RDP 9812: An Optimising Model for Monetary Policy Analysis: Can Habit Formation Help? Appendix A: First-order Conditions for the Nonlinear Model

Beginning with the definition of period utility:

the overall utility function:

and the habit-formation reference consumption level:

The derivative of U with respect to C_t is:

Noting that that and that then we can express:

which collapses to a more compact discounted summation:

Defining:

then we have the derivatives of utility U with respect to C_t and C_t+1

Combining these with a standard budget constraint (with time-varying real interest rate r_t) in a Lagrangian, we obtain the first-order conditions:

Both of these expressions must equal zero for an optimum, yielding an Euler equation: