$©$ September 21, 2020, Christopher D. Carroll BrockMirman

The Brock-Mirman Stochastic Growth Model

Brock and Mirman (1972) provided the ﬁrst optimizing growth model with unpredictable (stochastic) shocks.

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

[ ∞ ] ∑ n max 𝔼 β log Ct+n n=0

(1)

s.t. Kt+1 = Yt − Ct α Yt+1 = At+1K t+1

where $A t$ is the level of productivity in period $t$ , which is now allowed to be stochastic (alternative assumptions about the nature of productivity shocks are explored below). Note the key assumption that the depreciation rate on capital is 100 percent.

In this model the capital stock is not useful as a state variable: Because capital has a 100 percent depreciation rate, all that matters to the consumer when choosing how much to consume is how much income they have now, and not how that income breaks down into a part due to $K$ and a part due to $A$ .

The ﬁrst step is to rewrite the problem in Bellman equation form

V (Y ) = max log C + β 𝔼 [V (Y )] t t Ct t t t+1 t+1

(2)

and take the ﬁrst order condition:

u ′(C ) = β 𝔼 [A αK α− 1u ′(C )] t t[ t+1 t+1 ] t+1 1 At+1 αK α− 1 --- = β 𝔼t ---------t+1-- Ct Ct+1 ⌊ ⌋ 1 = β 𝔼 | αA K α − 1-Ct--| t⌈ ◟---t+1◝ ◜-t+1◞ Ct+1 ⌉ ≡Rt+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 the consumption function $Ct = κYt$ , where $κ = 1 − α β$ . To see this, note ﬁrst that the proposed consumption rule implies that $Kt+1 = (1 − κ )Yt$ .

The ﬁrst order condition says

[ ] A K α κY 1 = β 𝔼t α --t+1---t+1 ----t-- Kt+1 κYt+1 [ ] = β 𝔼 α -Yt+1---κYt-- t Kt+1 κYt+1 [ ] -Yt--- = β α K [ t+1 ] ---Yt---- = β α [ Yt − Ct ] Yt = β α ----------- Yt(1 − κ ) [ 1 ] = β α --------- (1 − κ) (1 − κ) = αβ κ = 1 − α β.

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 − κ)Yt α = α βAtK t kt+1 = log α β + at + αkt

(3)

which tells us that the dynamics of the (log) capital stock have two components: One component ( $a t$ ) mirrors whatever happens to the aggregate production technology; the other is serially correlated with coeﬃcient $α$ equal to capital’s share in output.

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

y = a + αk t+1 t+1 t+1 = α (log Kt+1 ) + at+1 = α (log α βYt ) + at+1 = α (yt + log αβ ) + at+1

(4)

so the dynamics of aggregate output, like aggregate capital, reﬂect a component that mirrors $a$ 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:

at+1 = at + 𝜖t+1.

(5)

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 $ˇy$ in the prior period. Then the expected dynamics of output would be given by

y = ˇy + a t t 𝔼t [yt+1] = ˇy + at + αat 2 𝔼t [yt+2] = ˇy + at + αat + α at

(6)

and so on, as depicted in ﬁgure 1.

Figure 1:Dynamics of Output With a Random Walk Shock

Also interesting is the case where the level of technology follows a white noise process,

a = ˇa + 𝜖 . t+1 t+1

(7)

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

Figure 2:Dynamics of Output With A White Noise Shock

pict

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 $A = 1 ∀ t t$ . The consumption Euler equation is

Ct+1 ------ = (βRt+1 )1∕ρ 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 $R = 1∕β$ .

We can further derive the steady state level of capital of a nonstochastic version of the model in which $at = a ∀ t$ from (3):

k = log α β + a + αk (1 − α )k = log α β + a ( ) log-α-β- k = + a 1 − α

(8)

The nonstochastic version of the model is of course not very interesting, except as a point of comparison to the stochastic version of the model. But what could be meant by the ‘steady state’ of a stochastic mdoel that never settles down? We can deﬁne a ‘stochastic steady state’ for such models in a number of (potentially) diﬀerent ways:

The location (if one exists) to which the model will converge after an arbitrarily long period in which no shocks occurred $at+1 = at ∀ t$ (even if in every period agents expect that shocks will occur)
The mean value of some variable in the model (say, $K$ )
The value of some state variable, say $ˇK$ , such that $𝔼t[Kt+1 ] = Kt$ if $K = Kˇ t$ .

We consider here the last of these, which we will show reduces (in this special case) to the same equation as for the nonstochastic version of the model, $Kt+1 = Kt$ . To see this, rewrite the Euler equation as:

⌊ ⌋ C 1 = β 𝔼t |⌈ αAt+1K α − 1---t-|⌉ ◟----◝ ◜-t+1◞ Ct+1 ≡Rt+1 [ ] α− 1-Yt-- 1 = β 𝔼t αAt+1K t+1 Y [ t+1 ] α− 1--AtK--αt--- 1 = β 𝔼t αAt+1K t+1 α [ At+]1K t+1 AtK α 1 = β 𝔼t αK αt+−1 1---α-t- K t+1 [ α] 1 = β αK α− 1 AtK-t- t+1 K tα+1

where the expectations operator disappears because no variables are stochastic (the $′ A t+1s$ in the numerator and denominator cancel, and $Kt+1$ is directly chosen in $t$ so is known. For any given $At$ , the steady state where $ˇ Kt+1 = Kt = K$ is then where

ˇ α− 1 1 = βαAt K t ( log βα ) kˇ = -------- + at 1 − α

which is the generalization of the nonstochastic solution derived in (8).

The result that the nonstochastic and stochastic steady states are the same is special to the Brock-Mirman model; it is NOT true of many other models of growth; it occurs here because the linearity of the consumption function, among other special assumptions. Furthermore, the third of our possible deﬁnitions of a steady state will generally diﬀer at least a little bit from either of the ﬁrst two.

References (click to download .bib ﬁle)

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