2. 2
Capital
• The capital stock of an economy is the
sum total of machines, buildings, and
other reproducible resources in existence
at a point in time
– these assets represent some part of the
economy’s past output that was not
consumed, but was instead set aside for
future production
3. 3
Rate of Return
Time
Consumption
t1 t2 t3
c0
s
At period t1, a decision is made to hold s from
current consumption for one period
s is used to produce additional
output only in period t2
x
Output in period t2 rises by x
Consumption returns to
its long-run level (c0) in
period t3
4. 4
Rate of Return
• The single period rate of return (r1) on
an investment is the extra consumption
provided in period 2 as a fraction of the
consumption forgone in period 1
11
s
x
s
sx
r
5. 5
Rate of Return
Time
Consumption
t1 t2 t3
c0
s
At period t1, a decision is made to hold s from
current consumption for one period
s is used to produce additional
output in all future periods
y
Consumption rises to
c0 + y in all future
periods
6. 6
Rate of Return
• The perpetual rate of return (r) is the
permanent increment to future
consumption expressed as a fraction of
the initial consumption foregone
s
y
r
7. 7
Rate of Return
• When economists speak of the rate of
return to capital accumulation, they have
in mind something between these two
extremes
– a measure of the terms at which
consumption today may be turned into
consumption tomorrow
8. 8
Rate of Return and
Price of Future Goods
• Assume that there are only two periods
• The rate of return between these two
periods (r) is defined to be
1
0
1
c
c
r
• Rewriting, we get
rc
c
1
1
1
0
9. 9
Rate of Return and
Price of Future Goods
• The relative price of future goods (p1) is
the quantity of present goods that must
be foregone to increase future
consumption by one unit
rc
c
p
1
1
1
0
1
10. 10
Demand for Future Goods
• An individual’s utility depends on
present and future consumption
U = U(c0,c1)
and the individual must decide how
much current wealth (w) to devote to
these two goods
• The budget constraint is
w = c0 + p1c1
11. 11
Utility Maximization
c1
c0
w
w/p1
w = c0 + p1c1
U0
U1
The individual will maximize utility by
choosing to consume c0* currently and
c1* in the next period
c0*
c1*
12. 12
Utility Maximization
• The individual consumes c0* in the
present period and chooses to save
w - c0* to consume next period
• This future consumption can be found
from the budget constraint
p1c1* = w - c0*
c1* = (w - c0*)/p1
c1* = (w - c0*)(1 + r)
13. 13
Intertemporal Impatience
• Individuals’ utility-maximizing choices
over time will depend on how they feel
about waiting for future consumption
• Assume that an individual’s utility
function for consumption [U(c)] is the
same for both periods but period 1’s
utility is discounted by a “rate of time
preference” of 1/(1+) (where >0)
14. 14
Intertemporal Impatience
• This means that
)(
1
1
)(),( 1010 cUcUccU
• Maximization of this function subject to
the intertemporal budget constraint
yields the Lagrangian expression
r
c
cwccU
1
),( 1
010L
15. 15
Intertemporal Impatience
• The first-order conditions for a maximum
are
0)(' 0
0
cU
c
L
0
1
)('
1
1
1
1
r
cU
c
L
0
1
1
0
r
c
cw
L
16. 16
Intertemporal Impatience
• Dividing the first and second conditions
and rearranging, we find
)('
1
1
)(' 10 cU
r
cU
• Therefore,
– if r = , c0 = c1
– if r < , c0 > c1
– if r > , c0 < c1
17. 17
Effects of Changes in r
• If r rises (and p1 falls), both income and
substitution effects will cause more c1 to
be demanded
– unless c1 is inferior (unlikely)
• This implies that the demand curve for
c1 will be downward sloping
18. 18
Effects of Changes in r
• The sign of c0/p1 is ambiguous
– the substitution and income effects work in
opposite directions
• Thus, we cannot make an accurate
prediction about how a change in the
rate of return affects current
consumption
19. 19
Supply of Future Goods
• An increase in the relative price of
future goods (p1) will likely induce firms
to produce more of them because the
yield from doing so is now greater
– this means that the supply curve will be
upward sloping
20. 20
Equilibrium Price of
Future Goods
c1
p1
S
D
c1*
p1*
Equilibrium occurs at p1*
and c1*
The required amount of
current goods will be put
into capital accumulation
to produce c1* in the
future
21. 21
Equilibrium Price of
Future Goods
• We expect that p1 < 1
– individuals require some reward for waiting
– capital accumulation is “productive”
• sacrificing one good today will yield more than
one good in the future
22. 22
The Equilibrium Rate of Return
• The price of future goods is
p1* = 1/(1+r)
• Because p1* is assumed to be < 1, the
rate of return (r) will be positive
• p1* and r are equivalent ways of
measuring the terms on which present
goods can be turned into future goods
23. 23
Rate of Return & Real and
Nominal Interest Rates
• Both the rate of return and the real
interest rate refer to the real return that
is available from capital accumulation
• The nominal interest rate (R) is given by
rate)inflationexpected1)(1(1 rR
)1)(1(1
eprR
24. 24
Rate of Return & Real and
Nominal Interest Rates
• Expansion of this equation yields
ee prprR 11
small,isthatAssuming
epr•
eprR
25. 25
The Firm’s Demand for Capital
• In a perfectly competitive market, a firm
will choose to hire that number of
machines for which the MRP is equal to
the market rental rate
26. 26
Determinants of Market
Rental Rates
• Consider a firm that rents machines to
other firms
• The owner faces two types of costs:
– depreciation on the machine
• assumed to be a constant % (d) of the machine’s
market price (p)
– the opportunity cost of the funds tied up in
the machine rather than another investment
• assumed to be the real interest rate (r)
27. 27
Determinants of Market
Rental Rates
• The total costs to the machine owner for
one period are given by
pd + pr = p(r + d)
• If we assume the machine rental market
is perfectly competitive, no long-run
profits can be earned renting machines
– the rental rate per period (v) will be equal to
the costs
v = p(r + d)
28. 28
Nondepreciating Machines
• If a machine does not depreciate, d = 0
and
v/p = r
• An infinitely long-lived machine is
equivalent to a perpetual bond and must
yield the market rate of return
29. 29
Ownership of Machines
• Firms commonly own the machines they
use
• A firm uses capital services to produce
output
– these services are a flow magnitude
• It is often assumed that the flow of
capital services is proportional to the
stock of machines
30. 30
Ownership of Machines
• A profit-maximizing firm facing a
perfectly competitive rental market for
capital will hire additional capital up to
the point at which the MRPk is equal to
v
– under perfect competition, v will reflect
both depreciation costs and the opportunity
costs of alternative investments
MRPk = v = p(r+d)
31. 31
Theory of Investment
• If a firm decides it needs more capital
services that it currently has, it has two
options:
– hire more machines in the rental market
– purchase new machinery
• called investment
32. 32
Present Discounted Value
• When a firm buys a machine, it is buying
a stream of net revenues in future
periods
– it must compute the present discounted
value of this stream
• Consider a firm that is considering the
purchase of a machine that is expected
to last n years
– it will provide the owner monetary returns in
each of the n years
33. 33
Present Discounted Value
• The present discounted value (PDV) of
the net revenue flow from the machine to
the owner is given by
n
n
r
R
r
R
r
R
PDV
)1(
...
)1(1 2
21
• If the PDV exceeds the price of the
machine, the firm should purchase the
machine
34. 34
Present Discounted Value
• In a competitive market, the only
equilibrium that can prevail is that in
which the price is equal to the PDV of the
net revenues from the machine
• Thus, market equilibrium requires that
n
n
r
R
r
R
r
R
PDVP
)1(
...
)1(1 2
21
35. 35
Simple Case
• Suppose that machines are infinitely
long-lived and the MRP (Ri) is the same
in every year
• Ri = v in a competitive market
• Therefore, the PDV from machine
ownership is
...
)1(
...
)1(1 2
n
r
v
r
v
r
v
PDV
36. 36
Simple Case
• This reduces to
...
)1(
1
...
)1(
1
1
1
2 n
rrr
vPDV
1
1
r
r
vPDV
r
vPDV
1
38. 38
General Case
• We can generate similar results for the
more general case in which the rental
rate on machines is not constant over
time and in which there is some
depreciation
• Suppose that the rental rate for a new
machine at any time s is given by v(s)
• The machine depreciates at a rate of d
39. 39
General Case
• The net rental rate of the machine will
decline over time
• In year s the net rental rate of an old
machine bought in a previous year (t)
would be
v(s)e -d(s-t)
40. 40
General Case
• If the firm is considering the purchase of
the machine when it is new in year t, it
should discount all of these net rental
amounts back to that date
• The present value of the net rental in
year s discounted back to year t is
e -r(s-t)v(s)e -d(s-t) = e(r+d)tv(s)e -(r+d)s
41. 41
General Case
• The present discounted value of a
machine bought in year t is therefore the
sum (integral) of these present values
dsesvetPDV sdr
t
tdr )()(
)()(
• In equilibrium, the price of the machine at
time t [p(t)] will be equal to this present
value
dsesvetp sdr
t
tdr )()(
)()(
42. 42
General Case
• Rewriting, we get
• Differentiating with respect to t yields:
t
sdrtdr
dsesvetp )()(
)()(
t
tdrtdrsdrtdr
etvedsesvedr
dt
tdp )()()()(
)()()(
)(
)()()(
)(
tvtpdr
dt
tdp
43. 43
General Case
• This means that
dt
tdp
tpdrtv
)(
)()()(
• dp(t)/dt represents the capital gains that
accrue to the owner of the machine
44. 44
Cutting Down a Tree
• Consider the case of a forester who
must decide when to cut down a tree
• Suppose that the value of the tree at
any time t is given by f(t) [where f’(t)>0
and f’’(t)<0] and that l dollars were
invested initially as payments to workers
who planted the tree
45. 45
Cutting Down a Tree
• When the tree is planted, the present
discounted value of the owner’s profits is
PDV(t) = e-rtf(t) - l
• The forester’s decision consists of
choosing the harvest date, t, to maximize
this value
0)()('
)(
tfretfe
dt
tdPDV rtrt
46. 46
Cutting Down a Tree
• Dividing both sides by e-rt,
f’(t) – r f(t)=0
• Therefore,
)(
)('
tf
tf
r
• Note that l drops out (sunk cost)
• The tree should be harvested when r is
equal to the proportional growth rate of
the tree
47. 47
Cutting Down a Tree
• Suppose that trees grow according to the
equation
t
etf 4.0
)(
• If r = 0.04, then t* = 25
• If r rises to 0.05, then t* falls to 16
ttf
tf 2.0
)(
)('
48. 48
Optimal Resource
Allocation Over Time
• Two variables are of primary interest for
the problem of allocating resources over
time
– the stock being allocated (k)
• the capital stock
– a control variable (c) being used affect
increases or decreases in k
• the savings rate or total net investment
49. 49
Optimal Resource
Allocation Over Time
• Choices of k and c will yield benefits
over time to the economic agents
involved
– these will be denoted U(k,c,t)
• The agents goal is to maximize
T
dttckU
0
),,(
where T is the decision time period
50. 50
Optimal Resource
Allocation Over Time
• There are two types of constraints in
this problem
– the rules by which k changes over time
),,( tckfk
dt
dk
– initial and terminal values for k
k(0) = k0
k(T) = kT
51. 51
Optimal Resource
Allocation Over Time
• To find a solution, we will convert this
dynamic problem into a single-period
problem and then show how the solution
for any arbitrary point in time solves the
dynamic problem as well
– we will introduce a Lagrangian multiplier (t)
• the marginal change in future benefits brought
about by a one-unit change in k
• the marginal value of k at the current time t
52. 52
A Mathematical Development
• The total value of the stock of k at any
time t is given by (t)k
• The rate of change in this variable is
kk
dt
d
K
dt
dk
dt
ktd )(
• The total net value of utility at any time
is given by
kktckUH ),,(
53. 53
A Mathematical Development
• The first-order condition for choosing c
to maximize H is
0
),,(
c
k
c
tckU
c
H
• Rewriting, we get
0
c
k
c
U
54. 54
A Mathematical Development
• For c to be optimally chosen:
– the marginal increase in U from increasing
c is exactly balanced by any effect such an
increase has on decreasing the change in
the stock of k
55. 55
A Mathematical Development
• Now we want to see how the marginal
valuation of k changes over time
– need to ask what level of k would maximize
H
• Differentiating H with respect to k:
0
),,(
k
k
k
tckU
k
H
56. 56
A Mathematical Development
• Rewriting, we get
k
k
k
U
• Any decline in the marginal valuation of
k must equal the net productivity of k in
either increasing U or increasing
k
57. 57
A Mathematical Development
• Bringing together the two optimal
conditions, we have
0
c
k
c
U
c
H
0
k
k
k
U
k
H
• These show how c and should evolve
over time to keep k on its optimal path
58. 58
Exhaustible Resources
• Suppose the inverse demand function
for a resource is
p = p(c)
where p is the market price and c is the
total quantity consumed during a period
• The total utility from consumption is
c
dxxpcU
0
)()(
59. 59
Exhaustible Resources
• If the rate of time preference is r, the
optimal pattern of resource usage will
be the one that maximizes
T
rt
dtcUe
0
)(
60. 60
Exhaustible Resources
• The constraints in this problem are of
two types:
– the stock is reduced each period by the
level of consumption
ck
– end point constraints
k(0) = k0
k(T) = kT
61. 61
Exhaustible Resources
• Setting up the Hamiltonian
kcUekkUeH rtrt
)()(
yields these first-order conditions for a
maximum
0
c
U
e
c
H rt
0
k
H
62. 62
Exhaustible Resources
• Since U/c = p(c),
e-rtp(c) =
• The path for c should be chosen so that
the market price rises at the rate r per
period
63. 63
Important Points to Note:
• Capital accumulation represents the
sacrifice of present for future
consumption
– the rate of return measures the terms at
which this trade can be accomplished
64. 64
Important Points to Note:
• The rate of return is established
through mechanisms much like those
that establish any equilibrium price
– the equilibrium rate of return will be
positive, reflecting both individuals’
relative preferences for present over
future goods and the positive physical
productivity of capital accumulation
65. 65
Important Points to Note:
• The rate of return (or real interest
rate) is an important element in the
overall costs associated with capital
ownership
– it is an important determinant of the
market rental rate on capital (v)
66. 66
Important Points to Note:
• Future returns on capital investments
must be discounted at the prevailing
real interest rate
– use of present value provides an
alternative way to study a firm’s
investment decisions
67. 67
Important Points to Note:
• Capital accumulation (and other
dynamic problems) can be studied
using the techniques of optimal
control theory
– these models often yield competitive-
type results
