# Duality of household's and firm's Euler equations when firm owns capital stock

A very basic result of the RBC literature is that, assuming the firms own the capital stock, the household’s FOC w.r.t. bonds and the firm’s FOC w.r.t. bonds are identical because of the way the firm’s (future) cash flow is discounted and translated into utility units by the stochastic discount factor. For instance, a typical Euler equation could read

lambda = beta * R(+1) * lambda(+1)

using conventional notation. The Modigliani-Miller theorem holds, the firm’s debt is indeterminate. So far, so good. Please let me refer you to Eric Sim’s lecture notes for details on what I mean: https://www3.nd.edu/~esims1/rbc_model.pdf

My question is what happens to this duality when we modify the households problem such that the Euler equation changes for the household but not for the firm? For instance, what if wealth (bonds) enter the utility function, say in the form log(B). The the household’s Euler equation will read:

lambda * 1/B = beta * R(+1) * lambda(+1)

but the firm’s Euler equation will still read:

lambda = beta * R(+1) * lambda(+1)

The wealth in the utility is just an example. You can think of other modifications of the households problem such as a wealth tax.

Do I need to somehow change the stochastic discount factor in the firms problem to reflect these changes in the household’s problem? As I understand it, the purpose of the discount factor is to translate profits measured in goods into utility units which would be done through multiplying with lambda (the Lagrangian of the household’s budget constraint), or do I miss something regarding the correct discount factor in the firm’s objective?

If inconsistent Euler equations is the way it is, I’m struggling to understand the economics of that. Does it mean, for instance in the case of wealth in the utility, that firms will borrow and infinite number of bonds and pay out an infinite cash flow as dividends since wealth creates utility while debt does not come at a cost? Or in case of the households wealth tax, does that mean wealth is always zero in the optimum and consumption is smoothed by the firm’s profit retention ratio? If the Modigliani-Miller theorem does not hold anymore, is there a way to compute an optimal debt-capital ratio?

Apologies for these rather basic questions! I would very much appreciate any advice or links to related literature! Thank you very much for your time!

Christian

Think about a firm manager hired by the owners of the firms, i.e. the households. They would take the objective of the owners into account and would use exactly the same discount factor. So it should not matter and you should end up with the same FOC (at least if you are dealing with a representative household). Formally, you should be able to show this when computing the equilibrium of the decentralized economy using an Arrow-Debreu setup.