Stochastic discount factor SOE

Hi all,

I have a question related to the real SDF in a model like Adolfson et al. (2007). I know that the SDF is equal to the ratio of marginal utilities of consumption, i mean, in a standar NK model, the real SDF used in the proccess of Calvo pricing is \beta^s\frac{\lambda_{t+s}}{\lambda_t}\frac{P_t}{P_{t+s}} with \lambda_t as the nominal lagrange multiplier of the households budget constraint. But in a model like Adolfson, where there are different prices for consumption and investment goods, the FOC with respect to consumption is C_t^{-\sigma}=\frac{P_t^c}{P_t}\lambda_t would the real SDF be the same (\beta^s\frac{\lambda_{t+s}}{\lambda_t}\frac{P_t}{P_{t+s}})?

Thanks in advance
Best regards

Yes, that is the correct FOC for the cash flows denominated in terms of the final good. If you have other goods, there will be a relative price showing up to transform everything into the final good.

Thank you professor, another question related to this topic, as you said, we need to put everything in terms of the final good, that is:
-\frac{\lambda_t}{P_t}\left(P_t^cC_t+P_t^iI+S_tB_{t+1}^\ast+B_{t+1}-\ W_tL_t-R_{t-1}B_t-\mathrm{\Phi}_{t-1}S_tR_{t-1}^\ast B_t^\ast-R_t^kK_t\right)
but then i have another restriction of the law of motion of capital, however i don´t now if the multiplier of that restriction should be Q_t or \frac{Q_t}{P_t} because that makes a difference when calculating the steady state.

I know that usually marginal tobin´s Q is assumed to be 1 in the steady state, so, if I made the lagrange multiplier Q_t, the steady state FOC of investment would be {:\ \ \ \lambda}_t\frac{P_t^i}{P_t}=Q_t that implies marginal tobin´s Q would be equal to \frac{P_t^i}{P_t}. I don´t know if that make sense. But in the other hand, if i assum the lagrange multiplier of the law of motion of capital is \frac{Q_t}{P_t} then things change and tobin´s Q would equals to P_t^i which I guess has a unit root.

Thanks in advance

This is again a matter of convenience. Depending on your setup, Q is a real or a nominal multiplier and correspondingly may inherit the unit root of the price level in the nominal case.