Pareto efficiency (optimality conditions) in simple New Keynesian model

Question

I am looking for the pareto-optimal equilibrium for a central planner's problem in a simple New Keynesian model. The planner's problem is to choose $\{ C_{t}, H_{t}, Y_{t}, \pi_{t}, \{h_{t}(j)_{j=0}^{\infty} \}, \{y_{t}(j)_{j=0}^{\infty} \} \}_{t=0}^{\infty}$ to maximize $$ U_{0} = E_{0} \sum_{t=0}^{\infty} \beta^{t}u(C_{t}, 1-H_{t}) $$

subject to: $$ C_{t} = (1-\frac{\emptyset}{2}\pi_{t}^{2})Y_{t} $$ $$ Y_{t} = \ [\int_{0}^{1}y_{t}^{\frac{\epsilon -1}{\epsilon}}(j)\;dj ] ^{\frac{\epsilon}{\epsilon-1}} $$ $$ y_{t}(j) = A_{t}h_{t}(j) $$ $$ H_{t} = \int_{0}^{1} h_{t}(j) \; dj $$

The first-order conditions for this problem imply that for all $j$ and $t$: $$ \frac{u_{l}(C_{t}, 1-H_{t})}{u_{c}(C_{t}, 1-H_{t})} = A_{t} $$ $$ y_{t}(j) = Y_{t}$$ $$h_{t}(j) = H_{t} $$ $$ Y_{t} = A_{t}H_{t} $$ $$ \pi_{t} = 0 $$ $$ C_{t} = Y_{t} $$

To arrive to the above implications I tried setting up the following Lagrangian: $$ L_{0}(j) = E_{0} \sum_{t=0}^{\infty} \beta^{t} u(C_{t}, 1-H_{t}) + \lambda_{t}[(1-\frac{\emptyset}{2}\pi_{t}^{2})Y_{t} -C_{t}] + \mu_{t}[\ [\int_{0}^{1}y_{t}^{\frac{\epsilon -1}{\epsilon}}(j)\;dj ] ^{\frac{\epsilon}{\epsilon-1}} - Y_{t}] + \omega_{t}[A_{t}h_{t}(j)- y_{t}(j)] + \alpha_{t}[H_{t} - \int_{0}^{1} h_{t}(j) \; dj ] $$

to obtain the FOCs:

$$ \frac{\partial L_{0}(j)}{\partial C_{t}}: u_{c}(C_{t}, 1 - H_{t}) - \lambda_{t} = 0 $$ $$ \frac{\partial L_{0}(j)}{\partial H_{t}}: -u_{H}(C_{t}, 1 - H_{t}) + \alpha_{t} = 0 $$ $$ \frac{\partial L_{0}(j)}{\partial Y_{t}}: \lambda_{t}(1 - \frac{\emptyset}{2}\pi_{t}^{2}) - \mu_{t} = 0$$ $$ \frac{\partial L_{0}(j)}{\partial \pi_{t}}: -\emptyset \pi_{t} Y_{t} \lambda_{t} = 0 $$ $$ \frac{\partial L_{0}(j)}{\partial h_{t}(j)}: \omega_{t}A_{t} - \alpha_{t} = 0 $$ $$ \frac{\partial L_{0}(j)}{\partial y_{t}(j)}: \mu_{t}[\int_{0}^{1} y_{t}^{\frac{\epsilon -1}{\epsilon}}(j)]^{\frac{1}{\epsilon -1}} y_{t}^{\frac{-1}{\epsilon}}(j) - \omega_{t} = 0 $$

Now, I have no idea what I am doing wrong or how to get to the implications above. Any help would be very much appreciated!

Answer 1

The Lagrangian is also a function of the multipliers $(\lambda_t, \mu_t, \omega_t, \alpha_t)$ so you are missing their FOC's (these will give you the constraints as FOC's so if you were already taking them into account, omit my comment, please). Also, $-u_{H}=u_l$ since $l=1-H$ (I would re-write the second FOC, in terms of $u_l$ in order to construct the first "target" equation that has the MRS=MRT. Finally, there are a couple of useful facts that should help you. First, note that in your 4th FOC, you have a product equal to zero, so at least one of the elements is equal to zero. Try to convince yourself, that the only possibility (so that you don't run into a contradiction with the other FOC's) is for $\pi_t=0$ as required by your second-to-last target equation, and in turn implies $C_t=Y_t$ (from your first constraint). Second, using the second constraint you can replace the integral in your last FOC with $Y_t$.

This should help for further simplifying the 10 FOC's you should have, into the 6 target equations you want.