EPP - Macroeconomics 1Lecture 4 - Endogenous growth

Zsofia L. Barany

Sciences Po

2014 October

Motivation

I we have seen three growth modelsI SolowI NGMI OLG

I we learnedI no long-run growth with constant technologyI growth only possible through technological change

I empirical literature: about 50% of GDP per capita diff acrosscountries is due to productivity diff, which can be the result of

I technological differencesI quality of institutionsI cultural attitudes

I now we focus on the determinants of technological change,the sequence of {A(t)}t

Ideas and increasing returns I.

I Solow, NGM: accumulation of capital (or human capital) isthe source of transitional growth

I if capital is accumulated, and it has decreasing returns ⇒perpetual accumulation and high marginal products are notpossible

I how can we keep the marginal product high?

I easy answer: ’AK’ models, where the marginal product of capis constant (see problem set 1)

I alternative: we need increasing returns as a wholeRomer (1980): increasing returns are a result of ideas beingnon-rival

I rival vs non-rival → technological characteristicI excludable vs non-excludable → institutional characteristic

Ideas and increasing returns II.

I non-rivalry naturally suggests IRSI movie production & DVD productionI discovery of drug & drug productionI fixed cost & CRS

I a way to think about this

Y (t) = F (A(t),K (t), L(t))

with

F (A(t), λK (t), λL(t)) = λF (A(t),K (t), L(t))

→ CRS in all rivalrous factors

F (λA(t), λK (t), λL(t)) > λF (A(t),K (t), L(t))

→ IRS in all factors

Perfect competition does not work

I take the economy as a whole in a perfect competition setupwith the previous production function

I the profit in the economy is zero:

Π = F (A(t),K (t), L(t))− w(t)L(t)− R(t)K (t)

F (A(t),K (t), L(t))−MPL(t)L(t)−MPK (t)K (t) = 0

I so how do we pay for the fixed cost of investment in ideas?who pays the research labs?

I there are two solutions:

1. ideas accumulate as a by-product of production → nobody hasto pay

2. give producers some market power → positive profits → canpay for research

Models of R&D

I previous model: growth, technology improvement is aby-product of neoclassical capital accumulationtechnological improvement is free

I however R&D is a purposeful activity, conducted by profitoriented firms

I most simple models of endogenous technological change arethose in which R&D expands the variety of inputs or machinesused in production (Romer, 1990)→ due to the increasing number of varieties, the division oflabor (specialization) increases

I alternative: R&D improves the quality of inputs or machinesused in production

I product innovation: increases the number of inputs or thequality of inputs

I process innovation: reduces the cost of existing products

The expanding varieties model

based on Romer (1990), we follow chapter 6 of BSM

three types of decision makers in the model:

1. householdsmaximize utility, subject to the usual budget constraint

2. final good producershire labor and intermediate inputs → combine them toproduce the final output, sold at unit price

3. R&D firms

I devote resources to invent new intermediate inputs (stage 1)I once a product has been invented, the innovating firm obtains

a perpetual patent→ allows the firm to sell at the price it chooses

I price is chosen to maximize profits (stage 2)

1. Households

I consumer side is exactly the same as in the Ramsey model

I continuous time, infinite lives

I representative household with CIES utility

I no population growth, labor supplied inelasticallyL is the labor supply (and the population), and is constant

I representative household owns a balanced portfolio of all thefirms in the economy

2. Final good

I final good is produced in perfect competition

Yi (t) = AL1−αi

N∑j=1

Xαij

I Xij is the amount of intermediate good j used by firm i inproduction

I note: production function features diminishing marginalproduct of each input ⇒ the firms will try to use all inputsthat have a finite price

I production function is CRS in L and all the Xjs

I changes in the quantity of one intermediate do not change themarginal productivity of another (but change the marginalprod of labor)

2. Final good

I source of technical change: increase in N – product innovationexample: add fertilizers to labor and tools in agriculture

I the more intermediate inputs there are, the more efficientproduction is

I modeling trick:

Yi (t) = AL1−αi

∫ N

j=0Xαij dj

I N is not a discrete number, but a measure of intermediategoods

I allows to work with N → the source of technological changethis is the object of interest

2. Demand for inputs

I let the final good be the numeraire, price = 1

I perfect competition in the final good production ⇒ firms areprice-takers

I demand for labor by firm i given wage w

∂Yi

∂Li= w ⇒ Li (w) =

((1− α)A

∫Xαij

w

) 1α

I demand for intermediate input j given price Pj :

∂Yi

∂Xij= Pj ⇒ Xij(Pj) =

(αAL1−α

i

Pj

) 11−α

note: decreasing in Pj , but independent of price of otherintermediates

3. Intermediate goods –assumptions

We have to specify three things

1. technology to produce already existing intermediate goodsI 1 unit of final good is needed to produce 1 unit of intermediate

goodI the intermediate goods are fully used up in production, they

are non-durable, δj = 1

2. technology to invent new intermediate goodsI η > 0 units of final good produce one new variety, or one

blueprint of a new intermediate input

3. market structure of intermediate-good marketsI basic economics of R&D: there is an initial outlay to invent a

new product (or process)I with perfect competition there would be no incentives to inventI some monopoly power critical for innovationI here: perpetual monopoly rightsI alternatives: secrecy, intangible, firm-specific know-how

3. Intermediate inputs – main interest

I ultimate interest: the decision to invent new varieties

I to do this potential inventors need to compute the presentvalue of profits they will make and evaluate against the cost

I to figure out the PV of profitswe need to figure out how much they will produce and sell ineach period

3. Intermediate inputs – stage 2

I due to full depreciation, the inventor’s price decision isindependent across periods

I thus the inventor can choose in each period the price of hisintermediate good, Pj , to maximize per period profit:

maxPj

Πj = PjXj(Pj)− Xj(Pj)

where Xj(Pj) =∑

i Xij(Pj) is the total demand forintermediate good j at price Pj from all final good producers

3. Intermediate inputs – stage 2

I optimal price

Pj =1

αI Pj > 1 = MC : they charge a constant mark-up over the

marginal cost

I all intermediate good producers charge the same price, P = Pj

I at this price the total demand for intermediate good j is

Xj = A1

1−αα2

1−α L

increasing in the market size, L

I intermediate producer j ’s profit is:

Πj =1− αα

A1

1−αα2

1−α L ≡ Π

larger market ⇒ larger profit, scale effectsnote: profit is constant across periods

3. R&D firms – stage 1

I so owning the patent of an intermediate good at date t is likeowning an asset that pays Π in each future period→ in continuous time: it has a flow payment of Π

I the cost of acquiring this asset is η

I an alternative use for these η units of consumption is to lendthem

I call r(v) the interest rate on loans at time v

I people have to be indifferent between these two optionsotherwise either too much supply of loans or too muchdemandthis would change the interest rate, until the value of the twooptions equalize

I there will be a positive amount of innovation as long ascost of innovation = return to innovation

3. R&D firms – stage 1, free entry

I thus at time t the free entry implies

I cost of innovation = return to innovation

η =

∫ ∞t

Πe−∫ vt r(s)dsdv ≡ V (t)

this equation gives us the equilibrium interest rate

I this has to hold for all t

I differentiating wrt time, we get that

r(t) =Π

V (t)=

Π

η

I interest rate has to be constant = market return on R&D

I ⇒ the market value of a firm that holds an intermediate goodpatent is η at any point in time (not just at the discovery ofthe new blueprint

Back to the households

I households maximize

U =

∫ ∞0

C (t)1−θ − 1

1− θdt

I what is the return on assets held by the household?

I as long as there is positive R&D in equilibrium, the rate ofreturn on loans has to equal the rate of return on R&D, whichis r = Π/η

I the inter-temporal BC is

d assets

dt= w(t)L + r · assets − C (t)

I we get the familiar Euler equation

C

C=

1

θ(r − ρ)

Equilibrium

I claim: there exists a unique balanced growth path, theeconomy jumps to the BGP at time 0, no transition

I total amount spent on producing intermediate inputs

X =

∫ N

j=0Xjdj = NXj = NA

11−αα

21−α L = constant1 · NL

I final good produced by firm i

Yi = AL1−αi NXα

i = AL1−αi N

(A

11−αα

21−α Li

)α= constant2·NLi

I and hence in the economy

Y =∑i

Yi = constant2 · N∑i

Li = constant2 · NL

Resources devoted to R&D

I whatever of the final good is not spent on producingintermediate goods or on consumption is spent on innovation:

Z = Y − X − C

I η units of final good give one unit of innovation, hence

N =Z

η

I we can re-arrange to get

ηN

N=

Y

N− X

N︸ ︷︷ ︸these are constants

−C

N

The balanced growth path

I for a BGP C and N have to grow at the same rate

I we know from the Euler equation that

gC =1

θ(r − ρ) =

1

θ

(π

η− ρ)

I Π/η is a known and exogenous constant

I intuition: when Π/η is large the returns from postponingconsumption are high, hence more resources are invested inR&Dmore decentralized intuition: inventors are willing to pay ahigher interest rate, so consumers are willing to postpone

The balanced growth path

I everything grows at the same rate:

N

N=

Y

Y=

X

X=

C

C=

1

θ

(π

η− ρ)

I Y is gross output

I Y − X is GDP (or net output) also grows at the same rate

I if r < ρ zero growth, as there will be no R&D

I no transition, the economy is always on the BGP

The balanced growth path

I to see why there are no transitional dynamics

Yi = AL1−αi NXα

i

I growth in output per worker comes fromI growth in quantity of intermediates per worker X/LI growth in N

I ’reduced form’ very similar to NGM, except that N isendogenous

I an important difference: X/L is always constantwhen output increases, there is extra ’investment’ inintermediate goods, but all of it goes into new varieties:all growth in X = NXj is absorbed by N

I intuition: there are decreasing returns to increasing Xj , butconstant returns to expanding N

Properties of the BGP 1.

the endogenous (!) growth rate is higher if

I θ and ρ are lower: people are more willing to deviate fromcons smoothing and to postpone cons → more investment inR&D

I A is higher: increases the productivity of X

I η is lower: increases the equilibrium rate of return, and hencethe returns to postponing consumption → more resourcesinvested in R&D

I L is larger: the invented good can be used in a non-rivalmanner across the whole economy: same cost, higher profits

Properties of the BGP 2.

not Pareto optimal – due to monopoly pricing

I Xj < X Sj → lower level of output (static efficiency loss)

I the social planner values innovation more than the monopolist(he is able to use them more intensively), he prescribes moreinnovation → growth effect (dynamic efficiency loss)

subsidies can help

I for the purchase/sale of intermediate goods → increase use ofintermediate goods → right growth rate and level of output

I for innovation (η) → increase rate of innovation → leads toright growth rate of N, but not to the right quantities of X

I details are very model dependent: probably a combination ofthe two can reach first best

The scale debate

I big debate in the 90s

I some people say this killed endogenous growth theory

I many models predict that the growth rate is increasing in thepopulation→ tied to the non-rivalry of ideas

I two possible interpretations

1. demand for innovation – more people can use any singleinnovation

2. supply of innovations – more people have more ideas, and youonly need one idea for one innovation

Problematic implications

1. the scale debateI countries with larger population don’t necessarily grow fasterI Kremer (1993):

I world pop is positively correlated with world growth over longstretches of time

I landmass area (and presumably landmass pop) is pos corr withsubsequent growth over a very long horizon

I → maybe world population is better to look atit is hard to think of innovations within country borders, aspeople can copy (see problem set)

2. no BGP with growing populationI the population of most nations has not been constantI with population growth as in the NGM (L(t) = L(0)ent) no

balanced growth, instead the growth rate of the economywould be increasing over time

Problematic implications

3. the model implies

N

N=

1

η

R&D spending

N= constant · R&D spending

Y /L

Jones (1995,1999) shows thatI measured R&D spending

Y/L has been steadily increasing in

industrialized countriesI but productivity growth has not

+1 knife-edged nature of steady state: consider η = φNσ

I σ > 0: ’fishing out of ideas’ ⇒ no growthI σ < 0: ’standing on the shoulder of giants’ ⇒ increasing

growth ratesI σ = 0: only one that gives BGP

Jones’ solution to scale effect (1995)

I two changes:I n > 0 – population growth gives more incentives to innovateI η = φNσ with σ > 0 – makes it increasingly costly to innovate

these tow things exactly cancel out → no explosive growthI results

I BGP growth rate = nI not all that surprising:

initially there are high returns (low cost) of R&D, so N/Lgrowsbecause of decreasing returns eventually growth in N is justenough to keep N/L constant

I offsetting forces: market size vs increasing costsI no other parameter matters for the growth rateI φ, σ, L, even subsidies only have level effects

Summary

I different models of endogenous growth

I key element 1: non-rivalry of ideas

I key element 2: externalities or monopolistic competition

I in models of purposeful innovations, the pace of growth isdetermined by incentivesmarket structure, competition policy, taxes, patents, propertyrights

I equilibrium typically not Pareto optimal (even withoutdistortionary policies)due to externality or monopolistic competitionin practice: barriers to R&D may be more important thanmonopoly distortions

Summary

I constant R&D costs: larger markets lead to explosionincreasing R&D costs: N/L converges to a constant

I policy variables have growth effects only in models withscale effects / no pop growthrazor-edge assumptions

I models which remove these unappealing features→ policy variables do not have growth effect, only level effect

I if there is a transition, and the transition takes long, the leveleffect maybe observationally equivalent to growth effect