2. Solow With Resoruces Flashcards
(19 cards)
2 types of natural resources. Which is of interest and why
Renewable - constant stock
Non-renewable - drags on growth…
When we add resources, what is added to our CD function (2)
B) expression
(On page 2)
Add
R: resources used in production
T: Amount of land
Yt = K(t) to the a R(t) to the β T(t) to the y (A(t)L(t)) to the 1-a-β-Y
Net increase in capital (K dot) , population growth Ldot, tech growth A.
Have they changed?
No still K.(t) = sY(t) - δK(t)
L.(t) = nL(t)
A.(t) = gA(t)
What about assumption for land T
B) what about assumption for resources used in prod R
Land is fixed, thus land growth T.(t) =0
B)
Resources decline!
R.(t) = -bR(t)
Negative growth rate i.e resources decline!
So summary: what does A, L, R and T grow at
Constant rates
g for A
n for L
-b for R
0 for T
What is needed for a balanced growth path
K and Y each grow at a constant rate (K/Y capital output ratio constant, in order to get k and y steady states!)
How can we show where growth rate of K to be constant PG 5
Use net increase in capital equation K. = sY - δK
Divide by K(t)
K.t(t) / K(t) = sY(t)/K(t) - δ
For growth rate of K to be constant, Y/K must be constant (since s and δ are constants. So Y and K must have equal growth rates in order to keep K growth (LHS) constant
Take logs of the CD function we did earlier
Then differeniate with respect to time, why?
Becasue differentiating with respect to time gives us the variable’s growth rate:
gY(t) = agK(t) + βgR(t) + ygT(t) + (1-a-β-y)[gA(t) + gL(t)]
g…. Is the growth rate of whatever variable!
Then, using our knowledge of growth rates of R,T,A and L, we can sub in our values and simplify it!
gY(t) = agK(t) + βgR(t) + ygT(t) + (1-a-β-y)[gA(t) + gL(t)]
Now get
gY(t) = agK(t) - βb + 0 + (1-a-β-y)[n+g]
So nowe have
gY(t) = agK(t) - βb + 0 + (1-a-β-y)[n+g]
We said growth rates of K and Y must be equal for balanced growth path. So what do we do? Pg8
gK = gY
replace gK in equation for gY, and rearrange for gY, and call it gy bgp : growth rate of Y on balanced growth path.
gY = (1-a-β-y)(n+g) - βb/ 1-a
Draw graph to show the gk and gy (growth rate of output and growth rate of capital) relationship and how they must be equal in steady state (balanced growth path)
PG 9 my own graph
Y axis gY
X axis gK
How to find growth rate of output/income per worker, on BGP? (gY/L) bgp
expand it twice
pg10
b) key result: is growth in income per worker positive or negative on balanced growth path?
gY/L bgp = gY bgp - gL bgp
sub in our gY bgp and gLbgp (which is just n!)
(1-a-β-y)(n+g) - βb /1-a - n
Which can simplify to
(1-a-β-y)g - βb -(β+y)n /1-a
b) this shows us growt in income per worker on BGP can be positive OR negative (since resources declining and land fixed!)
So gY/L bgp = (1-a-β-y)g - βb - (β+y)n /1-a
What causes negative growth to output per worker, and what about positive?
b) what has happened empircally, positive or negative
negative - resources declining and fixed land are drags on growth
but tech growth spurs growth.
b) positive growth of output per worker: tech progress spur has been larger than the drags on growth
To see how much resource and land effect growth, now assume no resource or land limitations now, and grow at the same rate as population growoth
So what would happen to T.(t) = 0 and R.(t) = -bR(t) now?
now let
T.(t) = n(T)
R.(t) = nR(t)
After doing same steps, what would the growth of output per worker on BGP be? (pg12)
b) how does final answer compare to the gY/L BGP with the previous assumptions T. =0 and declining resources?
final eq
gY/L bgp = (1-a-β-y)g / 1-a
b) its just the first term of the numerator
(1-a-B-Y)g -Bb -(B+y)n / 1-a
see
So the difference is Bb + (B+y)n/ 1-a
(note, now -Bb is positive and not -(β+γ)
What is this known as
the growth drag - the difference in growth between these cases where we change assumptions to resources and land
Growth drag = βb + (β+γ)n/ 1-a
What does growth drag increase with (5)
β - resource share
y - land share
b - rate of depletion of resources
n - rate of population
a - capital share
Nordhaus estimates these parameters
β = 0.1
y = 0.1
a = 0.2
b = 0.005 (economy uses 0.5% of its non-renewable resources)
also let n= 0.01, what is the drag, and how can we use it i.e what does the value mean
0.003125
0.3% : so per capita growth is lower by 0.3% due to fixed land and declining resources.
so overall effect of resources/land isn’t large
So we expect nonrenewable resource price to be high:
why may this not be the case
Elasticity of substitution
if easy to swap energy for capital or labour, demand for energy falls, and so price may not change/even fall.
Graphs show energy use stop rising, showing people found ways to reduce energy use. Suggesting high elasticity of substitution >1
