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Introduction: Basic Facts and NeoclassicalGrowth Model
Diego RestucciaUniversity of Toronto
and NBER
University of OsloAugust 14-18, 2017
Restuccia Macro Growth and Development University of Oslo 1 / 34
Overview
Restuccia Macro Growth and Development University of Oslo 2 / 34
Facts and the neoclassical growth model
Basic facts
Framework: Neoclassical growth model with distortions
Discussion, extensions
Some Facts
Restuccia Macro Growth and Development University of Oslo 3 / 34
Large income differences across countries at any point in time(recent history)
Growth not systematically related to the level of development
Strong positive relationship between real investment rates (or realcapital to output ratios) and real GDP per capita across countries
GDP per capita across Countries and Time
YearDecile 1960 1970 1980 1990 2000 2010 2014
1 3.4 3.2 3.1 2.3 1.7 1.8 2.02 5.5 5.1 4.5 3.2 2.7 3.0 3.33 7.4 6.8 6.4 4.5 3.9 4.8 5.74 9.6 8.9 8.0 6.8 6.2 9.2 10.95 12.7 11.5 11.5 10.2 10.0 14.8 18.56 14.9 15.8 17.4 15.4 16.1 21.9 25.67 22.0 22.9 24.1 22.3 22.1 29.9 35.08 30.5 33.4 39.1 38.0 45.3 53.0 56.79 49.9 56.0 62.2 65.1 72.6 77.0 79.510 80.1 81.3 82.6 79.3 83.5 98.3 105.0
R 10/1 23.4 25.4 27.1 34.0 49.5 53.6 51.3
Restuccia Macro Growth and Development University of Oslo 4 / 34
GDP per worker, ratio of 5% richest andpoorest countries
1960 1965 1970 1975 1980 1985 1990 199510
15
20
25
30
35
40
45
50
55
60
Source: Duarte and Restuccia (2006)
Restuccia Macro Growth and Development University of Oslo 5 / 34
GDP per capita across Countries and Time
Country 1960 1980 2000 2014
Botswana 2.3 7.5 21.5 29.1Ethiopia 2.5 2.5 1.2 2.6Malawi 4.7 4.3 2.1 1.9Indonesia 5.3 7.1 9.0 19.8China 5.6 5.7 9.5 24.6India 5.9 4.0 4.4 10.5Korea 6.2 18.3 50.5 68.2Zimbabwe 11.3 10.0 6.1 3.1Singapore 14.3 41.7 83.3 149.7Japan 30.8 63.2 73.9 68.2Mexico 32.0 38.1 25.4 31.1Austria 53.4 62.9 77.8 92.7France 59.4 75.4 68.3 76.5United Kingdom 68.0 64.7 74.9 75.3New Zealand 81.2 60.2 59.4 66.0
Restuccia Macro Growth and Development University of Oslo 6 / 34
GDP per capita 1960 and 2011
Source: Jones (2016)
Restuccia Macro Growth and Development University of Oslo 7 / 34
Growth in GDP per worker, 1960-1996
0.05 0.10 0.20 0.30 0.50 0.75 1−4
−2
0
2
4
6
8
AGO
ARG
AUS
AUT
BDI
BEL
BEN
BFA
BGD
BOL
BRA
BWA
CAF
CAN
CHE
CHL
CHN
CIV
CMR
COG
COLCRI
DNKDOMECU
EGY
ESP
ETH
FINFRAGAB
GBR
GHAGIN
GMB
GNB
GRC
GTM
HKG
HND
IDN
IND
IRL
IRN
ISRITA
JAM
JOR
JPN
KEN
KOR
LKALSO MAR
MGD
MEX
MLIMOZ
MRT
MUS
MWI
MYS
NAM
NER
NGA
NIC
NLD
NORNPL
NZL
PAK
PAN
PER
PHL
PNG
PRT
PRY
ROM
RWA
SEN
SGP
SLV
SWE
SYR
TCD
TGO
THA
TTO
TUR
TWN
TZA UGA URY
USA
VEN
ZAF
ZAR
ZMB
ZWE
Relative Output per Worker 1960 (in logs)
Ann
ualiz
ed G
row
th in
Lab
or P
rodu
ctiv
ity (
%)
Source: Duarte and Restuccia (2006)
Restuccia Macro Growth and Development University of Oslo 8 / 34
Real Investment to Output Ratio
Source: Restuccia and Urrutia (2001)
Restuccia Macro Growth and Development University of Oslo 9 / 34
Neoclassical Growth Model
Restuccia Macro Growth and Development University of Oslo 10 / 34
Understand factors behind choice of investment
By spelling out the economic forces determining investment wewould be in better position to understand investment ratedifferences across countries
And, therefore, part of income differences across countries
Start with optimal growth and then consider a competitive marketeconomy with distortions
A Model of Optimal GrowthPreferences and Endowments
Restuccia Macro Growth and Development University of Oslo 11 / 34
Large number of infinitely-lived homogeneous households
Preferences over consumption goods at each date, ct, described bythe following utility function:
∞∑t=0
βtu(ct) (1)
where β ∈ (0, 1)
Households are endowed with one unit of productive time perperiod and k0 > 0 units of the capital stock
Assume population is constant over time (L normalized to one)
A Model of Optimal GrowthTechnology
Restuccia Macro Growth and Development University of Oslo 12 / 34
At each date only one good produced with the followingproduction function,
Yt = AtKαt L
1−αt (2)
where At is total factor productivity (TFP), assumed to beconstant over time
Output can be allocated to consumption and investment
Ct +Xt ≤ Yt (3)
The capital accumulation is given by,
Kt+1 = (1− δ)Kt +Xt (4)
where one unit of investment Xt transforms into one unit ofcapital next period
Social Planner’s Problem
Restuccia Macro Growth and Development University of Oslo 13 / 34
A benevolent social planner cares about the utility of therepresentative household
Chooses sequences of consumption and investment to maximize(1) subject to (2), (3), and (4)
Formally,
max{Ct,Xt,Kt+1}∞t=0
∞∑t=0
βtu(Ct)
subject toCt +Xt ≤ AKα
t L1−αt t = 0, 1, 2, ...
Kt+1 = (1− δ)Kt +Xt t = 0, 1, 2, ...
Ct,Kt+1 ≥ 0 and K0 > 0 given
Characterization of Planner’s Problem
Restuccia Macro Growth and Development University of Oslo 14 / 34
Under mild conditions on u(·) the resource constraint is satisfiedwith equality (no output will be wasted) and the constraints thatCt and Kt+1 be non-negative are not binding
Can substitute the two constraints to eliminate Xt and Ct fromthe problem and express the maximization problem as a choice ofKt+1
The first order condition with respect to the capital stock nextperiod gives,
u′(Ct) = βu′(Ct+1)[αAKα−1
t+1 + (1− δ)]
t = 0, 1, 2, ... (5)
Planner is equating the marginal cost of postponing consumptionto the marginal benefit
Definition: Steady State
Restuccia Macro Growth and Development University of Oslo 15 / 34
A steady state solution to the planner’s problem is a Ks such thatK0 = Ks implies Kt+1 = Kt = Ks for all t
In a steady state the capital stock is constant over time
Notice that if the capital stock is constant then investment,consumption, and output are also constant
Applying the steady state definition to the equation (5)
1 = β[αAKα−1s + (1− δ)]
which implies
Ks =
(αA
1β − (1− δ)
) 11−α
Steady-State Implications
Restuccia Macro Growth and Development University of Oslo 16 / 34
In steady state Xs = δKs, then XsYs
= δKsYs
Therefore the investment to output ratio,
Xs
Ys=
αδ[1β − (1− δ)
]The investment rate is determined by technology and preferenceparameters that are assumed to be constant across countries
Explore competitive equilibrium version of the neoclassical modelwith distortions
Competitive Equilibrium
Restuccia Macro Growth and Development University of Oslo 17 / 34
Decentralization of the planner’s solution as the solution tooptimization problems of consumers and firms in competitivemarkets
Households own the capital stock and supply capital and laborservices to firms at competitive rental rates
Households are endowed with equal ownership shares of all firmsin the economy
Large number of firms operating constant returns to scale outputtechnology
Firms hire capital and labor services at competitive rates tomaximize profits
Let wt and rt be the rental rates of labor and capital at each datein terms of the consumption good at date t
Representative Household’s Problem
Restuccia Macro Growth and Development University of Oslo 18 / 34
Given prices, profits from firms πt, and k0, the problem of thehousehold is to choose sequences of ct, xt, and kt+1 to maximizethe present discounted value of utility
Formally,
max{ct,xt,kt+1}∞t=0
∞∑t=0
βtu(ct)
subject to
ct + xt = wt + rtkt + πt t = 0, 1, 2, ...
kt+1 = (1− δ)kt + xt t = 0, 1, 2, ... k0 > 0
Notice that since households do not value leisure they allocate alltheir time to work in the market
Representative Firm’s Problem
Restuccia Macro Growth and Development University of Oslo 19 / 34
Given prices, the firm’s problem is to choose capital and labor soas to maximize profits
Because the problem of the firm is static (the decisions of firmstoday do not affect their decisions tomorrow), we can write theirproblem as a sequence of one-period maximization problems,
maxKt,Lt>0
πt =[AKα
t L1−αt − wtLt − rtKt
]
Definition: Competitive Equilibrium
Restuccia Macro Growth and Development University of Oslo 20 / 34
A competitive equilibrium is a sequence of prices {wt, rt}∞t=0, allocationsfor the household {ct, xt, kt+1}∞t=0, allocations for firms {Kt, Lt}∞t=0,and profits {πt}∞t=0 such that:
(i) Given prices, profits, and k0 > 0, the allocations for the householdsolve the household’s problem
(ii) Given prices, the allocations for firms solve the firm’s problem
(iii) Markets clear:
Output ct + xt = YtCapital kt = Kt
Labor 1 = Lt
Characterization of Competitive Equilibrium
Restuccia Macro Growth and Development University of Oslo 21 / 34
The industrial organization of this economy is irrelevant because ofthe constant returns to scale assumption on the output technology
First order conditions from the firm’s problem
Kt : αAKα−1t L1−α
t = rt,
Lt : (1− α)AKαt L−αt = wt.
Market clearing conditions imply
rt = αAkα−1t , wt = (1− α)Akαt ,
and therefore, the competitive equilibrium wage and rental rate ofcapital are equated to the marginal product of labor and capital
Characterization of Competitive Equilibrium
Restuccia Macro Growth and Development University of Oslo 22 / 34
It is straightforward to show that at these prices, the demand ofcapital and labor from firms imply zero profits in equilibrium(πt = 0 for all t)
Also note that the the competitive assumption together with theCobb-Douglas specification of the output technology imply thatthe share of capital in income is equal to α
The household’s first order condition with respect to the capitalstock next period is given by,
u′(ct) = βu′(ct+1) [rt+1 + (1− δ)] t = 0, 1, 2, ... (6)
Definition: Steady State
Restuccia Macro Growth and Development University of Oslo 23 / 34
A steady state competitive equilibrium is a competitive equilibriumwith ks such that k0 = ks imply kt+1 = kt = ks for all t
Notice that at the equilibrium prices, equation (6) determines anallocation of capital that exactly corresponds to the solution of theplanner’s problem
Not surprising since in this environment the fundamental welfaretheorems hold, allocations of the planner’s problem and thecompetitive equilibrium coincide
Competitive Equilibrium with Distortions
Restuccia Macro Growth and Development University of Oslo 24 / 34
The determination of the investment to output ratio in theneoclassical growth model does not leave much room for thinkingabout investment rate differences across countries
Inspection of the Euler equation for capital accumulation of theconsumer –equation (6)– does give some hints about the factorsthat may be relevant in understanding investment rate differencesacross countries
Households would equate the marginal cost and benefit of tradingconsumption inter-temporally
Government policies such as taxes and other distortionarypractices can change the returns to investing – so do inefficienciesin producing investment goods
Many possibilities, here consider a tax to investment
Competitive Equilibrium with Distortions
Restuccia Macro Growth and Development University of Oslo 25 / 34
Consider a government that taxes household’s investment atproportional rate τ and rebates the proceeds back to householdsas a lump-sum subsidy transfer
The budget constraint of the household becomes,
ct + (1 + θ)xt = wt + rtkt + Tt t = 0, 1, 2, ...
where Tt is a lump-sum transfer
Definition: Competitive Equilibrium
Restuccia Macro Growth and Development University of Oslo 26 / 34
A competitive equilibrium is a sequence of prices {wt, rt}∞t=0, allocationsfor the household {ct, xt, kt+1}∞t=0, allocations for firms {Kt, Lt}∞t=0,and government transfers {Tt}∞t=0 such that:
(i) Given prices, transfers, and k0 > 0, the allocations for thehousehold solve the household’s problem
(ii) Given prices, the allocations for firms solve the firm’s problem
(iii) Markets clear: Output ct + xt = Yt, capital kt = Kt, and labor1 = Lt
(iv) The government’s budget is balanced every period
Tt = θXt
Solving for CE with Distortions
Restuccia Macro Growth and Development University of Oslo 27 / 34
The Euler equation for capital accumulation from households nowsatisfies,
u′(ct) = βu′(ct+1)
[rt+1
(1 + θ)+ (1− δ)
]t = 0, 1, 2, ...
Notice that with θ > 0 the allocation of capital in this version ofthe model will differ from the optimal planner’s solution
The tax will discourage investment as it lowers the return toinvesting relative to the cost of foregone consumption
Steady-State Implications
Restuccia Macro Growth and Development University of Oslo 28 / 34
It is straightforward to show that the steady state capital stock isinversely related to the tax rate
Substituting the rental rate for capital and imposing the steadystate condition in the Euler equation,
1 = β
[αAkα−1s
(1 + θ)+ (1− δ)
]which implies
ks =
αA
(1 + θ)[1β − (1− δ)
] 1
1−α
Steady-State Implications
Restuccia Macro Growth and Development University of Oslo 29 / 34
In steady state,
Ks
Ys=
Ks
AKαs
=K1−αs
A=
α
(1 + θ)[1β − (1− δ)
]
and hence does not depend on A.
Researchers exploit this property in growth and developmentaccounting to separate the contribution of TFP and capitalintensity to output differences by writing the production functionin intensive form
Y = A1
1−α
(K
Y
) α1−α
Steady-State Implications
Restuccia Macro Growth and Development University of Oslo 30 / 34
In steady state xs = δks, therefore,
Xs
Ys=
αδ
(1 + θ)[1β − (1− δ)
] ≡ sDifferences in tax rates may help in accounting for investment ratedifferences across countries
Higher taxes imply lower capital accumulation, lower investmentrates, lower capital to output ratios, and lower output per worker
A tax to capital income has the same qualitative implications thana tax on investment (see Restuccia and Urrutia, 2001 for anapplication to investment taxes and differential productivity oninvestment goods)
What matters is effective wedges that affect the return to capitalinvestment
Relative Price of Investment
Source: Restuccia and Urrutia (2001)
Restuccia Macro Growth and Development University of Oslo 31 / 34
Cross-Country Implications
Restuccia Macro Growth and Development University of Oslo 32 / 34
Let y = Y/L be output per worker
Then for any arbitrary countries i and j
yiyj
=
(AiAj
) 11−α
(sisj
) α1−α
Solow assumed Ai = Aj = 1 (i.e., no differences in technologyacross countries), then
yiyj
=
(sisj
) α1−α
Cross-Country Implications
Restuccia Macro Growth and Development University of Oslo 33 / 34
Can differences in investment to output ratios explain laborproductivity differences across countries?
yi/yj
αsisj
1/3 2/3
4 2 166 2.5 369 3 81
Large differences in investment rates imply small differences inoutput per worker if reproducible capital is physical capital
A broader notion of capital, e.g. human capital, may provideamplification (see Erosa et al 2010; Manuelli and Seshadri 2014)
Discussion
Restuccia Macro Growth and Development University of Oslo 34 / 34
What determines human capital differences across countries?Standard theory implies TFP an important factor
Development accounting or even modern approach to humancapital differences leaves a large role for TFP differences
What accounts for TFP differences?
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