advanced macroeconomics ii - isaac baley

Advanced Macroeconomics II

Lecture 2

Investment: Frictionless and Convex Adjustment Costs

Isaac Baley

UPF & Barcelona GSE

January 11, 2016

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Motivation

• Investment increases productive capacity of the economy

⇒ key to determine standards of living in the long-run.

• Investment is highly volatile

⇒ key to understand short run (business cycle) fluctuations.

• Investment depends on real interest rates

⇒ key to understand impact of monetary policy

• Investment is a channel through which many fiscal instruments act

⇒ key to understand impact of fiscal policy

• Some facts...

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Facts about investment

• Fact 1: Aggregate investment is relatively volatile.

• Fact 2: High correlation of aggregate investment with output.

I HP-detrended, quarterly time series from US during 1954 –1991.

Variable St Dev (%) Correlation with GNP

GNP 1.72 1

Consumption (non-durables) 0.86 0.77

Investment (gross private domestic) 8.24 0.91

Hours Worked 1.59 0.86

Cooley and Prescott (1996)

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• Fact 1: Aggregate investment is relatively volatile.

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• Fact 2: High correlation of aggregate investment with output.

I Investment gradually increases and decreases along the business cycle.

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• Fact 3: Investment to Capital Ratio is relatively stable

I Law of motion: Kt+1 = (1− δ)Kt + ItI In steady state, Kt+1 = Kt =⇒ It

Kt= δ.

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• Fact 4: Investment at the plant level is “lumpy”.

• Lumpiness: Spikes (infrequent and large changes) and Inaction

• Doms and Dunne (1998), Cooper and Haltiwanger (2005)

I Longitudinal Research Database, 7000 US plants during 1972-1988.

I 18% of plants report investment rates of 20% (relative to capital).

I 50% of plants experience a 1 year capital adjustment of ≥ 37%.

I 80% of plants in a given year change their net capital stock <10%.

• Becker et al (2006)

I Between 9 and 28% of plants have exactly zero investment in a year.

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• Fact 4: Investment at the plant level is “lumpy”.

Source: Cooper and Haltiwanger (2005), On the Nature of Capital Adjustment Costs, REStud.8 / 45

Roadmap

1 Frictionless investment

I Rental model

I Ownership model and user cost of capital

I Some empirical evidence and a quick fix

2 Convex adjustment costs

- Tobin’s q theory

- Quadratic adjustment cost (microfounds q theory)

- Empirical Evidence

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Frictionless investment

• Start from model without adjustment costs.

• First, we assume that firms rent capital every period from households.

I Static problem.

I No cost to change the level of capital they rent.

I Same as in the Solow and Ramsey models.

• Second, we assume that the representative firm owns capital.

I Dynamic problem: depreciation and future production.

I No costs to change their investment (capital they purchase).

• Note: Without adjustment costs, who owns the capital does not matter.

• In both models: marginal benefit︸︷︷︸marginal productivity of capital

= marginal cost︸︷︷︸rental rate or user cost

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Frictionless investment: Rental model

• A firm rents capital Kt every period to produce.

• Suppose we can write profits, after optimizing over other inputs, as

Π(Kt , xt), where xt are other inputs’ costs.

• Let rK the rental cost of a unit of capital, then the firm solves:

maxKt

Π(Kt , xt)− rKKt

• The first order condition (FOC) for the demand of capital is:

ΠK (Kt , xt) = rK (1)

• If profit function exhibits decreasing returns to capital and the usual

Inada conditions, then LHS is decreasing in K and RHS is constant ⇒unique K ∗ that solves (1).

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Frictionless investment: Ownership model (1)

• A (risk neutral) representative firm owns capital Kt , rents labour Lt ,

produces output Yt and maximizes profits Πt .

• Let V0 be the expected discounted value of profits for the firm.

V0 = max{Kt ,Lt}∞t=0

E0

[ ∞∑t=0

(1

R

)t

Πt

](2)

where period profits and investment are given by

Πt = Yt − pt It − wLt

It = Kt − (1− δ)Kt−1

I pt = price of the capital goods.

I w = wage (constant, partial eq.)

I R = 1 + r = gross risk free rate, assumed constant (partial eq.).

I δ = depreciation rate.

• Think: Why does the firm discount with R? How would the discount change

if firm was owned by a household? 12 / 45


• The production function F transforms inputs (Kt , Lt) into output Yt :

Yt = F (Kt , Lt)

where : FK > 0, FL > 0, FKK < 0, FLL < 0, FLK > 0

*Note 1: Capital is productive in the same period it is purchased. It can be

modelled also with a time to build, where Yt = F (Kt−1, Lt).

*Note 2: In partial equilibrium, no need to impose constant returns to scale.

• We restate the problem net of the flexible factors (only labor in this case):

L∗t (Kt ,w , pt) = arg max F (Kt , L∗t )− pt It − wL∗t

and substitute into the value function:

V0 = max{Kt}∞t=0

∞∑t=0

(1

R

)t

E0 [F (Kt , L∗t )− pt It − wL∗t ] (3)

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• Substituting the expression for investment It , the problem becomes:

V0 = max{Kt}∞t=0

∞∑t=0

(1

R

)t

E0

F (Kt , L∗t )− pt

It︷︸︸︷[Kt − (1− δ)Kt−1]− wL∗t

(4)

• Optimality: First order condition for capital at a generic time t is:

Fk (Kt , L∗t )− pt +

1

R(1− δ)Et [pt+1] = 0

• Rearrange to express as:

pt = Fk (Kt , L∗t ) +

1

R(1− δ)Et [pt+1] (5)

I Pay pt today (units of output)

I Produce today Fk (units of output)

I Future resale value of undepreciated capital (1− δ)Et [pt+1]

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• Iterate on (5), use the law of iterated expectations and a transversality

condition to express the price of capital as discounted marginal product:

pt =∞∑j=0

(1− δR

)j

E[Fk

(Kt+j , L

∗t+j

)]

• Comparing FOC to that of rental model, we define the user cost of capital:

UKt ≡ pt −1

R(1− δ)Et [pt+1]

• UKt is an estimate of the rental rate of capital rK .

• UKt increases with riskless rate R, depreciation δ, and current price pt , and

decreases with future price of capital goods E[pt+1].

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• Particular case: Cobb-Douglas production with productivity θt :

Yt = θtKαt L

βt (6)

Then the FOC reads:

αθtKα−1t Lβt = UKt

• Solve for Kt , we get the “desired” level of capital without adustment costs:

K∗t = (Lt)β

1−α

(αθtUKt

) 11−α

(7)

• If labour L is fixed, K∗t follows closely θt and UKt .

K∗t =(L) β

1−α

(αθtUKt

) 11−α

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Frictionless investment: Consequences

• The UK model determines the stock of capital:

K∗t =(L) β

1−α

(αθtUKt

) 11−α

• Fluctuation in “desired” capital K∗t are matched by equal fluctuations in

observed capital Kt .

• Empirical studies: Not rejected as a long run relation.

• But it does not explain short-run fluctuations:

I K∗t does not depend on past or expected future levels of capital.

I K∗t is a jump variable ⇒ Investment adjusts immediately.

I Excessive volatility of It against data (Cooper & Haltinwanger, 2000).

• Need for something that slows down adjustment of capital stock.

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Frictionless investment: A quick fix (1)

• Distinction between net investment I nt (after depreciation) and replacement

investment I rt = δKt−1

• The flexible accelerator model:

I nt = β (K∗t − Kt) , β < 1

Net investment closes the gap between desired and current capital stock.

• Delayed adjustment of Kt to K∗t generates a negative correlation over time

between UKt and net investment rate.

• But such correlation is not strong in the data!

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Frictionless investment: A quick fix (2)

• Low empirical correlation between user cost UKt and investment rate

(equipment spending).

• Better strategy: micro-founded model with convex adjustment costs.

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Roadmap

1 Frictionless investment

I Rental model

I Ownership model and user cost of capital

I Some empirical evidence and a quick fix

2 Convex adjustment costs

I Tobin’s q theory

I Quadratic adjustment cost (microfounds q theory)

I Empirical Evidence

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Tobin’s q

• Define the discounted return of installed capital at t, which assumes no

further investments, as:

W (Kt) ≡∞∑j=0

(1

R

)j

F ((1− δ)jKt)

where the argument of the production function F (·) assumes capital evolves

as Kt+j = (1− δ)jKt since It+j = 0 ∀j > 0.

• For simplicity we do not write explicitly labor and productivity but they do

affect firms production.

• Define average return on installed capital Q as follows:

Qt ≡W (Kt)/Kt

pt

I Note: we must divide by pt to convert units, since the numerator is

measured in terms of output and the denominator in terms of capital.

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Tobin’s q

• Define marginal return on capital q as follows:

qt ≡Et [∂W (Kt)/∂Kt ]

pt=

Et

[∑∞j=0

(1−δR

)jFK ((1− δ)jKt)

]pt

(8)

• Note the derivative FK in the previous expression.

• In other words:

q =Market value of installed capital

Replacement cost of installed capital

• q measures the perpetual return from one marginal unit of installed

capital.

• Tobin’s q theory states that investment is a function of q and r :

I = I (q, r) with∂I

∂q> 0 and

∂I

∂r< 0

• Simple rule: Invest if q > 1.

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Tobin’s q and frictionless model

• Is q > 1 consistent with frictionless model (i.e. Fk = UKt)? No!!

• From the definition of q we have that

qt − 1 =Et [∂W (Kt)/∂Kt ]− pt

pt(9)

where Et [∂W (Kt)/∂Kt ]− pt is the NPV of marginal profits net of the cost

of investment pt .

• From (8) we have that:

Et

[∂W (Kt)

∂Kt

]− pt = Et

[∞∑j=0

(1− δR

)j

FK ((1− δ)jKt)

]− pt

= FK (Kt)− pt +1− δR

Et [FK (Kt+1)]

+

(1− δR

)2

Et [FK (Kt+2)] + . . .

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Tobin’s q and frictionless model

• Adding and subtracting prices, we recover user cost UKt at every period:

Et

[∂W (Kt)

∂Kt

]− pt = FK (Kt)− pt +

1− δR

Et [pt+1]︸︷︷︸UKt

+1− δR

Et

FK (Kt+1)− pt+1 +1− δR

Et [pt+2]︸︷︷︸UKt+1

+

(1− δR

)2

Et

FK (Kt+2)− pt+2 +1− δR

Et [pt+3]︸︷︷︸UKt+2

+ ...

• Define the net marginal profits at time t, denoted πt , as:

πt ≡ FK (Kt)− pt +1

R(1− δ)Et [pt+1] = FK (Kt)− UKt

and rewrite as:

Et

[∂W (Kt)

∂Kt

]− pt =

∞∑j=0

(1− δR

)j

Et [πt+j ] (10)

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Tobin’s q and frictionless model (cont...)

• Summarizing, the expression for q under the frictionless model reads:

qt − 1 =Et

[∂W (Kt )∂Kt

]− pt

pt=

∑∞j=0

(1−δR

)j Et

[FKt+j − UKt+j

]pt

=

∑∞j=0

(1−δR

)j Et [πt+j ]

pt(11)

• But the profit maximizing condition of the frictionless model implies that:

FKt+j = UKt+j for any j ≥ 0 =⇒ πt+j = 0 for any j ≥ 0

• It follows that without adjustment costs qt = 1 always.

• Why? Since investment is frictionless, any situation with qt > 1 triggers an

immediate increase in investment until qt = 1.

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Reviving Tobin’s q

• Tobin’s q theory sounds appealing, and intuitively right.

• Empirically we observe a positive (even though weak) correlation between ItKt

and average Qt , both at the aggregate and at the firm level.

• How can we modify the model to make it compatible with the q − theory?

• Adjustment costs!

I They reduce the volatility of investment, and are consistent with the

realistic feature that the cost of installing capital adds up to the cost of

purchasing it.

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Convex adjustment costs (1): Idea

• Convex adjustment costs: small investments can be easily integrated in the

current structure of the firm, while big investments create larger disruptions.

• Idea: adjustment costs generate positive relation between q and I .

I Suppose that q = 1 and suddenly future net revenues are expected to

increase.

I Et

[∂W (Kt)∂Kt

]increases (one marginal unit of capital installed today

generates more return in the future).

I Investment goes up, but spread over time to minimize adjustment costs.

I Since investments goes up a little today, Et

[∂W (Kt)∂Kt

]falls only little,

and q is still larger than 1.

I Therefore, q and I become positively related.

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Convex adjustment costs (2): Problem

• Formally we write the problem with convex adjustments costs as:

max{It}∞t=0

E0

{ ∞∑t=0

(1

R

)t

[yt − pt It − C (It)]

}

• It is equivalent to write the problem in terms of investment or capital.

• Adjustment costs C (It):

I Convex function of investment (CI > 0,CII > 0)

I Measured in units of output.

I We specialize to quadratic adjustment costs:

C (It) ≡γ

2I 2t , γ > 0

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Convex adjustment costs (3): Quadratic costs

• For simplicity we omit labour (think of a per capita production function):

yt = θtKαt

• Substitute production function and adjustment costs to obtain:

max{It}∞t=0

E0

{ ∞∑t=0

(1

R

)t [θtK

αt − pt It −

γ

2I 2t

]}

• First order condition (with respect to Kt) is:

αθtKα−1t − pt − γIt +

1

R[Et [pt+1] (1− δ) + γ (1− δ)Et [It+1]] = 0

• Solving for It :

It =1

γ(FKt − UKt) +

1

R(1− δ)Et [It+1] (12)

where:

UKt = pt −1− δR

Et [pt+1] and FKt = αθtKα−1t

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Convex adjustment costs (4): Policy

• Iterating (12) forward:

It =1

γ

∞∑j=0

(1− δR

)j

Et

[FKt+j − UKt+j

](13)

• Comparing this to the expression for q in (11) above, which read:

qt − 1 =

∑∞j=0

(1−δR

)j Et

[FKt+j − UKt+j

]pt

we obtain that:

It =1

γ(qt − 1) pt (14)

• Now investment is a positive function of q, as argued by Tobin.

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Convex adjustment costs (5): Intuition

• Intuition for expression (14):

It =1

γ(qt − 1) pt

• Rewrite as:

pt + γIt = qtpt

I Since C (It) = γ2 I

2t , then the marginal cost is C ′ (It) = γIt

I From the definition of qt , we have that: qtpt = Et

[∂W (Kt)∂Kt

]• Therefore, expression (14) implies that optimal investment satisfies that:

pt + C ′ (It)︸︷︷︸Marginal cost of installing one unit of capital

= Et

[∂W (Kt)

∂Kt

]︸︷︷︸

Marginal profits expected from that unit of capital.

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Convex adjustment costs (5): Transversality

• Consider the two equations derived before:

It =1

γ(FKt − UKt) +

1

R(1− δ)Et [It+1]

It =1

γ(qt − 1) pt

• Assume for simplicity that pt = 1 and δ = 0 :

1

γ(qt − 1) =

1

γ

(FKt − 1 +

1

R

)+

1

Rγ(Et [qt+1]− 1)

• Simplifying:

qt = FKt +1

REt [qt+1]

• Substituting recursively forward:

qt =∞∑j=0

1

R jEt

[FKt+j

]+ lim

j→∞

1

R jEt [qt+j+1]

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Convex adjustment costs (5): Transversality

• Maximization requires the following transversality condition:

limj→∞

1

R jEt (qt+j+1) = 0

• This is not an exogenous constraint (e.g. “no Ponzi scheme” condition).

• It is a condition required from profit maximization.

I If this condition is not satisfied, q grows too fast over time.

I This cannot be compatible with profit maximization, it would be

optimal to invest more.

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Lagrange Multiplier Method

• We can also solve the model using the Lagrange multiplier (LM) method.

• Recall the investment equation:

Kt = It + (1− δ)Kt−1

• The Lagrange multiplier is the shadow value of capital:

I What is the increase in firm value V when we relax constraint by 1 unit?

I 1 unit of investment good has a value of p outside the firm and of pq

inside the firm.

I Therefore the LM associated to this constraint will measure exactly pq!

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Lagrange Multiplier Method (cont...)

• Set the Lagrangian, using as multiplier λt = qtpt :

max{It}∞t=0

E0

{ ∞∑t=0

(1

R

)t [θtK

αt − pt It −

γ

2I 2t + qtpt(It + (1− δ)Kt−1 − Kt)

]}

• FOC with respect to It :

−pt − γIt + qtpt = 0

• And we recover back our expression for investment:

It =1

γ(qt − 1) pt

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Implications of the q model

• Investment is a linear function of marginal qt

It =1

γ(qt − 1) pt

where qt is a sufficient statistic: summarizes all relevant information about

the future that is relevant for It

qt = 1 +

∑∞j=0

[1R (1− δ)

]j Et

[FKt+j − UKt+j

]pt

• Implications:

I It is much smoother than FKt − UKt

I It reacts to Et

[FKt+j − UKt+j

]even if FKt − UKt does not change.

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Tobin’s q: Empirical evidence (1)

• Marginal q is the value of one marginal unit of capital in the firm divided by

its purchasing price.

I Problem: It is not observable!

• Average Q is the value of the whole firm divided by the replacement value of

its assets.

I Solution: Average Q can be estimated

Q̂ =Stock market value

Book value

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Tobin’s q: Empirical Evidence (2)

• Hayashi (1982) provides a set of sufficient conditions for q = Q:

1 The firm is price-taker (additional units of output do not imply selling

at a lower price).

2 Production technology and adjust. costs with constant returns to scale.

For example: y(K ) = θK , C (i ,K ) = c(I/K )K

3 Adjustment costs are convex in ratio I/K .

For example: c

(I

K

)=γ

2

(I

K

)2

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• Assume Q = q holds. Model implies that:

I

K= β0 + β1Q + ε, ε ∼ is an observational error

• Obtain an empirical counterpart for Q, called Q̂, usually:

Q̂ = stock market value / book value

• Estimate with OLS the following regression using aggregate data(I

K

)t

= β0 + β1Q̂t + ε̂t

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• Estimate(

IK

)t

= β0 + β1Q̂t + ε̂t with yearly data.

• β > 0, but Qt is not a sufficient statistic for IK

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• Even worse at a quarterly frequency.

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• More and more good firm level data available.

• Estimate a panel regression with firm (or plant) level data:(I

K

)it

= βi0 + β1Q̂it + β2

(Xit

Kit

)+ vit

I Q̂it is an empirical counterpart for Qit

I Individual fixed effects βi0

I Other firm characteristics Xit (should be insignificant)

• Results:

I β1 is positive ⇒ Evidence for q-theory

I β2 not zero ⇒ Qt is not a sufficient statistic for I/K

I β2 > 0 when Xit is cash flow ⇒ evidence of financing constraints

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Why is the q model rejected?

• Possible explanations for empirical rejection:

1 Q̂t is a noisy measure of Qt .

2 Assumptions that make Qt = qt do not hold

• Imperfect competition (Cooper and Ejarque, 2001)

• Decreasing returns to scale

3 Adjustment costs not quadratic

• Model misspecification

4 Finance matters (borrowing constraints, capital market imperfections)

• The literature has explored these possibilities both at the aggregate and at

the firm level and has found strong evidence for all the points above.

• Example, cash flow and acceleration of output predict investment very well.

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Other factors beyond q?

• Cash flow and acceleration of business output explain much better

investment (PDE: producers durable equipment)

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Two main avenues

• The empirical failure of the Q model started two fields of research:

1 Optimal Investment with financing constraints

2 Optimal Investment with non convex adjustment costs

• Plenty of evidence has been found that both factors matter at firm level, but

still unclear wether they matter at the aggregate level.

• Active debate about the Q model, especially in Finance, where cash flow,

credit lines and other measures of liquidity affect investment beyond q

I Abel and Eberly, ReStud (2011), Bolton, Chen, and Wang, JF (2011)

• In Macro, the debate is whether or not we should care about non-convex

adjustment costs when we model aggregate investment.

I Khan and Thomas, Econometrica (2008): we should not care.

I Bachmann, Caballero and Engel, AEJ: Macro (2013): we should care.

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advanced macroeconomics ii - isaac baley

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