# Are these two specifications of household utility different

Dear Johannes,
I have seen two types of specifications of household utility,
The first one:
U_0=\sum_{t=0}^{\infty}E_t[\ln C_t+\frac{H_t^{1+\gamma}}{1+\gamma}],
household optimization behavior is choosing optimal consumption C_t in period t and labor supply H_t in period t to maximize U_o in period 0,
The second one:
U_t=\sum_{s=0}^{\infty}E_t[\ln C_{t+s}+\frac{H_{t+s}^{1+\gamma}}{1+\gamma}],
household optimization behavior is choosing optimal consumption C_t in period t and labor supply H_t in period t to maximize U_t in period t,
Are these two expected utilities different or not?
Thank you very much and look forward to hearing from you.
Best regards,
Jesse

The way you wrote it down, the two would be different because of the different conditioning set of the expectations. To be identical, the first one needs to be E_0 instead of E_t. I guess that is a typo.

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Dear Johannes,
Yes, I made a typo, and the first one is Eo instead of Et, However, I am wondering that do household in both the first one and the second one have the same decision variables: Ct and Ht? or the second one’s decision variables should be Ct+s and Ht+s?
Thank you very much and look forward to hearing from you.
Best regards,
Jesse

In the first case, they optimize with respect to C_t \: \forall \: t, in the second case with respect to C_{t+s} \: \forall \: s. That this is identical can be seen from setting t_0=0 and using s=t to indicate the index.