 7; 3 5 7 9 11 13 15 17 19 21 23 25 27 29 4. h A, 0 , h B, 0 as in 1, but γ = 0 . 05; 3 5 7 9
The role of capital mobility • A competitive equilibrium { h i,t , a i,t , k i,t } i = A,B,t> 0 verifying k i, 0 = β/λh i, 0 is char- acterized by: λ (1 − α ) Ah µ i,t − 1 k α � � h i,t = i,t − 1 − γa i,t − 1 , 1 + δ 1 (1 − α ) Ah µ i,t − 1 k α � � a i,t = i,t − 1 + γδa i,t − 1 , 1 + δ k i,t = s i,t − 1 = β λh i,t , i = A, B. • The SS is identical to the case with capital mobility. • Hopf bifurcation at γ 1 .
• But convergence speed is slower. • Moreover , capital mobility enlarges the basin of attraction of the SS and promotes synchronization. Proposition 2 [No capital mobility] With no capital mobility, the steady state is locally stable for γ ∈ [0 , γ 1 ) . For γ in a neighborhood on the left of γ 1 , the speed of convergence is slower than with perfect mobility of capital. 1.2 1 0.8 0.6 3 5 7 9 11 13 15 17 19 21 23 25 27 29 0.4 0.2 2 3 4 5
Case study: Belgium - the Netherlands Simulate Belgium (B)- the Netherlands (NL) relative GDP from 1500- 2000. Initial conditions for Belgium and the Netherlands in 1500: year 1500 Belgium (B) the Netherlands (NL) B/NL Knowledge ( h i, 0 ) 0.598 0.491 1.219 Habit stock ( a i, 0 ) 4.033 3.147 1.281 GDP per capita* 875 754 1.16 *Source: Maddison (1995)
Simulated (dots) vs Actual (solid) Relative GDP per capita – B/NL 1.1 1600 1700 1800 1900 2000 0.9 0.8 0.7 0.6 0.5
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