Why Lagrangean multiplier appears as stochastic discount factor in agent's optimization equation?

Dear Johannes,
In equation (11) on page 15 in Christiano, Motto and Rostagno’s paper ‘FINANCIAL FACTORS IN ECONOMIC FLUCTUATIONS’ (please refer to attached PDF document)
3_CMR 2010.pdf (2.9 MB)
multiplier \lambda on the household’s budget constraint appears in capital producers’ profit maximization in equation (11), I am wondering is \lambda acting like a stochastic discount factor in capital producers’ profit maximization equation (11)?
Thank you very much and look forward to hearing from you.
Best regards,

Hi Jesse,

exactly, the stochastic discount factor is equal to the marginal utility of consumption of the households since it is assumed that the households are the owners of the capital producers.

Assumptions like these are very common.

Thank you! But my question is why they use Lagrangean multiplier as the stochastic discount factor in capital producers’ optimization behavior? why Lagrangean multiplier is equivalent to stochastic discount factor here? Anyone please help me! Thank you very much!

You are confusing something. The SDF appears in the first order conditions and follows from intertemporal optimization. The SDF is usually
As you can see, the SDF results from the marginal valuation of consumption streams of the respective agents. This marginal utility is equal to the Lagrange multiplier on the households budget constraint. In the paper you mention, households own the capital producers and therefore take their valuation into account. Their profits \Pi_t appear in the household’s budget constraint (see equation (30)).