where $ℸ = (1 − δ)$ is the amount of capital left after one period of depreciation at rate $δ$ .¹ $et$ is the value of the proﬁt-maximizing ﬁrm: If capital markets are eﬃcient this is the equity value that the ﬁrm would command if somebody wanted to buy it.

∑∞ e (k ) = max βn (f − i − j ) t t {i}∞t t+n t+n t+n n=0 [ ] ∑∞ = max f − i − j(i ,k ) + β max βn (f − i − j ) {it} t t t t {i}∞t+1 t+1+n t+1+n t+1+n n=0 = max ft − it − j(it,kt) + βet+1 ((kt + it)ℸ ) {it}

Deﬁne $ji t$ as the derivative of adjustment costs with respect to the level of investment.

0 = − 1 − ji+ ℸ βek (kt+1 ) t t+1 1 + jit = ℸ βekt+1 (kt+1 )

(3)

In words: The marginal cost of an additional unit of investment (the LHS) should be equal to the discounted marginal value of the resulting extra capital (the RHS).

The Envelope theorem says

--- ℸ --- ◜( ◞ ◟ )◝ ek(k ) = fk(k ) − jk + βek (k ) ∂kt+1-- t t t t t+1 t+1 ∂k t = fk(kt) − jkt + β ℸekt+1 (kt+1) ◟-----◝◜-----◞ =(1+jit) from (3)

So the corresponding $t + 1$ equation can be substituted into (3) to obtain

( ) (1 + jit) = fk(kt+1 ) + (1 + jit+1 − jkt+1) ℸ β

(4)

which is the Euler equation for investment.

Now suppose that a steady state exists in which the capital stock is at its optimal level and is not adjusting, so costs of adjustment are zero: $i i k k jt = jt+1 = jt = jt+1 = jt = jt+1 = 0$ .

If $ji = ji = jk t t+1 t+1$ then (4) reduces to

=R −1 ◜◞◟◝ [ k ˇ ] 1 = β ℸ f (k) + 1 R = ℸ (fk(ˇk))

(5)

so that the capital stock is equal to the value that causes its marginal product to match the interest factor, after compensating for depreciation.

Another way to analyze this problem is in terms of the marginal value of capital, $λt ≡ ek(kt). t$

Rewrite (4) as

k k λt = f (kt) − jt + β ℸ (λt + λt+1 − λt ) k k = f (kt) − jt + β ℸ (λt + Δ λt+1 ) k k (1 − β ℸ )λt = f (kt) − jt + Δ λt+1 fk(k ) − jk + Δ λ λt = -----t-----t-------t+1- (1 − β ℸ)

(6)

and the phase diagram is constructed using the $Δ λt+1 = 0$ locus. In the vicinity of the steady state, we can assume $jk ≈ 0 t$ in which case the $Δ λt+1 = 0$ locus becomes

k --f-(kt)--- λt = (1 − β ℸ)

(7)

which implies (since $fk(k ) t$ is downward sloping in $k t$ ) that the $Δ λt = 0$ locus (that is, the $λt(kt )$ function that corresponds to $Δ λ = 0 t$ ) is downward sloping.

The phase diagram is depicted below.

The steady state of the model will be the point at which $ˇ kt+1 = kt = k$ , implying from (2) a steady-state investment rate of

ˇk = (ˇk + ˇi)ℸ ˇi = (1 − ℸ )ˇk∕ ℸ = (δ ∕ℸ )ˇk

(8)

and solving (5) for $fk(ˇk )$

( ) (1-−--β-ℸ-) k ˇ = f (k) β ℸ

(9)

which can be substituted into (7) to obtain the steady-state value of $λ$ :

( ) ˇ R- λ = ℸ .

(10)

We now wish to modify the problem in two ways. First, we have been assuming that the ﬁrm has only physical capital, and no ﬁnancial assets. Second, we have been assuming that the manager running the ﬁrm only cares about the PDV of proﬁts; suppose instead we want to assume that the ﬁrm is a small business run by an entrepreneur who must live oﬀ the dividends of the ﬁrm, and thus they are maximizing the discounted sum of utility from dividends $u (ct)$ rather than just the level of discounted proﬁts. (Note that we designate dividends by $c t$ ; dividends were not explicitly chosen in the $ϙ$ -model version of the problem, because the Modigliani-Miller theorem says that the ﬁrm’s value is unaﬀected by its dividend policy).

We call the maximizer running this ﬁrm the ‘entrepreneur.’ The entrepreneur’s level of monetary assets $mt$ evolves according to

That is, next period the ﬁrm’s money is next period’s proﬁts plus the return factor on the money at the beginning of this period, minus this period’s investment and associated adjustment costs, minus dividends paid out (which, having been paid out, are no longer part of the ﬁrm’s money).

Value is simply the discounted sum of utility from future dividends:

∑∞ v (k ,m ) = max βnu (c ) t t t {i,c}∞t t+n n(=0 ) ∑∞ = max u (c) + β βnu (c ) {i,c}∞t t t+1+n n=0 = max u (ct) + βvt+1 (kt+1, mt+1 ). {it,ct}

Assume that $f$ and $j$ do not depend directly on $mt$ . That is, their partial derivatives with respect to $mt$ are zero.

FOC wrt $ct$ :

′ m u (ct) = R βv t+1

(12)

Envelope wrt $mt$ :

vmt = Rβvmt+1

(13)

and combining the FOC with the Envelope theorem we get the usual

m m vt = Rβv t+1 = u′(ct) = Rβu ′(ct+1 ) ′ = u (ct+1)

where the last line follows because we have assumed $Rβ = 1$ .

This holds because maximizing with respect to $mt+1$ (subject to the accumulation equation) is equivalent to maximizing with respect to the components of $mt+1$ .

This holds because the derivative of the RHS of (14) with respect to $it$ is

( ( ∂f ∂k ) ∂i ∂j ) ( ∂k ) u ′(ct) ---t+1----t+1-- ∕R − ---t− ---t + β ---t+1- vk (kt+1, mt+1 ) ∂kt+1 ∂it ∂it ∂it ∂it t+1

(15)

(remember that $mt+1$ is a control variable and thus its derivative with respect to investment is zero) so the FOC translates to

u ′(ct)(fkt+1ℸ ∕R − 1 − jit) + β ℸvkt+1 = 0

(16)

which reduces to (14).

This can be seen by directly taking the derivative of the RHS of (14) with respect to $k t$ :

( ( ) ) ( ) ∂ft+1 ∂kt+1 ∂jt ∂kt+1 u ′(ct) -------------- ∕R − ---- + β ------- vkt+1 ∂kt+1 ∂kt ∂kt ∂kt

(18)

and noting that the Envelope theorem tells us the derivatives with respect to the controls $mt+1$ and $it$ are zero while $∂kt+1 ∕ ∂kt = ℸ$ .

To see this, start with the Envelope theorem,

=u ′(c)((1+ji)− f k ℸ∕R) from (14 ) t t◜ t+◞1◟ -◝ k ′ k k k vt = u (ct)(ft+1ℸ ∕R − jt) + ℸ βv t+1 = u′(c )(f k ℸ ∕R − jk) + u ′(c )((1 + ji) − f k ℸ ∕R ) t ( t+1 ) t t t t+1 = u′(ct) 1 + ji− jk t t

(20)

which means that we can rewrite (14) substituting the rolled-forward version of (20)

′ i k k u (ct)((1 + jt) − f t+1 ℸ ∕R ) = ℸ βv t+1 ′ ( i k ) = ℸ βu[ (ct+1) 1 + jt+1 − jt+1 ] (1 + ji) = ℸ β fk(k ) + (1 + ji − jk ) t t+1 t+1 t+1

where the last line follows because with $R β = 1$ we know that $c = c t+1 t$ implying $u ′(c ) = u ′(c ) t+1 t$ .

Since behavior (for either a ﬁrm manager or a consumer) is determined by Euler equations, and the Euler equations for both consumption and investment are identical in this model to the Euler equations for the standard models, there is no observable consequence for investment of the fact that the ﬁrm is being run by a utility maximizer, and there is no observable consequence for consumption of the fact that the consumer owns a business enterprise with costly capital adjustment.

Now consider a ﬁrm of this kind that happens to have arrived in period $t$ with positive monetary assets $m > 0 t$ and with capital equal to the steady-state target value $kt = ˇk$ .

The consequences for the ﬁrm are depicted below in ﬁgure 2.

Dividends follow a random walk. Thus, there is a one-time downward adjustment to the level of dividends to reﬂect the stolen money. Thereafter dividends are constant, as are monetary assets (which are constant at zero forever).

The theft of the money has no eﬀect on investment or the capital stock, because the ﬁrm’s investment decisions are made on the basis of whether they are proﬁtable and the theft of the money has no eﬀect on the proﬁtability of investments.

Now consider another kind of shock: The ﬁrm’s main building gets hit by a meteor, destroying some of the ﬁrm’s capital stock.

The results are depicted below in 3.

Again, because dividends follow a random walk, what the ﬁrm’s managers do is to assess the eﬀect of the meteor shock on the ﬁrm’s total value and they adjust the level of dividends downward immediately to the sustainable new level of dividends. Thereafter there is no change in the level of dividends.

Investment is more complicated. The ﬁrm’s capital stock is obviously reduced below its steady-state value by the meteor, so there must be a period of high investment expenditures to bring capital back toward its steady state. However, the ﬁrm started out with monetary assets of zero. Therefore the high initial investment expenditures will be paid for by borrowing, driving the ﬁrm’s monetary assets to a permanent negative value (the ﬁrm goes into debt to pay for its rebuilding). Gradually over time the capital stock is rebuilt back to its target level, and investment expenditures return to zero (or the level consistent with replacing depreciated capital).