FIN 30220: Macroeconomic
Analysis
Capital Markets
Recall that production is a function of labor, capital and technology.
Y  F ( A, K , L)
Capital Markets determine investment, which affects the evolution of the
capital stock over time
Annual Depreciation Rate
K '  (1   ) K  I G
Tomorrow’s
capital stock
Remaining
portion of current
capital stock
Purchases of New
Capital
t=0
The US Economy
Capital Stock =
K
Productivity =
A
Output/Income Determined
Y  F ( A, K , L* )
w
p
s
l (NLI )
 w
 
 p
Y  C  I G  G  NX
Production is allocated to
various uses
*
l d ( A, K )
L*
Labor Markets
Determine
employment
t=1
L
K '  (1   ) K  I G
New capital is
added to existing
stock
Capital Stock =
K'
Productivity =
A'
The labor market
in t=1 begins
2010
2011
The US Economy
GDP = $14,500B
Private Nonresidential Fixed
Assets
Private Nonresidential Fixed
Assets
$17,346B
$16,946B
2010
w
p
l s (NLI )
 w
 
 p
Consumption
$10,200B
Investment
$1,800B
Government
$3,000B
Net Exports
-500B$
Total Employment
132M
*
l d ( A, K )
L*
L
K '  K  IG  K 
2010
Total Employment
Current Capital
$16,946
Depreciation
$1,400B (8%)
130M
Gross Investment
$1,800B
Net Investment
$400B
w
p
l s (NLI )
 w
 
 p
*
l d ( A, K )
L*
L
How was our net investment of $400B financed?
w
p
Gross Domestic Product = $14,500B
l s (NLI )
Net Factor Payments ($200B)
Gross National Product = $14,700B
l d ( A, K )
130M
L
We used 130M people
and $16.9T worth of
private capital to produce
$14.5T in output!!
Depreciation/Indirect Taxes ($2,200)
National Income = $12,500B
Financial Markets allocate saving to finance borrowing
Saving
Households: $500B
Business: $800B
Current Account
-$500B
$1,800B
Financial Markets
Government
Deficit
$1,400 B
S  CA  I N  G  T 
Net Investment
$400B
Financial
Markets
Commercial Banks accept
deposits from one group
(savers) and lends those
funds out to others
(borrowers)
r
Investment Banks buy bonds
from one group (borrowers)
and sell those bonds to
others (savers)
Bond Price
Real interest
rate
PB
I  G  T 
S
r*
PB *
I  G  T 
S  I  G  T 
S, I
S
S  I  G  T 
Bonds
Financial
Markets
Suppose that government
runs a large deficit. The
increase in the demand for
loanable funds should
rise...this increases the
interest rate
r
The government borrows
money by selling bonds. The
increased supply of bonds
should lower bond prices
PB
I  G  T 
S
r*
PB *
I  G  T 
S  I  G  T 
S, I
S
S  I  G  T 
Bonds
Alexander Hamilton was appointed by George
Washington as the first Secretary of the Treasury in
1789. The US government has had outstanding
debt securities in global financial markets ever
since.
As of February 2015, total debt of the US government was equal to
$18,141,000,000,000.00
$13,038,000,000,000.00
Debt held by the public
(net debt) measures
outstanding government
securities in financial
markets
$5,103,000,000,000.00
Intergovernmental Debt
(one branch of
government borrowing
from another – not “real”
debt)
US Government Securities can be broadly categorized marketable and
non-marketable
Marketable Debt includes all
Treasuries that can be resold after
their initial purchase. Virtually all
US debt is in marketable
securities
Bills (< 1 year maturity)
Notes (1- 10 year maturity)
Bonds ( > 10 year maturity)
Inflation Indexed Notes/Bonds
Non Marketable Debt includes all
Treasuries securities that can’t be
traded after initial purchase
Savings Bonds
State and Local Government Series
(Slugs)
Rural Electrification Administration
series
Treasury Bills make one payment of principal upon maturity. Consider a 90
Day T-Bill with a face value of $1,000 selling for $997
today
90 days
Pay $997
Receive $1,000
What is your (annualized) return on the bond?
Bond Equivalent Yield
 FV  P  365 
BEY  

 *100
 P  n 
 $1,000  $997  365 


 *100  1.22%
$997

 90 
Discount Yield
 FV  P  360 
DY  

 *100
 FV  n 
 $1,000  $997  360 


 *100  1.20%
$1000

 90 
We could do the same calculation in reverse. Consider a 90 Day T-Bill with a
face value of $1,000
today
Pay Price P Today
90 days
Receive $1,000
Suppose you required a 2% annualized return (Bond Equivalent Yield) . What
would you be willing to pay?
 FV 
P

1 i 
 BEY  n 
i


 100  365 
 2  90 
i

  .0049
 100  365 
 $1,000 
P
  $995.12
 1.0049 
( 995 4/32)
We could do the same calculation in reverse. Consider a 90 Day T-Bill with a
face value of $1,000
today
Pay Price P Today
90 days
Receive $1,000
Suppose you required a 2% annualized return (Discount yield). What would you
be willing to pay?
P  FV 1  i 
 DY  n 
i


 100  360 
 2  90 
i

  .005
 100  360 
P  $1,0001  .005  $995
Bond Prices vs. Bond Yields
today
90 days
Pay Price P Today
Receive $1,000
Note that there is a negative relationship between prices and yields
 $1,000 
P

 1 i 
or
P  $1,0001  i 
As yields go up, prices go
down!
Interest Rates tend to rise during expansions and fall during recessions
9.00
100
90 Day T-Bill: Secondary Market
8.00
99.5
7.00
Price
6.00
99
5.00
98.5
4.00
3.00
98
Yield
2.00
97.5
1.00
0.00
97
Jan-90 Jan-92 Jan-94 Jan-96 Jan-98 Jan-00 Jan-02 Jan-04 Jan-06
Recession
Recession
A yield curve represents the average annual returns for securities of different
maturities.
i4 yr  3.50%
i2 yr  2.5%
i3 yr  3%
i1 yr  2%
Now
Average annual
return on a 1 year
bond purchased
today
1 Year
2 Years
Average annual
return on a 2 year
bond purchased
today
3 Years
4 years
Average annual
return on a 3 year
bond purchased
today
These interest rates are referred to as “spot interest rates”.
Average annual
return on a 4 year
bond purchased
today
Suppose we observe a current set of spot rates
i2
i1
Now
2.5%
2%
1
2
3
year
years
years
Consider two investment strategies
Strategy #1: Invest $1 in a
2 year Bond. Your 2 year
cumulative return is
1.0252  1.05
For these
strategies, to
pay the same
return:
1.021  i1,1   1.0252
4
years
Strategy #2: Invest $1 in a 1 year bond
and then reinvest in a one year bond in
one year. Your 2 year cumulative return
is
1.021  i1,1   ??
1  i
1,1
2

1.025

1.02
 1.03
Forward rates are not observed, but can be calculated given any two spot
rates
i2
i1
Now
2.5%
2%
1
year
2
years
i1,1
3
years
3%
1  i1,1
Return on a
1 year
security
4
years
Purchase date
is 1 year from
today
2
2


1  i2 
1.025


1  i1
1.02
i1,1  3%
 1.03
Given any yield curve, we can calculate an expected path for forward rates:
i4  3.5%
i3  3%
i2  2.5%
i1  2%
Now
1 Year
2 Years
i1,1  3%
2
2


1  i2 
1.025
1  i1,1 

1.02
1  i1
 1.03
3 Years
i1, 2  4%
3
3


1  i3 
1.03
1  i1, 2 

2
1  i2  1.0252
4 years
i1,3  5%
 1.04
4
4


1  i4 
1.035
1  i1,3 

1  i3 3 1.033
 1.05
Alternatively, suppose we know the path of forward rates…
2%
i1, 0
Now
2.5%
i1,1
1
Year
2
Years
3
Years
4 years
Again, consider two investment strategies
Strategy #1: Invest $1 in a 1 year bond
and then reinvest in a one year bond in
one year. Your 2 year cumulative return
is
1.021.025  1.0455
For these
strategies, to
pay the same
return:
1.021.025  1  i2 
2
Strategy #2: Invest $1 in a
2 year Bond. Your 2 year
cumulative return is
1  i2 2  ??
1  i2   1.021.025
1
2
Given a set of forward rates, we can calculate the implied spot rates
i1,1  3%
i1  2%
Now
1 Year
i1, 2  4%
2 Years
i1,3  5%
3 Years
4 years
2%
1.02
2.5%
1.021.03
1
2
 1.025
3%
1.021.031.04
1
3
 1.03
3.5%
1.021.031.041.05
1
4
 1.035
Spot rates are based on the geometric average of expected future rates
What can we learn from the US yield curve?
Suppose that expected future rates were expected to be constant
i1,1  3%
i0,1  3%
Now
1 Year
i1, 2  3%
2 Years
i1,3  3%
3 Years
4 years
3%
1.03
3%
1.031.03
1
2
The yield curve is
flat!
 1.03
3%
1.031.031.03
1
3
 1.03
3%
1.031.031.031.03
1
4
 1.03
An upward sloping yield curve suggests that the market expects interest
rates to rise in the future…with one small problem..
Strategy #1: Invest $1 in a
2 year Bond. Your 2 year
cumulative return is
1.0252  1.05
Strategy #2: Invest $1 in a 1 year bond
and then reinvest in a one year bond in
one year. Your 2 year cumulative return
is
1.021  i1,1   ??
For these strategies, to pay the
same return..
i1,1  3%
Are these two strategies actually equivalent??
Strategy #1 is less flexible and, hence, a bit riskier! Therefore, the
180 day rate has two components: and expected future rate AND a
liquidity premium
Finally, any security with “default risk” will offer a higher rate of return to compensate investors for
the possibility of default.
20.00
18.00
Large Spread
16.00
14.00
12.00
10.00
8.00
6.00
4.00
Small Spreads
2.00
0.00
4/1/53
4/1/61
4/1/69
10 Year Treasury
4/1/77
4/1/85
Aaa Corporate
4/1/93
Baa Corporate
4/1/01
Suppose that you invest $945 in a one year, $1,000 T-Bill. Prices are listed below
as well
Now
Pay $945
1 Year
Receive $1,000
CPI = 100
CPI = 104
 $1,000  $945 
BEY  
 *100  5.82%
$945


Now, lets convert your $1,000 to current prices and redo the yield
 CPI 
 100 
FV  FV 
  $1, 000 
  $962
 CPI ' 
 104 
 $962  $945 
BEY  
 *100  1.79%
$945


This is your inflation adjusted,
or, real return
Recall that nominal (currency) variables are meaningless without some
mention of prices. The same hold for interest rates.
Now
Pay $945
1 Year
Receive $1,000
CPI = 100
CPI = 104
Inflation Rate = 4%
Exact Method
1 i
1 r 
1 
1.0582
1.0175 
1.04
Approximation
r  i 
1.82%  5.82%  4.00%
US Real Returns (1948-2014)
At the time you bought the asset, you did not know what inflation
would be.
Now
Pay $945
1 Year
Receive $1,000
CPI = 100
Your Expectation: CPI = 102
Actual: CPI = 104
Expected
inflation
Ex Ante
3.82%  5.82%  2.00%
r  i  e
Actual inflation
Ex Post
r  i 
1.82%  5.82%  4.00%
We can only measure ex post real interest rates!!
Every interest rate can be broken up into (at least) three
components
Interest
=
Rate
“Base” Rate +
We will explain this
rate
Liquidity
Risk
+
+
Premium
Premium
These are
explained
elsewhere
Inflation
Premium
Households and Capital Markets
From the household’s perspective, capital markets provide an important
service. They allow households to reallocate their wealth across time.
Note that wealth is not the same as income. Suppose that you earn
$50,000 per year (income). You expect to work for 40 years. Your
wealth is defined as the present value if your lifetime income.
$50,000 $50,000
$50,000
W

 .... 
2
40
1  i  1  i 
1  i 
Suppose that the interest rate is 4% (i = .04).
$50,000 $50,000
$50,000
W

 .... 
 $989,638
2
40
1.04 1.04
1.04
Saving and Consumption
C, Y
C Y
$50,000
Without capital
markets,
consumption
equals income at
every point in time
time
C, Y
Savings < 0 (Borrowing)
With capital markets,
total lifetime
consumption equals
total lifetime wealth
C
$50,000
Y
Savings > 0
time
Generally, it’s your income that fluctuates over time. Your goal is to use
capital markets to maintain a constant stream of consumption
Peak earning power
occurs just prior to
retirement
C, Y
C
S>0
S<0
Y
S<0
time
$0
Young
(0 – 30)
Middle Age
(30 – 60)
Old
(60 - ? )
Consider the following example. You currently are earning $80,000, but expect to
earn $20,000 next year. You can borrow and lend at 5% interest. Further,
assume that P = 1 and there is no inflation.
Today, you can either save (S>0) or borrow (S<0)
C  S  $80,000
Current saving influences future consumption
C'  $20,000  1.05S
C'
$20,000
C
 $80,000 
 $99,047
1.05
1.05
PV of Lifetime
Consumption
Wealth
Future
Consumption
C'
$20,000
C
 $80,000 
1.05
1.05
All your wealth
spent next year
C'
Consumption
equals income
104,000
S>0
S<0
20,000
All your wealth
spent this year
Slope = 1.05
C
80,000
99,047
Current
Consumption
What you choose to do depends on your preferences!
Total Utility
(Happiness)
U  U (C , C ')
Current
Consumption
Save $1 today, what does it
cost you?
Save $1 today, what do you
gain next year?
Future
Consumption
Value of current
consumption
The consumption that dollar
could’ve purchased? What’s
that lost consumption worth
to you?
The dollar saved plus the
interest? What’s that extra
consumption worth to you?
MU C
Value of future
consumption
1  r  MUC '
Real return on savings
Recall that maximizing anything requires equating costs and benefits at the
margin
Benefits of saving
1  r  MUC '
Cost of saving
=
MU C
Let’s rewrite this…
Marginal Rate
of Substitution
MU C
 MRS
1  r  
MU C '
Marginal Rate of substitution measures
the value of current consumption in
terms of future consumption
Utility
MU C Current Pizzas
Future Pizzas
MRS 


Utility
MU C '
Current Pizzas
Future Pizzas
C'  20,000  1.0530,000
Future
Consumption
Real Interest
Rate
r
C'
S (Y  80,000, W  99,047)
1  r  MRS
104,000
r  .05
51,500
20,000
50,000
80,000 99,047
C
S
S  30,000
Savings
Savings = 30,000
Current
Consumption
Suppose that the interest rate increases to 8%...
Substitution effect: As
interest rates rise, current
consumption becomes
more expensive – spend
less today! (Save More)
C'
Income Effect: As
interest rates rise,
you earn more
interest income –
spend more today!
(Save Less)
104,000
Real Interest
Rate
r
Income
Effect
Substitution
effect
r  .08
r  .05
51,500
20,000
50,000
80,000 99,047
C
S  20,000 S  30,000 S  40,000
Savings
40,000
60,000
Current
Consumption
S
We typically assume that the substitution effect is dominant…a rise in the real
interest rate increases savings.
C'
C'  20,000  1.0540,000
Real Interest
Rate
r
S (Y  80,000, W  99,047)
r  .08
104,000
r  .05
62,000
20,000
40,000
80,000 99,047
C
S  30,000 S  40,000
Savings
Savings = 40,000
S
Suppose that your current income increases to $100,000…
Future
Consumption
Real Interest
Rate
S (Y  100,000, W  119,047)
r
C'
S (Y  80,000, W  99,047)
104,000
r  .05
62,000
51,500
20,000
50,000 60,000
100,000
C
Savings = 40,000
A rise in current income increases savings
S  30,000
S  40,000
S
Savings
Suppose that your current income stays at $80,000, but your future
income rises to $40,000
Future
Consumption
C'  40,000  1.0520,000
Real Interest
Rate
r
S (Y  80,000, W  118,095)
C'
104,000
r  .05
61,000
S (Y  80,000, W  99,047)
51,500
40,000
20,000
50,000
60,000
80,000
C
S  20,000 S  30,000
Savings = 20,000
A rise in future income lowers savings
S
Savings
Suppose that your current income rises to $100,000, AND your future
income rises to $40,000
C'
C'  40,000  1.0530,000
Real Interest
Rate
r
S (Y  100,000, W  138,095)
104,000
r  .05
71,500
S (Y  80,000, W  99,047)
51,500
40,000
50,000
70,000
100,000
C
S  30,000
Savings = 30,000
A permanent rise in income has no effect on savings
S
Savings
Recall the production function discussed earlier
Y  F ( A, K , L)
Labor
Output
Capital
Productivity
Typically the production function used is Cobb-Douglas
Capital’s Share of
Income
1
3
2
3
Y  AK L
Labor’s Share of
Income
The investment decision is based on changing the capital stock holding
employment fixed. Note that capital can’t be adjusted instantaneously.
Y
MPK=2
F ( A, K , L)
2
The Marginal Product of
Capital (MPK) measures the
change in production
associated with a small
change in the capital stock
10
MPK=10
As capital increases
(given a fixed labor
force), capital
productivity declines!!
K
K'
1
1
K
Suppose that an investment opportunity costs $100. This project will generate $25 in revenues
per year (MPK), but will depreciate at 10% per year. Assume that the interest rate is 5%
What’s the cost of capital?
Borrow $100
Buy Capital (or use
retained earnings)
Capital is worth $90
Interest owed = $5
Install Capital
Now
1 Year
Use capital to produce output
Cost:
.05($100) + .10(100) = $15
Lost interest
Depreciation expense
Pk  r   
Suppose that an investment opportunity costs $100. This project will
generate $25 in revenues (MPK), but will depreciate at 10% per year. The
interest rate is 5%
We will purchase capital as long as the benefits at the margin are greater than
the cost.
(Expected) future
marginal product
of new capital
once installed
MPK  Pk  r   
User cost of
capital
The optimality condition for capital gives us our “target” capital stock. To
find investment, we need to remember that capital evolves according to
Target
Capital
Stock
MPK = UC
K '  1   K  I
Annual
Depreciation
Rate
Current
Capital
Stock
Current
Investment
We need to solve for the level of investment needed to
reach our target capital stock
I  K '1   K
Example:
F ( A, K , L) FK ( A, K , L)
MPK
r  .05
Capital
Output
1
140
2
180
40
PK  $100
3
210
30
K 2
4
230
20
5
245
15
6
250
5
For the production function given
above, at a user cost of 15, 5 units
of capital are needed
  .10
PK r     15
I  5  1  .102  3.2
Investment demand records the by
the firm at every real interest rate
Real
Interest
Rate
r
I  MPK 
Capital
Output
1
140
2
180
40
3
210
30
4
230
20
5
245
15
6
250
5
r  .05
r  .05
  .10
PK  $100
I  3.2
FK ( A, K , L)
F ( A, K , L)
MPK
PK r     15
K 5
K 2
I
Investment
I  5  1  .102  3.2
Changing the interest rate allows us
to sketch out investment demand
F ( A, K , L) FK ( A, K , L)
Real
Interest
Rate
r
r  .10
Capital
Output
1
140
2
180
40
3
210
30
4
230
20
5
245
15
6
250
5
r  .05
r  .10
I  MPK 
I  2.2
I  3.2
  .10
PK  $100
MPK
PK r     20
K 4
K 2
I
Investment
I  4  1  .102  2.2
Changing production values allows
us to sketch out shifts in investment
demand
F ( A' , K , L) FK ( A' , K , L)
Real
Interest
Rate
r
Capital
Output
1
120
2
140
20
3
155
15
4
165
10
5
170
5
6
172
2
r  .05
r  .05
I  MPK 
I  MPK 
I  0.2
I  3.2
I
Investment
  .10
PK  $100
MPK
PK r     15
K 3
K 2
I  3  1  .102  0.2
Anything that raises (lowers) the productivity of capital will increase (decrease)
investment demand
The productivity of capital is
influenced by
Employment (+)
Technology (+)
r
r
I  MPK 
I
I
Finally, we need to find an equilibrium in the capital market – a real interest rate
that equates savings (inflow into financial sector) and Investment (outflow from
financial sector)
r
Y
S Y ,W 
F ( A, K , L)
Y
*
r*
I  MPK 
*
L
Labor Markets
determine current
output (Income)
L
SI
S, I
Given Current Income and the
current capital stock, Capital markets
determine Savings, Investment, and
the real interest rate
We need to make assumptions about the evolution of productivity (and, hence,
income) to know what happens to savings. Let’s suppose that productivity
evolves according to an autoregressive process
At 1   At   t
Persistence parameter
At
Productivity shock
At
   1
At  0    1
At    0
L
Suppose that the economy experiences a temporary increase in productivity
   0
For a given level of
capital and labor, a rise
in productivity raises
output
r
Y
S Y ,W 
F ( A, K , L)
Y
*
Increase in
productivity
r*
I  MPK 
*
L
L
SI
S, I
Suppose that the economy experiences a temporary increase in productivity
r
Y
S Y ,W 
F ( A, K , L)
Y
*
r*
I  MPK 
*
L
L
SI
S, I
With an increase in both the supply of loanable funds (savings) and the demand
for loanable funds (investment), the interest rate change is ambiguous, and the
quantity of both savings and investment increase
An increase in productivity that is perceived to be permanent will have minimal
effect on savings (permanently higher income raises consumption), but
investment increases    1
Y
r
S Y ,W 
F ( A, K , L)
Y*
Increase in
productivity
L*
r*
L
Interest Rates increase
I  MPK 
SI
S, I
Capital Markets and the business cycle
Given the mechanics of capital markets,
what relationships would we expect to see
between savings, investment, interest rates,
and output?
Just the facts ma’am.
Correlation
Output
Savings
+
Consumption
+
Investment
+
Interest Rates
?
GDP vs. Savings (% Deviation from trend)
GDP (% Deviation from Trend)
Savings (% Dev. From trend)
Correlation = .77
GDP vs. Consumption (% Deviation from trend)
GDP/Consumption (% Dev. From trend)
Correlation = .78
GDP vs. Investment (% Deviation from trend)
GDP (% Dev. From Trend)
Investment (% Dev. From Trend)
Correlation = .83
GDP vs. Interest Rates (% Deviation from trend)
Correlation = .42
The high, positive correlation suggests shocks that are more permanant
Example: Oil Price Shocks in the 1970’s
Dollars per Barrel
1979 Iranian
Revolution
(Temporary Shock)
1973 Arab Oil
Embargo
(Permanent Shock)
We can view the rapid rise in the price of oil as a decline in productivity…
S Y ,W 
r
S Y ,W 
r
r*
r*
I  MPK 
I  MPK 
SI
I, S
With a temporary decline, we get a drop
in savings and investment. The new
equilibrium interest rate is ambiguous.
SI
I, S
With a temporary decline, we get a drop
in investment. The new equilibrium has
lower interest rates
r
r
S
I
I
I
Real Interest Rate
Investment (% Dev. From Trend)
I
S
1973 Arab Oil
Embargo
1979 Iranian
Revolution