You use investment and consumption to replace output to introduce one more notation I_t for investment. I did not do it before, but technically speaking, this should works because you only introduce another one static variable here.

Technology Shocks in the New Keynesian Model, This paper should be very good paper for you to follow up using Dynare to solve the model. You do not need to loglinearize the model (you do not need a linearized NPKC to solve the model), just input the level variable and the FOCs into Dynare, and let it do the 1st or 2nd approximation for you.

上面的人水平高

这不过是把厂商定价过程与生产过程的决定弄到一起来算了而已。
解的时候分两步走比较简单：
一，N、K的选择解决生产成本最小化问题
二，引入价格调整成本真正起作用是在定价过程中，所以在定价决定时加上价格调整成本

我不赞同楼上换掉产出的说法，这个明显是在乱讲，消掉产出还怎么去得NKPC线？在没有消掉资本存量的基础上，可以考虑消掉投资I，使用变分法对Kt+1（i）求导，事实上这里确实也有多个一阶条件。从实用的角度来讲，确实是不用引入Kt(i)的，至少我所见过的NK极少引入Kt（这是RBC的特点而非NK），因此只用考虑最优定价。