©April 29, 2008, Christopher Carroll BrockMirman

The Brock-Mirman Stochastic Growth Model

Brock and Mirman (1972) provided the first optimizing growth model with unpredictable (stochastic) shocks. This handout presents the basic elements of their model.

The social planner’s goal is to solve the problem:

∞ ∑ s- t max β log Cs s=t
(1)

s.t. α Yt = ZtK t (2) Kt+1 = Yt - Ct (3)
where Zt is the level of productivity in period t. (We describe the assumptions about Zt below). Note the key assumption that the depreciation rate on capital is 100 percent.

The first step is to rewrite the problem in terms of Bellman’s equation and take the first order condition:

V (K ) = max log C + βV (K ) t t {Ct} s t+1 t+1
(4)

such that α Kt+1 = ZtK t - Ct

The first order condition says

′ [ α- 1 ′ ] u (Ct ) = β𝔼t Zt+1 αK t+1 u (Ct+1 ) 1 [ Z αK α - 1] --- = β𝔼t --t+1----t+1-- Ct Ct+1 ⌊ ⌋ | α - 1-Ct--| 1 = β𝔼t ⌈Zt◟+1--α◝K◜-t+1◞ C ⌉ ≡ℜt+1 t+1
where our definition of ℜt+1 helps clarify the relationship of this equation to the usual consumption Euler equation (and you should think about why this is the right definition of the interest factor in this model).

Now we show that this FOC is satisfied by a rule that says Ct = γY t, where γ = 1 - αβ. To see this, note first that the proposed consumption rule implies that Kt+1 = (1 - γ)Y t.

The first order condition says

[ ] 1 = β 𝔼 α Yt+1---γYt--- t Kt+1 γYt+1 [ ] --Yt-- = β 𝔼t α K [ t+1 ] ---Yt---- = β 𝔼t α [ Yt - Ct ] Yt = β 𝔼t α ----------- [ Yt(1 - γ]) 1 = β 𝔼t α --------- (1 - γ) (1 - γ ) = α β γ = 1 - αβ.

An important way of judging a macroeconomic model and deciding whether it makes sense is to examine the model’s implications for the dynamics of aggregate variables. Defining lower case variables as the log of the corresponding upper case variable, this model says that the dynamics of the capital stock are given by

Kt+1 = (1 - γ)Yt (5) α = α βZtK t (6) kt+1 = log α β + zt + αkt (7)
which tells us that the dynamics of the (log) capital stock have two components: One component (zt) mirrors whatever happens to the aggregate production technology; the other component is serially correlated with coefficient α equal to capital’s share in output.

Similarly, since log output is simply y = z + αk, the dynamics of output can be obtained from

yt+1 = zt+1 + αkt+1 (8) = α (log Kt+1 ) + zt+1 (9) = α (log α βY ) + z (10) t t+1 = α (yt + log α β ) + zt+1 (11)
so the dynamics of aggregate output, like aggregate capital, reflect a component that mirrors z and a serially correlated component with serial correlation coefficient α.

The simplest assumption to make about the level of technology is that its log follows a random walk:

zt+1 = zt + εt+1. (12)

Under this assumption, consider the dynamic effects on the level of output from a unit positive shock to the log of technology in period t (that is, εt+1 = 1 where εs = 0 ∀ s≠t + 1). Suppose that the economy had been at its original steady-state level of output y¯ in the prior period. Then the expected dynamics of output would be given by

yt = y¯+ zt (13) 𝔼t [yt+1 ] = y¯+ zt + αzt (14) 2 𝔼t [yt+2 ] = y¯+ zt + αzt + α zt (15)
and so on, as depicted in figure 1.

Figure 1: Dynamics of Output With a Random Walk Shock

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The other interesting possibility to consider is the case where the level of technology follows a white noise process,

zt+1 = z¯+ εt+1. (16)

The dynamics of income in this case are depicted in 2.


Figure 2: Dynamics Of Output With A White Noise Shock

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The key point of this analysis, again, is that the dynamics of the model are governed by two components: The dynamics of the technology shock, and the assumption about the saving/accumulation process.

For further analysis, consider a nonstochastic version of this model, with Zt = 1 ∀ t. The consumption Euler equation is

Ct+1-- 1∕ρ = (βRt+1 ) Ct

But this is an economy with no technological progress, so the steady-state interest rate must take on the value such that Ct+1∕Ct = 1. Thus we must have βR = 1 or R = 1∕β.

In the usual model the net interest rate r is equal to the marginal product of capital minus depreciation, r = F′(K) - δ, so the gross interest rate is R = 1 + F′(K) -δ. But in this case we have δ = 1 so R = F′(K).

The unconditional expectation of the interest rate, 𝔼(log[F′(K)]), is given by

𝔼(log [F ′(K )]) = (α - 1) log [β] + α 𝔼 (log [F ′(K )]) + 𝔼 [ε] ′ 𝔼 (log[F (K )])(1 - α ) = (α - 1) log [β] (17 ) ′ 𝔼(log [F (K )]) = log [1∕β ] (18 ) 𝔼 (log [F ′(K )β ]) = 0 (19 )

Previously we derived the proposition that

R β = 1 (20) ′ F (K )β = 1 (21) ′ log [F (K )β ] = 0 (22)
but since this is a model in which R = F′(K), this is effectively identical to the steady-state in the nonstochastic version of the model. Thus, in this special case, the modified golden rule that applies ‘in expectation’ is identical to the one that characterizes the perfect foresight model. The only difference that moving to the stochastic version of the model makes is to add an expectations operator 𝔼 to the LHS of the nonstochastic model’s equation.

References

   Brock, William, and Leonard Mirman (1972): “Optimal Economic Growth and Uncertainty: The Discounted Case,” Journal of Economic Theory, 4(3), 479–513.