Topic 3: II (OLG Models and Endogenous Growth Models)

Yulei Luo

SEF of HKU

September 27, 2013

Luo, Y. (SEF of HKU) Macro Theory September27,2013 1/38 Basic Ideas

We have assumed that there exists a representative agent in the economy when discussing infinite-horizon growth models. However, this assumption is not appropriate in some situations: In reality, we observe new households arrive in an economy over time, which introduces a range of new interactions between new and old generations. Decisions by older generations will affect the prices faced by younger generations. We therefore need overlapping generations models (OLG) to better capture these observations. The OLG models are more suitable to address some insights about the role of national debt and social security in the economy.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 2/38 The Baseline OLG Model

Time is discrete and runs to infinity. Each individual lives for two periods: young and old. The utility function for individuals born at t is

u (c1 (t)) + βu (c2 (t + 1)) , (1) where u ( ) satisfies the regular conditions. Factors markets are competitive.· Individuals only work in the first period of their lives (i.e., when they are young) and supply one unit of labor inelastically, earning the equilibrium wage rate w (t). The production side of the economy is the same as before (here we set A = 1): Y (t) = F (K (t) , L (t)) , where L (t) increases as follows: L (t) = L (0)(1 + n)t . For simplicity, assume δ = 1 such that

1 + r (t) = R (t) = f 0 (k (t)) and w (t) = f (k (t)) f 0 (k (t)) k (t) , − (2) where k = K /L.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 3/38 Consumption Decisions

Savings by an individual of generation t, s (t), are determined by:

max u (c1 (t)) + βu (c2 (t + 1)) , (3) c(t),c(t+1),s(t) { } s.t.c1 (t) + s (t) w (t) , (4) ≤ c2 (t + 1) R (t + 1) s (t) , (5) ≤ where we assume that young individuals rent their savings as capital to final good producers at the end of t and receive the return at t + 1. Since u ( ) > 0, both constraints hold as equalities. 0 · Optimization means that

u0 (c1 (t)) = βR (t + 1) u0 (c2 (t + 1)) . (6)

Luo, Y. (SEF of HKU) Macro Theory September27,2013 4/38 (conti.) Combining the Euler equation with the constraints yields

s (t) = s (w (t) , R (t + 1)) . (7)

Total savings in the economy is

S (t) = s (t) L (t) , (8)

where denotes the size of generation t, who are saving for t + 1. δ = 1 means that

K (t + 1) = L (t) s (w (t) , R (t + 1)) . (9)

Luo, Y. (SEF of HKU) Macro Theory September27,2013 5/38 Equilibrium

Definition A competitive equilibrium in the OLG economy can be represented by ∞ sequences: c1 (t) , c2 (t) , K (t) , w (t) , R (t) t=0, such that the factor prices are given{ by (2), individual consumption} decisions are given by (6) and (7), and the aggregate capital stock evolves according to (9).

Definition The steady state equilibrium is defined as an intertemporal equilibrium in which k = K /L is constant.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 6/38 The fundamental law of motion of the OLG economy:

s (w (t) , R (t + 1)) k (t + 1) = (10) 1 + n s (f (k (t)) f (k (t)) k (t) , f (k (t + 1))) = − 0 0 . (11) 1 + n In the steady state,

s (f (k∗) f 0 (k∗) k∗, f 0 (k∗)) k∗ = − , (12) 1 + n which depends on the form of the saving function s ( ). ·

Luo, Y. (SEF of HKU) Macro Theory September27,2013 7/38 Restrictions on Utility and Production Functions

Assume that 1 θ c (t) − 1 u (c (t)) = − and f (k) = kα, (13) 1 θ − where θ > 0. The production function is f (k (t)) = k (t)α. The Euler equation for consumption is then

c2 (t + 1) = [βR (t + 1)]1/θ , (14) c1 (t) which means that the saving function is w (t) s (t) = , (15) ψ (t + 1) where 1/θ (1 θ)/θ ψ (t + 1) = 1 + β− R (t + 1)− − > 1, (16) which ensures that positiveh consumption. i

Luo, Y. (SEF of HKU) Macro Theory September27,2013 8/38 The Effects of Factor Prices on Saving

∂s (t) 1 sw = = (0, 1) , (17) ∂w (t) ψ (t + 1) ∈

∂s (t) 1 θ 1/θ s (t) sR = = − [βR (t + 1)]− , ∂R (t + 1) θ ψ (t + 1)  

which means that sR < 0 if θ > 1, sR > 0 if θ < 1, and sR = 0 if θ = 1. The effects are determined by the interactions of income and substitution effects of a change in the .

Luo, Y. (SEF of HKU) Macro Theory September27,2013 9/38 The Canonical OLG Model

Here we consider a special case in which θ = 1, i.e., we have log utility. In this case

c2 (t + 1) = βR (t + 1) , (18) c1 (t) β s (t) = w (t) , (19) 1 + β

which means the capital accumulation equation should be

s (t) β w (t) β (1 α) k (t)α k (t + 1) = = = − , (20) 1 + n 1 + β 1 + n 1 + β 1 + n

where we use the fact that w (t) = (1 α) k (t)α in the competitive factor market. −

Luo, Y. (SEF of HKU) Macro Theory September27,2013 10/38 (conti.) It is straightforward to show that there is a unique steady state in which 1/(1 α) β (1 α) − k∗ = − . (21) (1 + n)(1 + β)   Just like the Solow model, this OLG model can lead to globally stable steady state equilibrium. [Insert figure here.]

Luo, Y. (SEF of HKU) Macro Theory September27,2013 11/38 Capital Overaccumulation Problem in the OLG Model

We now can compare the competitive equilibrium of the OLG economy to the choice of a social planner who maximizes a weighted average of all generations’utilities: ∞ ∑ ξt [u (c1 (t)) + βu (c2 (t + 1))] , (22) t=0 s.t.F (K (t) , L (t)) = K (t + 1) K (t) + L (t) c1 (t) + L (t 1) c2 (t) , − − where ξt is the weight that the planner places on generation t0s utility. Note that in per capita term:

c2 (t) f (k (t)) = (1 + n) k (t + 1) k (t) + c1 (t) + , − 1 + n The Euler equation is thus

u0 (c1 (t)) = βf 0 (k (t + 1)) u0 (c2 (t + 1)) , (23) which is the same as that obtained in the equilibrium OLG model because R (t + 1) = f 0 (k (t + 1)).

Luo, Y. (SEF of HKU) Macro Theory September27,2013 12/38 Capital Overaccumulation

However, the competitive equilibrium is not Pareto optimal. Why? In the steady state of the OLG economy, we have

c2∗ f (k∗) nk∗ = c∗ + = c∗, − 1 1 + n which means that ∂c∗ = f 0 (k∗) n, (24) ∂k∗ − which determines the golden rule k:

f 0 (kgold ) = n. (25)

∂c ∗ It is clear that if k∗ > kgold , then < 0, which means that reducing ∂k ∗ savings can increase total consumption for everybody. If this is the case, the economy is said to be dynamically ineffi cient (i.e., it overaccumulates capital).

Luo, Y. (SEF of HKU) Macro Theory September27,2013 13/38 (conti.) Imagine that introducing a social planner into the OLG economy that is on its balanced growth path with k∗ > kgold . If the planner does not change k∗, the available consumption for per worker each period is just: f (k ) nk . ∗ − ∗ Suppose now that in some period (t0), the planner allocates more resources to consumption and fewer on savings so that capital per worker the next period is just kgold , and thereafter remains k at kgold . Under this plan, in t0, consumption per worker could be

f (k∗) + k∗ (1 + n) kgold , (26) − which can be rewritten as f (k ) + (k kgold ) nkgold and is ∗ ∗ − − greater than f (kgold ) nkgold , and in each subsequent period, consumption per worker− could be

f (kgold ) nkgold , − which is greater than f (k∗) nk∗ by the definition of the Golden rule of consumption. − This policy can thus make more resources available for consumption in every period and improve everyone’swelfare.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 14/38 An OLG Model with Infinitely-lived Agents

Weil (1989): All agents are infinitely-lived, but newborns from their own households and do not receive any bequests. In the Weil model, agents of different vintages turn out to own different amounts of capital in equilibrium and thus effi ciency production requires trade in factor services. For simplicity, we assume log utility and no technological progress. An individual born on date v (the individual’svintage) maximizes: ∞ v s t v Ut = ∑ β − ln (cs ) , (27) s=t

on t. Total population grows at n: Lt+1 = (1 + n) Lt . We normalize the size of the initial vintage v = 0 (i.e., initial population) at L0 = 1. Each period, agents earn income from wages and from renting out capital. The period budget constraint for a family of vintage v is: v v v k = (1 + rt ) k + wt c . (28) t+1 t − t Luo, Y. (SEF of HKU) Macro Theory September27,2013 15/38 Maximization yields:

v ct+1 v = β (1 + rt+1) . (29) ct Next, we need to aggregate the capital accumulation and Euler equations across different vintages to drive the dynamic equations governing aggregate per capita capital and consumption:

f (kt ) ct nkt kt+1 kt = − , (30) − 1 + n − 1 + n ct+1 = 1 + f 0 (kt+1) [βct n (1 β) kt+1] , (31) − − t+1  where we use the facts that kt+1 = 0, rt kt + wt = f (kt ) by Euler’s theoren, and

0 1 2 t 1 t xt + nxt + n (1 + n) xt + + n (1 + n) − xt xt = ··· . (32) (1 + n)t

Luo, Y. (SEF of HKU) Macro Theory September27,2013 16/38 (conti.) Note that to calculate aggregate consumption or capital per capita, we must sum them of all vantages born since t = 0. Vintage v = 0, born at t = 0, has L0 = 1 members. Total population on t = 1 is L1; of this population, L1 L0 = (1 + n) 1 = n are of vintage v = 1. Similarly, vintage v−= 2 contains − 2 L2 L1 = (1 + n) (1 + n) = n(1 + n). − − Therefore, for any vintage v > 0, the number of their members is: v 1 n(1 + n) − , which implies (32). Note that (32) is simply total t consumption divided by the total population: Lt = (1 + n) .

Luo, Y. (SEF of HKU) Macro Theory September27,2013 17/38 Basic Ideas

The neoclassical growth models we have discussed attribute economic growth to exogenous technology progress, and they say nothing about the factors that drive technological progress itself. The rate of technology progress is assumed to be beyond the control of a country — It just happens. However, in reality, countries can do something to increase their technology level. New growth theories (or endogeoneous growth theories) extend neoclassical growth theory to incorporate market-driven innovation and therefore allow for endogeneously driven growth. The pioneers include Romer (1986, 1990), Lucas (1988), Rebelo (1990), and Aghion and Howitt (1992). We now consider three types of endogenous growth models: the AK model, the Romer externality model, and the model.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 18/38 The AK Model

In the Solow or RCK models, there are diminishing in capital, holding effi ciency labor constant. Consequently, the economy eventually settles down to a steady state growth path in which capital-labor ratio is constant. The AK model assume that at the aggregate level, output is linear in capital: Yt = AKt , A > 0. (33)

where Kt is interpreted to mean all capital including human capital. Here technical progress can be embodied in new capital investment, thereby making new capital more productive than old capital. (e.g., computers) The key assumption here is that there is no exogenous technical progress and there are constant returns to scale w.r.t. Kt . Output per capita should be

yt = Akt . (34)

Luo, Y. (SEF of HKU) Macro Theory September27,2013 19/38 Model Setting

Consider an economy in which the standard engines of neoclassical growth are absent: there is no technological progress and the population size is constant. The infinitely-lived representative consumer-manager with the standard isoelastic utility solves: ∞ 1 1/σ c − max ∑ βt t , (35) ct 1 1/σ { } t=0 − where σ > 0 is the EIS. Given that the interest rate is rt+1, at optimum the following Euler equation must hold: 1/σ ct+1 β (1 + r ) = . (36) t+1 c  t  Each worker manages his own firm, and the production technology is yt = Akt . In each period, firms invest up to the point where the net marginal product of capital equals the interest rate

rt+1 = A.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 20/38 Note that for any other interest rate, firms invest either an infinite amount or 0. Finally, the model is closed by the following goods market equilibrium condition

ct + it = yt = Akt , (37)

where it = kt+1 (1 δ) kt is investment. − − Equilibrium growth is determined by

ct+1 σ = [β (1 + A)] , 1 + g. (38) ct It is clear that there will be long-run growth, i.e., g > 0 provided that [β (1 + A)]σ > 1. Unlike the neoclassical growth models, here g is independent of the initial level of capital stock, which means that all countries with different starting points, can achieve persistent growth, and the growth rate only depends on model parameters: β, σ, A, and δ.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 21/38 Some Remarks

Such a steady state is feasible provided capital stock and output grow at the same constant growth rate, g. If k grows at g, we have g it = kt+1 kt = gkt = yt − A

where for simplicity assume that δ = 0. Since ct + it = yt , A g ct = − yt A Note that the model can also be formulated in the following central planner economy:

kt+1 = 1 + A δ kt ct , − −   A | {z } given k0. e

Luo, Y. (SEF of HKU) Macro Theory September27,2013 22/38 Some Remarks (2)

A key difference between this model with the neoclassical model is that a change in the saving rate (e.g., an increase in β) now has a permanent effect on the growth rate. A second difference is that the economy reaches its steady state growth path immediately; there is no transition period. The intuition is that the linear production function ties down the interest rate independently of the economy’scapital stock. The problem with the AK model is that labor is not productive. However, in the data labor is a significant component of factor input.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 23/38 Romer’sExternality Model

Romer (1986): There are externalities to capital accumulation, so that individual savers do not realize the full return on their investment. Each individual firm adopts the following production function:

α 1 α ρ F K, L, K = AK L − K , (39)

where K is individual firm’scapital  stock and K is the aggregate capital stock in the economy. We assume that ρ = 1 α such that a central planner faces an AK model (for the central planner,− K = K). Note that if we assume that α + ρ > 1, balanced growth path would not be possible. The rationale behind this specification is that the production process generates knowledge externalities. The higher the average level of capital intensity in the economy, the greater the incidence of technological spillovers that raise the marginal productivity of capital throughout the economy.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 24/38 In the competitive equilibrium, the wage rate is determined by

α α 1 α wt = (1 α) AK L− K − . (40) − t t t For simplicity, we now assume that leisure is not valued and normalize Lt = 1. Assume that there is a measure one of the firms, so that the equilibrium wage is wt = (1 α) AK t . (41) − Similarly, the rental rate of capital is given by

Rt = αA. (42)

Luo, Y. (SEF of HKU) Macro Theory September27,2013 25/38 (conti.) Therefore, the Euler equation in the decentralized economy is

ct+1 1/γ 1/γ gCE = = (βRt+1) = (βαA) , (43) ct while the growth rate in the corresponding central planner economy is

ct+1 1/γ gCP = = (βA) > gCE , (44) ct which is consistent with the fact that capital accumulation has externality. This model overcomes the labor is irrelevant shortfall of the AK model. However, it is little evidence in support of a significant externality to capital accumulation. Note that here ρ = 1 α = 2/3 − means significant externality (α = 1/3).

Luo, Y. (SEF of HKU) Macro Theory September27,2013 26/38 The Human Capital Model

In this model we model physical capital and human capital separately. Let h denote human capital per capita and k physical capital and suppose that the production function per capita α 1 α y = Ak h − , α [0, 1] . (45) ∈ No exogenous technical progress. Real resources are also needed for human capital accumulation: k ∆kt+1 = it δkt , h − ∆ht+1 = i δht , t − k h where it and it are the levels of investment in physical and human capital, respectively. Assume no population growth. The national resource constraint satisfies k h yt = ct + it + it . 1 γ And the utility function is isoelastic: u(ct ) = c − 1 / (1 γ). t − − Luo, Y. (SEF of HKU) Macro Theory  September27,2013 27/38 Solving the Model

Set up the Lagrangian as follows

∞ 1 γ α 1 α t ct − 1 Akt ht − ct (kt+1 + ht+1) L = β − + λt − − . ∑ 1 γ + (1 δ)(kt + ht ) t=0 ( −  − ) (46)

The FOCs with respect to ct , kt+1, and ht+1 are

γ ct− = λt , (47) α 1 1 α λt = β αAkt+−1 ht+−1 + 1 δ λt+1, (48) α α− λt = β (1 α) Ak h− + 1 δ λt+1, (49) − t+1 t+1  − respectively. Combining them, we have 

γ α 1 ct+1 − kt+1 − kt+1 α β αA + 1 δ = 1, = . ct ht+1 − ht+1 1 α   "   # −

Luo, Y. (SEF of HKU) Macro Theory September27,2013 28/38 Economic Implications

Note that since kt+1 is a constant, the growth rates of the two types ht+1 of capital are the same. In equilibrium we have k k t+1 = , (50) ht+1 h and the Euler equation is

γ α 1 ct+1 − k − β αA + 1 δ = 1. (51) ct h −   "   # The rate of growth of consumption is thus

α 1 1/γ ct+1 k − gc = 1 = β αA + 1 δ 1. (52) ct − h − − ( "   #) Assume that the growth is balanced, kt+1 ht+1 gc = gk = 1 = gh = 1 . kt − ht − Luo, Y. (SEF of HKU)  Macro Theory  September27,2013 29/38 (conti.) If we now substitute kt+1 = k = α into the production ht+1 h 1 α function, we obtain the AK model: −

α 1 α 1 α α − yt = Ak h − = A kt . (53) t t 1 α  −  A∗ In the human capital model labor is| treated{z more} seriously.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 30/38 A Model of Endogenous Innovation and Growth

Consider an alternative endogeneous growth model proposed by Romer (1990), in which invention is a purposeful economic activity that requires real resources. By explicitly modeling the research and development (R&D) process, one can gain insights about the effects of government policy on growth. Key assumption: Ideas are nonrival. That is, there are no technological barriers preventing more than one firm from simultaneously using the same idea. Romer assumes that inventors can obtain patent licenses on the blueprints for their innovations. Final goods production:

At 1 α α Yt = LY−,t ∑ Kj,t , (54) j=1

where j 1, , At indexes the different types of capital goods Kj ∈ { ··· } that can be used in production, and At captures the number of types of capital that have been invented as of t.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 31/38 Note that in this specification, an increase in Kj has no effect on the marginal productivity of Ki , i = j. For simplicity, we assume the depreciation rate of capital goods6 is 100%. R&D production: At+1 At = θAt LA,t , (55) − where θ is a productivity shift parameter and LA,t is the amount of labor employed in R&D. Assume there exists a third sector that intermediates between the R&D sector and the final goods production sector. Firms in the R&D sell blueprints to an intermediate capital goods sector that manufactures the designs in t and then sells the machines to firms that in the final goods production sector in t + 1. That is, once an intermediate goods producer buys the blueprint to produce capital good j it becomes the monopoly supplier of that type of capital to the final goods sector.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 32/38 To solve the model, we guess that its equilibrium involves a constant interest rate, constant relative prices, and a constant allocation of labor across the two sectors. (We will confirm that this guess is correct.) The demand for immediate capital goods by the final goods sector is determined by: At At 1 α α max LY− ∑ Kj ∑ pj Kj , (56) Kj − { } j=1 j=1

where pj is the price of capital Kj in terms of final goods. Maximizing implies 1 α α 1 pj = αLY− Kj − . (57)

Luo, Y. (SEF of HKU) Macro Theory September27,2013 33/38 The intermediate goods producer sets Kj to

1 1 1 α α 1 max pj Kj Kj = pj αLY− Kj − Kj (58) Kj 1 + r − 1 + r − { }   where we assume that capital sold at t must be produced at t 1 and future sales must be discounted by 1 + r. Maximizing implies−

1/(1 α) α2 − K = L K, (59) j 1 + r Y ,   for any j. Combining (57) with (59): 1 + r pj = p. (60) α , Given that the cost of producing the capital good is 1 + r (in terms of the final consumption good), this expression means that optimal price is a constant markup over cost. This is just the usual formula for a monopolist facing a constant price elasticity of demand.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 34/38 (conti.) The present value profit on capital produced in t 1 for sale in t: −

1/(1 α) 1 1 α α2 − Π = pK K = − LY . 1 + r − α 1 + r     The next question is what a blueprint will sell for. Since there is a free entry into the intermediate goods sector, the value of a blueprint must equal the entire discounted present value of the profit stream an intermediate goods producer will enjoy after purchasing it:

∞ Π (1 + r) Π pA = ∑ s t = . (61) s=t (1 + r) − r

Luo, Y. (SEF of HKU) Macro Theory September27,2013 35/38 The final step is to find the equilibrium rate of growth. If LA is constant over time, the growth rate of A is

At+1 At g , − = θLA, At which means that in steady state the number of capital good types grows at g, whereas the quantity of each types of capital good remains constant at K. To solve for the optimal allocation of labor across the two sectors, we equalize the marginal product of labor in the two sectors:

∂ (pAθALA) α α ∂Y = pAθA = (1 α) LY− AK = , (62) ∂LA − ∂LY where we use the fact that in the symmetric equilibrium, A α α ∑j=1 Kj = AK . After using the expressions for p, K, and pA, we have r LY = . (63) θα

Luo, Y. (SEF of HKU) Macro Theory September27,2013 36/38 (conti.) We then obtain a technology-determined relationship between growth and the interest rate: r g = θL . − α So far we have only dealt with the supply side of the economy. To close the model, we model the demand side as usual:

∞ 1 1/σ s t cs − 1 max ∑ β − − , (64) cs 1 1/σ { } s=t − which means that

ct+1 σ 1 1/σ = [β (1 + r)] , 1 + g or 1 + r = (1 + g) . (65) ct β Because capital depreciates by 100%, the economy jumps immediately to a steady state in which K, Y , C, and A grow at the same constant rate, which verify our guess on constant equilibrium interest rate, relative price, and labor allocations.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 37/38 (conti.) In the special case in which σ = 1,

α (1 + θL β) r = − , (66) 1 + αβ αβθL (1 β) g = − − . (67) 1 + αβ

Since negative growth is not possible here, we require that

1 β θL > − , (68) αβ

which means that if the initial size of the economy is too small for this condition to be met, the profits from invention are insuffi cient to pay for the labor costs, and there will be no innovation growth.

Luo, Y. (SEF of HKU) Macro Theory September27,2013 38/38