economics of growth and globalization
TRANSCRIPT
Economics of Growth and Globalization
Lecture 5: Growth Models
Alessio Moro, University of Cagliari
November 29, 2017
Lecture 5: Growth Models Economics of Growth and Globalization
Intro
In this lecture we will revise the growth model of exogenousgrowth.
In this model, GDP growth is driven by an exogenoustechnological progress that grows at a certain rate.
Next, we will modify the model to allow for multiple sectors.
Lecture 5: Growth Models Economics of Globalization 2 of 24
Kaldor Facts (Kaldor, 1961)
1 Labor productivity (Y/L) has grown at a sustained rate
2 Capital per worker (K/L) has also grown at a sustained rate
3 The real interest rate, or return on capital, has been stable.
4 The ratio of capital to output (K/Y) has also been stable.
5 Capital and labor have captured stable shares of nationalincome.
6 Among the fastest growing countries in the world there isappreciable variation in the rate of growth.
Lecture 5: Growth Models Economics of Globalization 3 of 24
One-sector exogenous growth model: centralized solution
There is a representative household with preferences
∞∑t=0
βt logCt .
with the subjective discount factor β < 1. The objective of the householdis to maximize the discounted sum of an infinite stream ofconsumption. These preferences imply that consumption in time periodscloser to zero are given more weight than periods further away.
The household is endowed with one unit of labor that shesupplies inelastically in the market.
There is a representative firm producing the output good usingcapital and labor
Yt = K θt (AtNt)
1−θ,
Lecture 5: Growth Models Economics of Globalization 4 of 24
Two-sector exogenous growth model: centralized solution
The evolution of the capital stock in the economy is
Kt+1 = Kt(1− δ) + Xt
with the depreciation rate 0 < δ < 1. Here Xt is the amount of outputthat is invested to produce new capital.
Lecture 5: Growth Models Economics of Globalization 5 of 24
Social planner
Assume that there is a benevolent social planner that is willing tomaximize the utility of the household.
This planner will allocate resources efficiently to do this. Theresource constraint of the planner is then
Ct + Xt = Yt
orCt + Kt+1 = K θ
t (AtNt)1−θ + Kt(1− δ)
Lecture 5: Growth Models Economics of Globalization 6 of 24
Lagrangean
The maximization problems requires to choose consumption Ct
and capital in the next period Kt+1.
Labor is supplied inelastically so there is no decision on it.
The Lagrangean of the planner is
L =∞∑t=0
βt logCt +∞∑t=0
µt
[K θt (AtNt)
1−θ + Kt(1− δ)− Ct − Kt+1
]The planner maximizes with respect to two variables, Ct and Kt+1.
Lecture 5: Growth Models Economics of Globalization 7 of 24
FOCs
βt
Ct= µt (1)
µt+1[θK θ−1t+1 (At+1Nt+1)1−θ + (1− δ)] = µt (2)
From (1) and (2) we can get (3)
βCt [θKθ−1t+1 (At+1Nt+1)1−θ + (1− δ)] = Ct+1 (3)
which is the consumption Euler Equation.
Lecture 5: Growth Models Economics of Globalization 8 of 24
Steady State
If technological change is zero At = At+1 = A we can find a steadystate for this economy:
In steady state all variables are constant over time, thusCt = Ct+1 = C and Kt = Kt+1 = K .
From (3)βC [θK θ−1A1−θ + (1− δ)] = C
θK θ−1A1−θ + (1− δ) = 1/β
K =θ1/(1−θ)A
(1/β − 1 + δ)1/(1−θ)
Lecture 5: Growth Models Economics of Globalization 9 of 24
Steady State
And using the constraint of the planner we can find consumption
C + K = K θ (A)1−θ + K (1− δ)
C =θθ/(1−θ)A
(1/β − 1 + δ)θ/(1−θ)− θ1/(1−θ)Aδ
(1/β − 1 + δ)1/(1−θ)
Lecture 5: Growth Models Economics of Globalization 10 of 24
Balanced growth path
If technological change is non-zero, but grows at a constant rate(1 + γa) = At+1/At this economy has a balanced growth path.
On a balanced growth path all variables grow at a constant rate.Consider again (3)
β
[θ
(At
Kt
)1−θ
+ (1− δ)
]=
Ct+1
Ct(4)
If Ct+1/Ct=1 + γc is constant it must be that A and K grow atthe same rate.
β
[θ
(At(1 + γa)
Kt(1 + γk)
)1−θ
+ (1− δ)
]= (1 + γc)
Thus γk=γa.
Lecture 5: Growth Models Economics of Globalization 11 of 24
Balanced growth path
How about the growth rate of Y and C?
Consider the production function and take logs
logYt = θlogKt + (1− θ)logAt ,
Now take the difference between two periods t+1 and t
logYt+1 − logYt = θ (logKt+1 − logKt) + (1− θ) (logAt+1 − logAt) ,
log (Yt+1/Yt) = θlog (Kt+1/Kt) + (1− θ)log (At+1/At) ,
log(1 + γy ) = θlog(1 + γk) + (1− θ)log(1 + γa) ,
log(1 + γy ) = θlog(1 + γa) + (1− θ)log(1 + γa) ,
log(1 + γy ) = log(1 + γa)
γy = γa .
Thus also Y grows at the same rate as A.
Lecture 5: Growth Models Economics of Globalization 12 of 24
Balanced growth path
Consider now the feasibility constraint
Ct + Kt+1 = K θt (At)
1−θ + Kt(1− δ)
after one period
(1 + γc)Ct + (1 + γa)Kt+1 = (1 + γa)K θt (At)
1−θ +Kt(1 + γa)(1− δ)
The only way for this constraint to hold after one period is thatγc=γa. Thus in this economy all variables grow at the same rate,which is the rate of technological change.
Lecture 5: Growth Models Economics of Globalization 13 of 24
Decentralized equilibrium
In a decentralized equilibrium there is no social planner.
The household takes market prices as given and makes optimaldecisions to maximize utility.
The firm takes market prices as given and makes optimaldecisions to maximize profits.
The interaction of household and firm determines the marketequilibrium.
Lecture 5: Growth Models Economics of Globalization 14 of 24
Household
There is a representative household with preferences
∞∑t=0
βt logCt .
with the subjective discount factor β < 1.
The household is endowed with one unit of labor that she suppliesinelastically in the market.
The household also holds the capital stock of the economy.She uses capital to move resources from the present to the future.Thus, the budget constraint of the household is
Ct + Kt+1 = wt + Kt(1 + rt − δ)
Here wt is the wage rate received by the household for providinglabor in the market and rt is the rental rate that the householdreceives by renting capital to firms.
Lecture 5: Growth Models Economics of Globalization 15 of 24
Household
The Lagrangean of the household is
L =∞∑t=0
βt logCt +∞∑t=0
µt [wt + Kt(1 + rt − δ)− Ct − Kt+1] .
The household maximizes with respect to two variables, Ct andKt+1.
Lecture 5: Growth Models Economics of Globalization 16 of 24
Firm
There is a representative firm producing the output good usingcapital and labor
Yt = kθt (AtNt)1−θ
,
Objective of the firm is to maximize profits by choosing theamount of capital and labor to use in production.
Note that we assume perfect competition. In PC firms takeprices as given. Assuming also constant returns to scale impliesthat the number of firms is irrelevant and we can assume only onerepresentative firm.
The representative firm then solves
maxNt ,Kt
Π = maxNt ,Kt
[ptk
θt (AtNt)
1−θ − wtNt − rtkt].
Note that the problem of the firm is static (and no constraints),while that of the consumer is dynamic (and with a constraint).
Lecture 5: Growth Models Economics of Globalization 17 of 24
Equilibrium Definition
An equilibrium for this economy is a set of prices {pt ,wt , rt}∞t=0,allocations for the consumer {Ct ,K t+1}∞t=0 and allocations for thefirm {Yt , k t+1,Nt}∞t=0 such that:
1 Given prices,{Ct ,K t+1}∞t=0 solves the household’s maximizationproblem;
2 Given prices,{Yt , k t+1,Nt}∞t=0 solves the firm’s maximizationproblem;
3 Markets clear:kt = Kt ,
Nt = 1,
Yt = Ct + Xt ,
where Xt = Kt+1 − Kt(1− δ) is investment in capital.
Lecture 5: Growth Models Economics of Globalization 18 of 24
Firm FOCs
rt = ptθ
(Kt
Nt
)θ−1A1−θt (5)
wt = pt(1− θ)
(Kt
Nt
)θA1−θt (6)
Using (5) and (6) we obtain
Kt
Nt=
θ
1− θwt
rt(7)
Lecture 5: Growth Models Economics of Globalization 19 of 24
Household FOCs
βt
Ct= µt (8)
µt+1[rt+1 + (1− δ)] = µt (9)
From (8) and (9) we can get (10)
βCt [rt+1 + (1− δ)] = Ct+1 (10)
which is the Euler Equation for the household.
Lecture 5: Growth Models Economics of Globalization 20 of 24
Solving the model
By setting pt = 1as the numeraire, and considering that inequilibrium it must be Nt = 1
rt = θK θ−1t A1−θ
t
We can then substitute in the Euler equation to obtain
βCt [θKθ−1t+1 A
1−θt+1 + (1− δ)] = Ct+1 (11)
which is the same equation that we had in the centralized problem.
Lecture 5: Growth Models Economics of Globalization 21 of 24
Solving the model
Consider now the budget constraint of the consumer
Ct + Kt+1 = wt + rtKt + Kt(1− δ)
Substitute for wt and rt from the firm’s FOCs (again settingpt = 1as the numeraire, and considering that in equilibrium it mustbe Nt = 1)
Ct + Kt+1 = (1− θ)K θt A
1−θt + θK θ−1
t A1−θt Kt + Kt(1− δ)
Ct + Kt+1 = K θt A
1−θt + Kt(1− δ)
orCt + Kt+1 = Yt + Kt(1− δ)
which is the constraint of the social planner in the centralizedsolution.
Lecture 5: Growth Models Economics of Globalization 22 of 24
Solving the model
Finally, we determine the evolution of the interest rate along thebalanced growth path. Recall that:
rt = θ
(Kt
At
)θ−1As K and A grow at the same rate, the interest rate is constant. Tosee this compute rt+1
rt+1 = θ
(Kt(1 + γa)
At(1 + γa)
)θ−1= θ
(Kt
At
)θ−1= rt .
Lecture 5: Growth Models Economics of Globalization 23 of 24