An Entrepreneur's Problem Under Perfect Foresight

Consider a firm that wants to maximize the present discounted value of profits after subtracting off costs of investment.

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

i k
0 = - P(1 + jt) + ℸ βe t+1(kt+1 ) (3)
i k
(1 + jt)P = ℸ βe t+1(kt+1 ) (4)

In words: The marginal after-tax 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
◜(---◞ ◟--)◝
k k k k ∂kt+1--
et(kt) = /τ f (kt) - Pj t + βe t+1(kt+1 ) ∂k
t
= /τ fk(kt) - Pjkt + βℸekt+1 (kt+1) (5)
◟-----◝◜-----◞
=P (1+jit) from (4)

So the corresponding $t + 1$ equation can be substituted into (4) to obtain

(1 + ji)P = ( τfk(k ) + (1 + ji - jk )P ) ℸβ (6)
t / t+1 t+1 t+1

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: $j=j=ji = ji = jk = jk = 0 tt+1t t+1 t t+1$ .

If $ji=ji = jk tt+1 t+1$ then (6) reduces to

-1
=◜R◞◟ ◝ [ ]
P = β ℸ /τfk(ˇk ) + P (7)

PR = ℸ (P + /τfk( ˇk)) (8)

so that the capital stock is equal to the value that causes its after-tax 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 (5) as

λt = /τfk(kt) - Pjkt + β ℸ (λt + λt+1 - λt) (9)
k k
= /τf (kt) - Pj t + β ℸ (λt + Δ λt+1 ) (10)
k k
(1 - βℸ )λt = /τf (kt) - Pj t + Δ λt+1 (11)
/τfk(k ) - Pjk + Δ λ
λt = ------t-------t-------t+1- (12)
(1 - β ℸ )

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

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

The phase diagram is depicted in figure 1.

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

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 people running the firm only care about the level of profits; suppose instead we want to assume that they 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 $ct$ ; 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:

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

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

FOC wrt $ct$ :

Envelope wrt $mt$ :

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

vm = R βvm
t t+1
= u ′(ct)
′
= R βu (ct+1 )
′
= u (ct+1 )

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

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 (21) with respect to $it$ is

(( ) ) ( )
u′(c) ∂/τ-ft+1-∂kt+1-- ∕R - ∂Pit--- ∂Pjt-- + β ∂kt+1-- vk (k ,m )(23 )
t ∂kt+1 ∂it ∂it ∂it ∂it t+1 t+1 t+1

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

u′(c )(τf k ℸ ∕R - 1 - Pji ) + β ℸvk = 0 (24)
t / t+1 t t+1

which reduces to (22).

This can be seen by directly taking the derivative of the RHS of (21)with respect to $kt$ :

( ( ) ) ( )
′ ∂/τft+1-∂kt+1-- ∂Pjt-- ∂kt+1-- k
u (ct) ∂k ∂k ∕R - ∂k + β ∂k vt+1 (26)
t+1 t t t

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,

′ i k
=u (ct)(P(1+jt)- /τft+1ℸ-∕R) from (22)
k ′ k k ◜ ◞ ◟k ◝
vt= u (ct)(/τf t+1 ℸ ∕R - Pj t ) + ℸβv t+1 (28)
′ k k ′ i k
= u (ct)(/τf( t+1 ℸ ∕R - Pj) t ) + u (ct)(P (1 + j t) - /τf t+1 ℸ ∕R ) (29)
= u′(c )P 1 + ji - jk (30)
t t t

which means that we can rewrite (22) substituting the rolled-forward version of (30)

′ i k k
u(ct)(P (1 + jt) - /τft+1 ℸ ∕R ) = ℸβv t+1
′ ( i k )
= ℸβu[ (ct+1 )P 1 + jt+1 - j t+1 ]
P (1 + ji) = ℸβ /τfk(k ) + P (1 + ji - jk )
t t+1 t+1 t+1

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

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 $mt > 0$ and with capital equal to the steady-state target value $ˇ kt = k$ .

Suppose that an executive steals all the firm’s monetary assets and disappears.

The consequences for the firm are depicted 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 in figure 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).

The solution code uses the following definitions for the production and adjustment cost functions:²

The policy functions are obtained using the method of reverse shooting, which is based on recovering for a given $kt+1$ and $et+1(kt+1 )$ the values of $kt$ , $it$ and $et(kt)$ consistent with the first order conditions and transition equations.

The reverse-shooting equation for capital comes from a combination of the dynamic budget constraint and the Euler equation. For convenience defining

With the values of $it$ and $jt$ obtained from these reverse-shooting equations, the reverse-shooting equation for value is very simple: It is the Bellman equation

The reverse shooting routine starts its backwards iterations from a $kˆt$ level very close to the steady state of the model and, as discussed in the methodological appendix to TractableBufferStock, the accuracy of the solution is improved if we approximate $i ˆt$ with a first order Taylor expansion using the derivative of investment at the steady state:

Finally since the problem is solved under perfect foresight the entrepreneur’s consumption function is simply: