An Entrepreneur’s Problem Under Perfect Foresight

Consider a firm characterized by the following:

With taxes:^[1]

Suppose that the firm’s goal is to pick the sequence $\iFunc_{t}$ that solves:

With taxes: $\vFirm(\kap_{t}) = \max_{\{\iFunc\}_{t}^{\infty}}~\sum_{n=0}^{\infty} \Discount^{n} \left(\rev_{t+n}-\xpend_{t+n}\right)$

subject to the transition equation for capital,

where $\DeprFac = (1-\depr)$ is the amount of capital left after one period of depreciation at rate $\depr$.^[2] $\vFirm_{t}$ 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.

Define $\adj_{t}^{\inv}$ as the derivative of adjustment costs with respect to the level of investment.

With taxes: $(1+j^{\inv}_{t})\kPriceAfterITC = \DeprFac\Discount \vFirm_{t+1}^{\kap}(\kap_{t+1})$, where the LHS is the after-tax marginal cost of investment.

With taxes: $(1+\adj_{t}^{\inv})\kPriceAfterITC = \DeprFac\Discount \left[ \TaxFree\fFunc^{\kap}(\kap_{t+1})+(1+\adj_{t+1}^{\inv}-\adj_{t+1}^{\kap})\kPriceAfterITC\right]$

1Steady State¶

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: $\adj_{t}=\adj_{t+1}=j^{\inv}_{t}=j^{\inv}_{t+1}=j^{\kap}_{t}=j^{\kap}_{t+1}=0$.

With taxes: $\kPriceAfterITC\Rfree = \DeprFac (\kPriceAfterITC+\TaxFree\fFunc^{\kap}(\check{k}))$, so the capital stock equals the value that causes its after-tax marginal product to match the interest factor.

2Phase Diagram Analysis¶

Another way to analyze this problem is in terms of the marginal value of capital, $\ek_{t} \equiv \vFirm_{t}^{\kap}(\kap_{t})$.

Solution to Exercise 5

Rewrite (8) as

$$\begin{gathered}\begin{aligned} \ek_{t} & = \fFunc^{\kap}(\kap_{t}) - \adj_{t}^{\kap}+\Discount \DeprFac (\ek_{t}+\ek_{t+1}-\ek_{t}) \\ & = \fFunc^{\kap}(\kap_{t}) - \adj_{t}^{\kap}+\Discount \DeprFac (\ek_{t}+\Delta \ek_{t+1}) \\ (1-\Discount\DeprFac) \ek_{t} & = \fFunc^{\kap}(\kap_{t}) - \adj_{t}^{\kap}+\Delta \ek_{t+1} \\ \ek_{t} & = \frac{\fFunc^{\kap}(\kap_{t}) - \adj_{t}^{\kap}+\Delta \ek_{t+1}}{(1-\Discount\DeprFac)} \end{aligned}\end{gathered}$$

(13)

and the phase diagram is constructed using the $\Delta \ek_{t+1} = 0$ locus. In the vicinity of the steady state, we can assume $\adj_{t}^{\kap} \approx 0$ in which case the $\Delta \ek_{t+1}=0$ locus becomes

$$\begin{gathered}\begin{aligned} \ek_{t} & = \frac{\fFunc^{\kap}(\kap_{t})}{(1-\Discount\DeprFac)} \end{aligned}\end{gathered}$$

(14)

which implies (since $\fFunc^{\kap}(\kap_{t})$ is downward sloping in $\kap_{t}$) that the $\Delta \ek_{t}=0$ locus (that is, the $\lambda_{t}(\kap_{t})$ function that corresponds to $\Delta \ek_{t}=0$) is downward sloping.

The phase diagram is depicted in Figure 1.

The steady state of the model will be the point at which $\kap_{t+1}=\kap_{t}=\check{\kap}$, implying from (2) a steady-state investment rate of

and solving (11) for $\fFunc^{\kap}(\check{k})$

which can be substituted into (14) to obtain the steady-state value of $\ek$:

3The Entrepreneur’s Problem¶

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 $\utilFunc(\cRat_t)$ rather than just the level of discounted profits. (Note that we designate dividends by $\cRat_{t}$; dividends were not explicitly chosen in the $\q$-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 $m_{t}$ 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).

With taxes: $m_{t+1} = \rev_{t+1}+\left(m_{t}-\xpend_{t}-\cRat_{t}\right)\Rfree$

Assume that $\fFunc$ and $\jFunc$ do not depend directly on $m_{t}$. That is, their partial derivatives with respect to $m_{t}$ are zero.

4Euler Equation for Dividends¶

5Alternative Formulation¶

6Investment FOC for Entrepreneur¶

With taxes: $\uP(\cRat_{t})(\kPriceAfterITC(1+\adj_{t}^{\inv})-\TaxFree f_{t+1}^{\kap}\DeprFac/\Rfree) = \DeprFac\Discount \vNum_{t+1}^{\kap}$

7Envelope Theorem for Capital¶

8Euler Equation for Investment (Entrepreneur)¶

With taxes: $\kPriceAfterITC (1+\adj_{t}^{\inv}) = \DeprFac\Discount \left[ \TaxFree\fFunc^{\kap}(\kap_{t+1})+\kPriceAfterITC(1+\adj_{t+1}^{\inv}-\adj_{t+1}^{\kap})\right]$

9Observational Equivalence¶

10Impulse Responses: Monetary Shock¶

Now consider a firm of this kind that happens to have arrived in period $t$ with positive monetary assets $m_{t}>0$ and with capital equal to the steady-state target value $\kap_{t}=\check{k}$.

Suppose that a thief steals all the firm’s monetary assets.

Solution to Exercise 13

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.

11Impulse Responses: Capital Shock¶

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

Solution to Exercise 14

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).

12Numerical Solution¶

The solution code uses the following definitions for the production and adjustment cost functions:^[3]

where $\Psi$ is the firm’s productivity and $\labor$ is the labor supplied by the entrepreneur (assumed equal to 1).

The policy functions are obtained using the method of reverse shooting, which is based on recovering for a given $k_{t+1}$ and $\vFirm_{t+1}(k_{t+1})$ the values of $k_t$, $i_t$ and $\vFirm_{t}(k_{t})$ 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

so that

we can rewrite the investment Euler equation as

so that

With the values of $i_{t}$ and $j_{t}$ obtained from these reverse-shooting equations, the reverse-shooting equation for value is very simple: It is the Bellman equation

where note that $f_{t}$ and $i_{t}-j_{t}$ are direct functions of $k_{t}$ and $i_{t}$ which we have already computed.

The steady-state level of capital can be obtained from (11):

while the steady-state value of $\ek$ comes from substituting $\check{\kap}$ into (14). Steady-state value is straightforward to compute, given that in the steady state the capital stock and amount of investment are constant:

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

where the $\check{~}$ identifies variables at the steady state and $\epsilon$ is the deviation from the steady state. $\check{i}^k$ is computed by differentiating the Euler equation with respect to $\kap$:

which at the steady-state where $i^{k}_{\hat{t}}=i^{k}_{t}=\check{i}^{k}$ becomes:

and given any particular set of parameter values $\check{i}^k$ can be found using standard numerical rootfinding methods.

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

because this corresponds to the solution to a perfect foresight consumption problem in which the consumer has monetary resources $m_{t}$ and net nonmonetary financial resources $\vFirm_{t}(k_{t})-f_t$ (see the Consumption Under Perfect Foresight and CRRA Utility).