Adding wage stickiness in a model with durable goods fixes the co- movement problem. This has been noted byErceg and Levin (2006)
and byCarlstrom and Fuerst (2010)among the others. In this section it is shown the effect of introducing wage stickiness in a model with durable goods, with and without financial frictions.
Wage stickiness is introduced and modeled in a way similar to
Kim and Oh (2014). The problem of the households in the presence of wage stickiness resembles also that for firms briefly outlined in the previous section. Households offer homogeneous labor services and rely on unions to set their wages as a mark-up over their marginal rate of substitution. Unions differentiate labor and set wages facing adjustment costs à la Rotemberg:
ACw,t= θw 2 Wt Wt−1 −1 !2 wt (3.28)
3.2. A DSGE Model with Durable Goods 119
whereθw is the wage adjustment cost parameter13 and the wage
refers to a single household. Labor services are then reassembled by labor packers that offer labor to goods producers. I assume that there is a union for each households group and calibrate the two elasticity of substitution between labor types, εw and ε˜w , in order to have a
mark-up of 20%in steady state. Solving the unions problem we get the following first order conditions:
(3.29) ϑw(πtw−1)πw =ϑwβEt λt+1 λt " (πtw+1−1)πtw+1wt+1 wt # + (1−εw)Nt+εwM RSt Nt wt (3.30) ˜ ϑw(˜πwt −1)˜πw = ˜ϑwγEt ˜ λt+1 ˜ λt " (˜πtw+1−1)˜πwt+1w˜t+1 ˜ wt # + (1−ε˜w) ˜Nt+ ˜εwM RS˜ t ˜ Nt ˜ wt
Notice that I assume borrowers and savers to be identical but for their impatience, this also entails that they supply the same labor types to unions and earn the same wage: wt = ˜wt, which implies
that πwt = ˜πwt . Notice also, that whenϑw = ˜ϑw = 0, there are no ad-
justment costs in setting the wage and therefore the condition states that the wage is set as a positive mark-up over the marginal rate of substitutions of the two groups.
Several simulations considering different degrees of wage rigidi- ties are considered. In the following figure, ϑw is set in order to ob-
tain an average frequency of wage adjustment of half a year for both groups. In order to calibrate the two parameters I mapϑw andϑ˜w to
get the same elasticity used in the Calvo model.14
13This specification of the problem implies that the adjustment cost parameter for borrowers and savers is going to be different if we want to have the same fre- quency of wage adjustment for the two households groups.
14In order to do so I compute ϑ
w as follows: ϑw = (εw−1)θw(1 +εφ))/((1−
120 Chapter 3. Durable Goods, Collateral Constraints and Co-movement Pr oblem in DSGE Models
FIGURE3.7: Baseline calibration, response to a monetary policy tightening. Average frequency of price adjustment 1 year for the non durable sector, while price stickiness in the durable sector varies from the full flexible case to 1 year average frequency. Habit in consumption parameter is:
3.2. A DSGE Model with Durable Goods 121
Interestingly, as already noted by Kim and Oh (2014), the intro- duction of wage stickiness seems to solve the co-movement problem for both specifications, irrespective of the combination of price stick- iness in the two sectors. This holds true even if we consider a rela- tively small degree of stickiness in the wage adjustment problem.15
The presence of financial frictions now has a strong effect on to- tal output, durable goods investment dynamics and aggregate con- sumption. Indeed, those variables reacts more strongly than in the representative agent model, for any combination of price rigidities considered. Therefore, financial frictions seem to play a role, but the presence of wage rigidity is crucial. From an intuitive point of view, the reason why durable investment collapses – bringing about an overall strong contraction in output – is given by the fact that the introduction of wage stickiness in the durable sector makes durable prices sticky even if they are flexible, for labor is the only factor of production for durables.
The introduction of wage stickiness seems not only to solve the co-movement problem, but also to restore the importance of finan- cial frictions. However, an issues seem to emerge, that is the strong response in durable investment following a monetary policy shock. The reaction of investment is the object of the following section, and is a well known effect of wage stickiness, already highlighted among the others byCarlstrom and Fuerst (2010).