# Dividing vacancy posting cost by the marginal utility of consumption Trigari 2009

Hi,

I am currently working on Trigari’s (2009, JMCB) paper and I have some trouble with her Bellman’s equation.

In this paper, the author builds a NK model with search and matching frictions on the labor market and an endogenous job separation margin. When writting the value of an open vacancy from firm’s viewpoint, she divide the utility cost of having an open vacancy (kappa in the paper) by the marginal utility of consumption (lambda in the paper). What could be the rationale to divide this cost, which is usually expressed in term of production losses, by lambda?

Best

This is no normalization choice, but a consequence of the model. Agents in the model do not care about output/production, but about utility. Thus, the production losses from vacancy costs are valued using marginal utility. Nevertheless, for the purpose of calibration, it is the share of costs relative to output that is targeted.

However, I have some other questions: if kappa is divided by lambda, this changes the vacancy posting condition (now we have lamba_t and lambda_{t+1} in this equation) but also the wage equation and the condition that defines the threshold value a_min at which a job is endogenously destroyed. As lambda usually does not appear in such equations (to the best of my knowledge Trigari’s paper is the only one with such a writting), is there any incidence for the dynamic of the model? What happens is we re-write the model without dividing kappa by lambda? Furthermore, is the fact that households ultimately own firms a justification of this operation?
I apologize if my questions do not seem very clear, best.

It is about firms being owned by workers and therefore use their stochastic discount factor for discounting future payment streams. This is similar to the price setting problem that firms in the New Keynesian model face. There we make the same assumption. You could of course assume that firms are risk-neutral and simply discount with \beta. That would of course change the dynamics of the model

I apologize to go back to you but actually I have still some trouble with this manipulation. Actually, when solving the firm’s problem (the maximization of its profits subject to the law of motion of employment) I do not retrieve the term kappa/lambda unless I impose it at the beginning…

Without you stating your computations, it is impossible to follow.

I attach a pdf file presenting my problem! Thank youfirm_pb.pdf (175.8 KB)

The FOC in question should follow from your equation (3). What is the problem when you try to solve for it? And what is the definition of \beta_{t,t+1}? That seems to be crucial. My guess is something like \beta\frac{\lambda_{t+1}}{\lambda_{t}}

Your guess about \beta_{t,t+1} is the good one! Could you just confirm that my writing of equation (3) is the good one? In particular, the way \kappa appears in this equation (divided by \lambda_t or not)

Thanks you!

As far as I can see the term in brackets is in terms of the final good. Thus, it must not be divided by marginal utility.