Pseudo Random Function
Well, yes, but you cannot use the key $k_1$ for anything else because knowledge of $k_1$ would mean that an attacker can simply calculate $k_2$. If an attacker can find a direct relation between parts of the output then the function is of course not pseudo random anymore. So although it is possible it seems pretty useless to me.

Even if you would not directly use $k_1$ as output you would need to prove that you use it in such a way that security is not compromised. In general you should try to not reuse keys for different purposes.