Stochastic Discount Factor (Household with deposit-in-advance constraint))


From what I have understood, the sdf is the expected marginal utility of consumption at t+1 on marginal utility of consumption at t. In models which the household has bonds or deposits or any kind of nominal contract, the sdf will also be equal to the inverse of interest rate. But, if the household has a deposit in advance constraint besides the budget constraint, the marginal utility of consumption at t+1 on t, considering the new second constraint, will not become as simple as the typical models with just a budget constraint which will have the following nominal sdf \Lambda_{t,t+1} = \frac{1}{R^d_t} = \beta\frac{\lambda_{h,t+1}}{\lambda_{h,t}}\frac{1}{\Pi_{t,t+1}}. In my instance, which I have a deposit in advance constraint that has both consumptions and deposit in it, the nominal sdf will become as the following \Lambda_{t,t+1} = \frac{1}{R^d_t} = \beta \frac{\varsigma_t \lambda_{t+1}}{\Pi_{t,t+1}\lambda_t - (1-\varsigma_t)R^d_t \lambda_{t+1}}
which may look strange, but if you put \varsigma_t=1 in the equation, it will act as if like the deposit in advance constraint is eliminated and therefore, we reach the upper typical nominal sdf. My question is that is it ok for the nominal sdf to be defined as such? I have just used the same method as constructing the nominal sdf for the typical household problems for my model, which is \Lambda_{t,t+1} = \frac{1}{R^d_t}
from what I have read from Schmitt-Grohe and Uribe, the sdf is defined as such.

I am not sure I understand the problem. In more complicated models where the consumption-savings decision is not just dependent on investment in a risk-free nominal bond the SDF will obviously be more complicated.

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Thank you for your response. So, no matter how complicated the household problem is, for real sdf we just calculate \Lambda_{t,t+1}=\frac{\Pi_{t,t+1}}{R^d_t}, no matter how complicated the result will be.

That’s not what I was saying. I said that the specific form will always depend on the particular setup, in particular the savings devices the agent can use.

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