Here, we propose a production function, which we will call the normalized Translog pro-
duction function, that exhibit complementarity-induced procyclical returns to scale in the
short run, fSR, and has conventional constant returns to scale in the long run, fLR.
We introduce some notations for illustrative purposes. Consider an arbitrary variable
value, and let ˆXt ≡ log xt ≡ log
(
Xt Xss
)
be a log-deviation of Xt from the steady state.
Additionally, define ˜Xt as a cross-sectional average level of Xt, which individual agents
take as exogenous, but in equilibrium Xt= ˜Xt. Thus, it follows that xt= ˜xtand ˆXt=Xˆ˜t
in equilibrium.
Our production function has the following form :
Yt= fSR(ks t, lt, et; εat, Φt)· fLR(Ksss, Lss, Ess; Φss) fLR(Ks ss, Lss, Ess; Φ) (3.3.1)
Here, Φtis a vector of endogenous parameters that governs potentially time-varying short-
run returns to scale of the economy. We say “endogenous” parameters because the time-
varying components are endogenously determined in equilibrium, although individual
agents take these values as given.¹⁷ Φis a vector of strictly exogenous parameters that govern (time-invariant) long-run returns to scale of the economy.¹⁸ The functional forms are given by
fSR(ks, l, e; εa, Φt) = εa(ks)αk · lαl+βellog ˜et · eαe+βellog ˜lt (3.3.2)
fLR(K, L, E; Φ) = (Ks)αkLαlEαe (3.3.3)
¹⁷Because the time-varying components in Φtare endogenously determined in equilibrium, the termi-
nology “parameter” can be somewhat misleading. Still, we call it a (endogenous) parameter because individ- ual agents take Φtas exogenous. The long-run value of Φt, Φ, is a vector of strictly exogenous parameters
in the sense that there are no endogenous components involved, and the vector consists of deep structural parameters.
where
Φt ≡ [αk, αl+βellog ˜et, αe+βellog ˜lt]′ (3.3.4)
Φ≡ [αk, αl, αe]′ (3.3.5)
Note that if we evaluate Φt at the steady-state (or, at the long-run horizon), we have
Φss= Φsince log ˜ess=log
(˜
Ess Ess
)
= 0and log ˜lss=log
(˜
Lss Lss
) = 0.
Properties of the Normalized Translog Production Function
Several features are worth mentioning. (i) The complementarity between energy input
and labor input is reflected by a single parameter βel > 0. If βel > 0, our model features
procyclical returns to scale in the short-run, provided that the dynamics of log ˜etand log ˜lt
are procyclical.
(ii) If we evaluate our production function (3.3.1)in the equilibrium (i.e., if we impose
Et = ˜Etand Lt = ˜Lt), then we can show that in the short run, our technology implies
the following Translog expression (which is why we call it normalized “Translog”) :
log yt=log εat + αklog kts+ αllog lt+ αelog et+βel· 2log(lt)log(et)
(iii) In equation (3.3.1), each input inside function fSR is normalized by its steady
state value (i.e., recall the definition xt≡ XXsst ), which is why we call our production func-
tion as “normalized” Translog. This kind of normalization can also be found in Koh and
production function, allowing a time-varying CES parameter in the short run. As in that
paper, the normalization of inputs with their steady state counterparts causes the produc-
tion function to collapse to a conventional Cobb-Douglas at the steady state. There are
two important advantages to doing so. First, it facilitates the calculation of the steady
state of the economy, as the steady state is identical to that of the model without com-
plementarity. Additionally, because the steady state is exactly the same as in the model
using the typical Cobb-Douglas technology, our model becomes directly comparable to
existing models in the literature.
Second, and more importantly, normalization allows the model to be compatible with
balanced-growth path. This captures the idea that the input complementarity we identify
is a short-run characteristic that does not affect the long-run growth of the economy.
(iv) Despite procyclical returns to scale, the production function becomes scale-free
up to the first order because log-linearizing the production function yields exactly the
same form as the log-linearized Cobb-Douglas production function: in the equilibrium
(i.e., impose ˜Et = Et and ˜Lt = Lt), the first-order approximation of our production
function is given by
ˆ
Yt = ˆεat + αkKˆts+ αlLˆt+ αeEˆt
Hence, the procyclicality of returns to scale does not generate any additional fluctua-
tion of output by itself and behaves exactly the same as conventional Cobb-Douglas up to
the first order. All the interesting dynamics arise through the first-order condition of the
returns to scale, in which propagation occurs because the fact that the economy exhibits
increasing returns to scale.
(v) Finally, we assume that when each firm solves the problem, it takes the returns
to scale of the economy as given. Remember that we introduced cross-sectional average
variables ˜Etand ˜Lt, and individual firms take these variables as exogenous. The returns
to scale of the economy is given by the sum of all powers in (3.3.1):
Short-Run Returns to Scale = αk+ αl+ αe+βel
[
log( ˜Et/Ess) +log( ˜Lt/Lss)
] Long-Run Returns to Scale = αk+ αl+ αe = 1
This assumption is technically required to guarantee that firms’ optimizing behavior
is well-characterized by the first-order conditions. If an individual firm can internalize
the change in returns to scale, then by choosing the larger amount of labor and energy,
each firm can make the returns to scale it faces arbitrarily large. This will induce firms to
choose an infinite amount of labor and energy. By assuming that individual firms do not
perceive that it can affect the economy’s returns to scale, this problem no longer arises.¹⁹
In addition, this assumption reflects the idea that the returns to scale is more of an
economy- or industry-wide characteristic than a firm-specific characteristic. Hence, a
single firm’s change of input does not affect the returns to scale, but when all firms jointly
increase (decrease) labor and energy inputs, then the returns to scale parameter changes
¹⁹The assumption we make renders our model similar to an internal increasing returns to scale (IRS) model. In contrast to an external IRS mode, an internal IRS model requires some degree of market power of individual firms. Following Benhabib and Farmer,1994, we also assume a monopolistic competitive model, but with a negligible amount of constant markup (2% markup). Hence, our model can be considered an “approximation” of a perfect competitive model.
Figure 3.1: Labor Market: Input Complementarity vs. Markup Countercyclicality Lt Wt L s L′s M P L(Lt, Kt−1; εat) W1 L1 W2 L2
(a) Standard RBC with CRS
Lt Wt L s L′s M P L(Lt,Kt−1;εa t) µt M P L(Lt,Kt−1;εa t) µ′t W1 L1 W2 L2
(b) Markup Countercyclicality with CRS
Lt Wt L s L′s M P L(Lt, Kt−1, Et; εat) M P L(Lt, Kt−1, Et′; ε a t) W1 L1 W2 L2 (c) Complementarity-induced Procyclical RTS
Note. Y-axis is real wage and x-axis is labor. The figures show how the labor market
reacts to positive demand shock (a) in a standard RBC model with constant returns to scale, (b) in a model with markup countercyclicality with constant returns to scale, and (c) in a model with complementarity-driven procyclical returns to scale.
toward IRS (DRS).
Key Intuition of Model Mechanism
In this section, we convey the basic intuition behind our mechanism. An important fea-
ture of the complementarity-induced procyclicality of returns to scale is that it generates
strong cyclical movement of input demand even with respect to demand shocks. Figure3.1 shows how both traditional countercyclical markup and complementarity-induced pro-
cyclical returns to scale explain an increase in wage and labor when firms face a positive
Figure3.1ashows the labor market under the standard RBC model with constant re- turns to scale technology. Because the production function only depends on labor, capital,
and productivity, the marginal product of labor only depends on labor, capital and pro-
ductivity. When firms experience a positive demand shock, they can only adjust labor
input since the capital input is a predetermined variable and productivity is not corre-
lated with positive demand shock. In other words, they cannot shift the labor demand
schedule. Only the labor supply schedule shifts to the right which results in increased
employment.²⁰ However, this effect leads to a decrease in wages and an increase in la- bor input because of a positive demand shock, which cannot be reconciled with empirical
evidence.²¹ Markup countercyclicality has been proposed to reconcile this seemingly con- tradictory prediction as in Figure3.1b. In models with nominal price rigidity, markups fall when firms face positive demand shocks because of rigid prices and increases in marginal
cost. The decrease in markups allows the labor demand schedule to shift to the right,
capturing both the increase in labor and the increase in wages (or constant wages) at the
equilibrium.
Input complementarity that induces procyclical returns to scale, however, can also
shift the labor demand schedule when firms face positive demand shock as shown in
Figure 3.1c. Suppose we allow other flexible inputs such as energy, which has strong
²⁰The labor supply shifts to the right with respect to demand shock for various reasons (Rotemberg and Woodford1991). For example, the labor supply can shift to the right because of an increase in the marginal utility of wealth resulting from an increase in government spending.
²¹Real wage procyclicality with respect to demand change follows the argument in Rotemberg and Wood- ford,1991; Rotemberg and Woodford,1992. There is evidence of weakly countercyclical real wages condi- tional on government spending (Nekarda and Ramey2011), but we are not aware of any paper that finds strong countercyclical real wages conditional on demand change predicted by conventional models with a perfectly competitive market
complementarity with the labor input, in the production function. In that situation, the
marginal product of labor depends not only on labor, capital, and productivity but also
on energy. Now consider a positive demand shock. First, this shifts the labor supply
schedule as in a standard RBC model, which increases labor input (movement along the
labor demand curve). This initial increase of labor increases the marginal productivity of
energy, which induces firms to hire more energy. This increase of energy input increases
the marginal productivity of labor, which eventually shifts the labor demand schedule
(shift of the labor demand curve). This interaction between energy input and labor input
is strong enough to make real wages increase when complementarity between these two
inputs generates the procyclicality of returns to scale.²²