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In this presentation at around 21:20, Ricardo Reis discusses a two-sector New Keynesian model to explain that the supply and demand decompositions of inflation by Adam Shapiro as not as informative as they are made out to be.

Specifically, he writes down the following set of equations for his two-sector NK model, with $j \in \{a, b \}$ being sector $j$.

$$ u(C) - v(L) \tag{1} $$ $$ C = (C_a^{1 - 1/\eta} + C_b^{1- 1/\eta})^{\eta/{\eta - 1}} \tag{2} $$ $$ C_j = \left(\int_0^1 C_j(i)\mathrm{d}i\right)^{\sigma/{\sigma - 1}} \tag{3} $$ $$ Y_j(i) = A^{\theta_j}L_j(i)^\alpha \tag{4} $$ $$ p_j = \lambda_j\left(\frac{w_tL_j(i)}{\alpha Y_j(i)}\right) + (1 - \lambda_j)\mathbb{E}\left(\frac{w_tL_j(i)}{\alpha Y_j(i)}\right) \tag{5} $$ $$ L = L_a + L_b, Y_a = C_a + G^{\gamma_a}, Y_b = C_b + G^{\gamma_b} \tag{6} $$

Here the parameters $\theta_j, \lambda_j, \gamma_j$ describe sectoral differences in technology, price stickiness, and government spending. Equations $(1)$-$(4)$ are pretty self-explanatory. Equation $(6)$ are market clearing conditions. I assume equation $(5)$ is an optimal pricing condition for a firm in sector $j$. He also adds "a monetary shock to nominal income (to keep the model static)," which honestly just confused me.

He then goes on to discuss one can generate any decomposition of supply and demand even with pure demand shocks. For example, a positive aggregate demand shock through monetary policy can raise output in sector $a$ while decreasing it in sector $b$ if prices are more sticky in sector $a$. Intuitively, this makes sense. Demand for sector $a$'s product rises more, it absorbs labour from sector $b$, enough to decrease output in sector $b$.

My question/request is: Could anyone fill in the details of this model? Specifically, how is equation $(5)$ derived? How is he keeping the model static? How specifically can one generate the supply/demand decompositions? A reference would be fine as well. Thanks!

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