Return on equity in a Calvo setting

Hello everybody,

Let’s assume that I have a general equilibrium model in which households provide labor as well as capital to firms. They are compensated in form of wage and remuneration of capital payments from firms.
I take the famous Calvo setting: firms are divided into final firms (they aggregate different intermediate goods and sell the aggregate to households) and intermediate firms (production of intermediate goods according to a standard production function and input factors provided by households). By definition, intermediate firms act under monopolistic competition because each of them produces a differentiated good. Therefore, intermediate firms make profit.

I wonder how to measure return on capital from households’ perspective.

When I do exactly the same like Boldrin, Christiano, and Fisher (1999) (they derive the return on equity from the value of a capital unit in combination with remuneration of capital), I get in combination with habit persistence preferences a substantial equity premium, like Jermann who does not use the remuneration of capital at all: instead, he uses dividend payments to households.

When I use Boldrin, Christiano, and Fisher’s approach, I do not use dividend payments (I think they indeed exist in terms of monopolistic competition on the intermediate goods market). When I include the dividend payments in combination with remuneration of capital, the equity premium explodes.

My main question is how to deal with gains from intermediate firms when I want to evaluate return on equity in my setting. So far, i assumed that gains from intermediate firms vanish in the process of aggregation in terms of price dispersion, but Im not sure…

Many thanks! :four_leaf_clover:

I am not sure I understand everything in this post, but I think you are confusing ownership of capital with ownership of firms/claims to the residual profits. In standard models, there are rental markets for capital and capital gets paid in relation to its marginal product (of course there is a markup involved). In contrast, owners of the firms get the residual, i.e. whatever is left after capital and labor have been paid. Thus, the equity return and its premium differs from the return on capital.

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Your confusion might stem from the fact that in the model by Jermann (1998) there exists an explicit financial market, whereas in Boldrin, Christiano, and Fisher (1999) this market is underlying the model equations. In other words, in Jermann (1998), households acquire equity by purchasing shares issued by firms which compensate investors by paying dividends. Since a share is a claim to the firm’s infinite sequence of dividends, the rate of return on equity is given by an asset pricing formula. In Boldrin, Christiano, and Fisher (1999), firms do not explicitly issue equity, so you cannot observe it from the equations of the model. But implicitly equity finances firms’ physical capital, as written in their footnote 17 :

The rate of return on equity assumes that, in the underlying market economy, capital accumulation is 100 percent equity financed.

So, in Boldrin, Christiano, and Fisher (1999), households receive the profits Pi - W*L that are entirely reinvested in firms to finance their investment in physical capital, I. Therefore, the rate of return on equity coincides with the rate of return on physical capital.

To grasp the difference between the explicit and implicit market setting, I suggest reading the section 3.2 of Boldrin, Christiano, and Fisher (1999) or Boldrin, Christiano, and Fisher (1997) for a version with an explicit market setting.

Regarding your main question, in standard models, a fixed cost is introduced such that intermediate firms do not make profits at steady state. Also, as explained by jpfeifer, intermediate firms rent capital on a competitive market. For a setting similar to yours, I recommend Alpanda (2013) where a framework with price rigidities à la Calvo (i.e. intermediate firms) and an explicit financial market is developed.

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To restate @apauli’s point. In a perfectly competive market as in Boldrin et al. (1999) and Jermann (1998) there are 0 pure profits, while the return to capital is equal to its marginal product. Thus, you do not need to worry about residual ownership and both the centralized version in Boldrin et al and Jermann are identical.
In contrast, in a monopolistically competitive economy, there are pure profits. Thus, there may be a difference between the centralized and the decentralized economy depending on your assumptions, i.e. whether the capital stock is owned by firms and the shares to the firms deliver claims to both the return to capital and pure profits.


@jpfeifer @apauli thank you very much for all your helpful feedback! As you have seen, my issue are the profits of intermediate firms.
When I assume that the households (HH) are the owners of intermediate firms in this Calvo setting and real profits enter directly HH’s buget constraint, do they appear anywhere in my model block? For the HH I maximize with respect to consumption, labor-effort, capital, investment, foreign- and domestic bond holdings and for the intermediate firm I maximize with respect to capital and labor for given wage and remuneration of capital. So the real profits in such a setting don’t enter the FOC conditions. At least in my opinion… So when I introduce a steady-state fix cost component to shut off profits of intermediate firms what else do I have to consider? Many thanks!

Profits of intermediate firms will not show up in the FOCs as they are exogenous from the perspective of the households. They only show up in the budget constraint, just like lump-sum taxes do in the standard RBC model. Once you consolidate the model, they then again do not show up in the resource constraint of the economy.
Note that a fixed cost only kills the profits in steady state, but not outside of it.

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Thank you for your comment: that all makes absolutely sense to me.
I’m struggling with the existence of a return on equity as households do not maximize asset holdings.

(Just as a reminder: in my model, the households maximize w.r.t. consumption, labor, investment, capital, domestic- and foreign bond holdings. The firms maximize w.r.t. capital, labor, and prices. Therefore, I work with a decentralized model.)

I would like to examine the equity premium in my model: I derive the return on capital out of first order conditions (similar expression to BCF 1999)… but as there is no asset holding decision, I simply do not know how to deal with dividend payments (before the steady-state).