Redes sociales como facilitadores de recursos y experiencias

We build a quantitative model that matches both the pricing moments found in the micro data and business cycle variations in the revenue distribution we find in the previous section. We augment a standard menu cost model similar to those found in Golosov and Lucas (2007) and Midrigan (2011) to match the shifts in the revenue distribution across the business cycle. We use this model to quantify the degree of state-dependence of monetary policy implied by our mechanism.

Households

Households maximize discounted expected utility,

C_tis the household consumption of the composite good, Ntis total labor supply, Btis one-period nominal bonds, Pt is the aggregate price level, Rt is the nominal rate of return, Wt is nominal wage, and Π_tis the firm profits that are transfered to the households. φ_tis an aggregate preference shock that represents a shock to the discount rate and affects the intertemporal substitution of households.

Households consume a continuum of a variety of products indexed by i. The composite good C_tis a Dixit-Stiglitz index of these products,

C_t=Z

ι_itc_it
^−1_ di_−1^

(1.13)

where ιit is the idiosyncratic preference for product i and  is the elasticity of substitution across goods. Households choose to consume variety cit to maximize Ctsubject to the constraint,

Z

q⁻¹_it p_itc_it = P_tC_t

where pit is the price of product i. The household’s subproblem of variety choice is standard ex-cept for qit. In this model qit serves as an exogenous demand shifter that is needed to match the documented changes in the revenue distribution across the business cycle. We treat q_it as exoge-nous in order to focus on the firm’s decision process. Furthermore, we wish to remain amenable

to different household processes that may generate the shifts in demand. Nevertheless, a possible interpretation of q_it is as the cost of shopping effort (see Coibion et al.,2015)). Another interpre-tation would treat q_it as the household’s valuation of product quality, defined as a function of the product’s idiosyncratic components (z_it, ι_it); this is in line with the findings of Jaimovich et al.

(2017).²⁰

We model qit as depending on the aggregate state of the economy as well as the idiosyncratic characteristics of the product.

q_it= q(˜q(z_it, ι_it), φ_t)

To be consistent with the empirical findings in section 4.1 we impose the condition that,

∂²lnq_t

∂ln˜q_t∂lnφ_t > 0

With a shopping cost interpretation, this condition means that in recessions, the opportunity cost of time falls and it becomes less costly to buy products that households usually purchase infrequently.

A product quality interpretation would imply that household sensitivity to quality differences de-creases in recessions.

The inclusion of idiosyncratic preferences ιit is important in our setting. The revenue effect implies that the revenue distribution is an important determinant of output fluctuations, and the inclusion of idiosyncratic demand shocks are critical in this regard. Studies such as Burnstein and Hellwig (2007) and Kumar and Zhang (2016) find that demand shocks constitute a large fraction of the idiosyncratic shocks that firms face, and ignoring this component may lead to an understate-ment of revenue dispersion. In the calibration of this model, we show that this is true in our sample as well.

20To interpret as valuation of product quality it may be more intuitive to specify qitas a part of the composite good index Ct= (R (qitι_itc_it)^−1 di)^−1 which gives almost identical results.

The demand curve for good c_it then follows,

c_it=pit

P_t

^−

q_it^ι^−1_it C_t (1.14)

and the expression for the aggregate price index can be derived as

Pt=

Following much of the literature on menu cost models, we assume that the monetary authority conducts monetary policy by targeting a path of nominal GDP

lnM_t= g + lnM_t−1+ ^m_t (1.16)

where rij is the transition probability from state i to state j. The parameter φtdetermines whether the economy is in a high output state or a low output state.

Firms

Firms produce a differentiated product y_itwith linear production technology

y_it= z_itn_it

where zit is firm i’s idiosyncratic productivity and nit is labor demand. The firm’s current period real profit is,

where the demand for y_itis given by equation (1.14).

Now consider the firm’s dynamic problem. Henceforth, for notational simplicity we omit the subscript i except when explicitly necessary. It is more convenient to express the firm’s dynamic problem in terms of real markup µt = ^p_W^tzt

t. Substituting in equation (1.14) for yit, the firm’s current period profit is

where ^z_t is a shock to productivity. The shock ^z_t follows a Poisson process,²¹

^z_t ∼

21Midrigan (2011) and Alvarez and Lippi (2014) argue that the process of idiosyncratic productivity shocks is important for the real effect of monetary policy as it determines the strength of the selection effect present in the models. In appendix C, we argue that an accurate representation of the revenue effect in the model could suggest that the selection effect in the model is accurately reflected as well.

and idiosyncratic demand evolves according to

lnι_t= lnι_t−1+ ^ι_t

where ^ι_tfollows the normal distribution N (0, σ²_ι). We assume that log productivity and log demand follow a random walk, which allows us to define the idiosyncratic state of a firm jointly as ω_t= z_tι_t. This reduces the number of state variables and decreases the computational burden of solving this model numerically. The idiosyncratic state ω_tthen follows

lnω_t= lnω_t−1+ ^z_t + ^ι_t

Since this variable follows a random walk, we include a probability of firm exit to keep the distribution of firms stationary. We assume that firms exit with probability α. When a firm exits, it is replaced by a new firm with zt = ιt = ωt = 1. We now write the firm’s dynamic problem, where each period it must pay a small menu cost b to change its price as,

V (µ_t, ω_t; Y_t, φ_t)

= max (

π(µ_t, ω_t, Y_t, φ_t) + (1 − α)βE_tΛ_t+1V

µ_t+1, ω_t+1; Y_t+1, φ_t+1 ,

maxµ˜

n

π(˜µ, ω_t, Y_t, φ_t) + (1 − α)βE_tΛ_t+1V

µ_t+1, ω_t+1; Y_t+1, φ_t+1o

− b )

where Λt+1is the stochastic discount factor. We use log linear approximations of the relationship between labor supply and output to allow convenient aggregation as in Nakamura and Steinsson (2010). We use the Krusell and Smith (1998) algorithm to solve the model numerically. We assume

that firms perceive the evolution of the price level _P^Pt

t−1 as a linear function of variables, lnPt+1

P_t



= β₁+ β₂lnY_t+ β₃lnφ_t+1+ β₄lnφ_t+ β₅^m_t+1.