Capital and Investment
2009
NIPA
• Recall income-expenditure identity  ≡  +  +  + 
• Saving is defined generally as income less expenditure on current needs
(consumption)
• So, if  represents government consumption, then domestic saving is
 ≡ − −
• Combining this identity with IE identity yields
 ≡  + 
• In the model economy studied so far, we abstracted from investment; so
that domestic saving could only take the form of net exports
• In reality, however, domestic saving may also be used to finance domestic investment spending (contributions to the domestic physical stock of
capital)
• Basic question: what determines the demand for domestic investment?
• We begin by assuming a small open economy (exogenous  )
• Later, we will consider a theory of the real interest rate (endogenous  )
The Aggregate Production Function
• Assume that output  is produced with two primary factors, capital ()
and labor (); i.e.,
 =  ( )
where  is TFP (total factor productivity) and  is a function
• Assume that  is increasing and concave in  and ; and that  is
linearly homogeneous in ( )
• Linear homogeneity implies  (  ) =  ( ) for any   0
(constant returns to scale or CRS)
• In simple terms: doubling all factor inputs results in a doubling of output
• Let  = 1; so CRS implies
 ( ) =  ( )
µ
¶
 ( )



1 =
=



• Output per worker  ≡ ( ) will depend on capital-labor ratio  ≡
()
• Define  () ≡  ( 1); so we can write
 =  ()
Example: Cobb-Douglas Production Function
 =  ( ) =   1−; where  ∈ [0 1]
• Note: in earlier chapter, I (implicitly) used this function and assumed
 = 0; i.e.,
 = 
• Sometimes, it is useful to consider the opposite extreme, so that  = 
• But more generally,  =   1−; so
µ ¶


=
=


Capital Accumulation
• Bathtub equation is
+1 =  −  + 
where  is depreciation (capital consumption) and  is gross investment
• Net investment is given by
+1 −  =  − 
• For no good reason, it is frequently assumed that capital depreciates at a
constant rate; i.e.,  =  for some 0 ≤  ≤ 1 (exogenous)
• If this is the case, then
+1 = (1 − ) + 
• In per worker terms,


+1 +1
= (1 − )
+
 +1
 
µ
¶
1
+1 =
[(1 − ) + ]
1+
where  is employment growth rate and  ≡ 
• So, if we ignore  (for the moment), the path of future capital (+1)
depends on the path of current investment () — what determines ?
Investment Demand
• Assume that employment is fixed at some level; i.e.,  =  (so  = 0);
this implies +1 = (1 − ) + 
• Initial capital stock 1 is exogenous; thus 1 ≡ 1 is exogenous
• Since 1 is exogenous, this implies current (per capita) GDP 1 = 1 (1)
is exogenous
• Assume 2-period model; so
1 = 1 (1)
2 = 2 (2) = 2 ((1 − )1 + 1)
• So, investment demand today 1 same as desired capital stock tomorrow
2
• Assume (for simplicity only) that capital depreciates fully; so +1 = 
• Let 1 denote expected present value of domestic business sector;
  (1)
1 = 1 − 1 + 2

• where  is gross real interest rate
• Note: future value is
2 = (1 − 1) + 2 (1)
• Question: given  1 and 2 which value of 1 maximizes 1 (or 2)?
• Answer: 1 determined by
2 0(1 ) = 
• Prediction: domestic investment demand depends positively on “good
news” (2) and negatively on real interest rate (); moreover, does not
depend on 1 (why?)
• Intuition: think of equation above as a “no-arbitrage-condition” (NAC)
— LHS is return on domestic investment; RHS is return on bond
— to eliminate arbitrage (riskless profit), LHS = RHS
Saving, Investment, and the Trade Balance
• Now, imagine that domestic agents own the domestic business sector
• Business sector chooses 1 such that 2 0(1 ) = ; which generates
wealth 1 = 1 − 1 + −12 0(1 ) for household sector
• Question: given this wealth, how should households formulate their consumption/saving plan? Easy (we’ve already done it)...
½
2
max  (1 2) : 1 +
≤ 1
12

¾
• Solution is characterized in the usual way
 (1  2 ) = 


1 + 2 = 1

• Desired domestic saving is then recovered by 1 = 1 − 1
• Trade balance is given by 1 = 1 − 1 ; where 1 denotes net claims
against foreigners (can be positive or negative)
• Several possibilities arise, depending on exogenous parameters (1 1 2    )
• Exercise: list all the endogenous variables
Business Cycle Facts (Mendoza, AER 1991)
[F1] Domestic saving does not fluctuate as much as domestic investment:
()  ()
[F2] Trade balance is countercyclical: ( )  0
[F3] Positive correlation between domestic saving and investment: ( ) 
0
• These facts consistent with our neoclassical model if productivity shocks
are persistent
Saving and Investment in a Closed Economy
• To this point, we have assumed a small open economy with exogenous 
• In a closed economy (e.g., the world economy),  cannot be exogenous
— more precisely, individuals will regard  as exogenous when formulating
their choice problems
— but  will ultimately be determined endogenously by market-clearing
conditions
• In particular, note that 1 = 0 in a closed economy (so that 1 = 1 );
the real interest rate will adjust to ensure that this condition holds
The Neoclassical Model
• In our neoclassical model, ∗ is determined by
1 (∗ 1 2∗) = 1 (∗ 2)
where 1 = 1 (1) and 2∗ = 2 (1 (∗ 2))
• In this version, I assumed 1 exogenous. But now, drawing on an earlier
chapter, imagine that 1 is determined, in part, by current period employment
• What determines current period employment? Usual optimality conditions.
• None of the basic conclusions are changed
A New-Keynesian Model
• Assume 1 is endogenous (determined by current employment)
• But unlike neoclassical view, imagine that employment is not determined
by usual optimality conditions (owing to some “friction,” which we need
not get into here)
• To simplify matters, assume that desired saving does not depend on future
income; hence we can write
1 ( 1) = 1 ( 2)
• This constitutes one equation in the two unknowns ( 1); the combinations of ( 1) that satisfy the restriction above is called an “IS Curve”
• In fact, one can interpret the IS Curve 1( 2) as an “aggregate demand”
function
• In the neoclassical model, 1 is determined by optimality conditions; with
1 so determined, interest rate is determined by
1 = 1(∗ 2)
• In New-Keynesian model, assume instead that  is determined by government policy (e.g., monetary policy)
• So “equilibrium” output is determined by
1∗ = 1( 2)
• That is, real GDP is “demand determined”
• Note how fluctuations in 2 (news about the future) can induce changes
in 1∗
• If these “aggregate demand shocks” are viewed as a “bad” thing, then
monetary policy may, in principle try to change  to stabilize the cycle
• In fact, this basic idea is a part of current “conventional wisdom” in central
banking circles