NK Model: ZLB Episodes with Forward Guidance Description and Resolution of the New Keynesian Anomalies When prices are more flexible, the Phillips line steepens, so the Euler line's required steepness increases: bounded rationality or marginal utility of wealth need to be larger. As the slope of the Euler line is determined by bounded rationality in the Gabaix model and by marginal utility of wealth in our model, these need to be large enough. The Euler-Phillips system remains a source at the ZLB as long as the Euler line is steeper than the Phillips line (figure 1D). Our phase diagrams illustrate the logic behind these results. The same is true here: when the marginal utility of wealth is high enough, such that condition ( 9) holds, the Euler-Phillips system is a source even at the ZLB and when the price-adjustment cost γ is lower, condition ( 9) imposes a higher threshold on the marginal utility of wealth. He also finds that when prices are more flexible, more bounded rationality is required to maintain the source property. Gabaix finds that when bounded rationality is strong enough, the Euler-Phillips system is a source even at the ZLB. The results that pertain to the WUNK model are close to those obtained by Gabaix ( 2016, proposition 3.1), although he does not use our phase-diagram representation. The results that pertain to the NK model in propositions 1 and 2 are well known (Woodford, 2001). In the WUNK model, by contrast, the Euler-Phillips system is a source in normal times and at the ZLB (B, D). The figure shows that in the NK model, the Euler-Phillips system is a source in normal times with active monetary policy (A) but the system is a saddle at the ZLB (C). At the ZLB, the natural rate of interest is negative, and the monetary-policy rate is 0. In normal times, the natural rate of interest r n is positive, and the monetary-policy rate is given by i = r n + ϕ π when monetary policy is active, ϕ > 1. The WUNK model is the same model, except that the marginal utility of wealth is not 0 but is sufficiently large to satisfy condition ( 9). The NK model is the standard New Keynesian model. The trajectories are solutions to the Euler-Phillips system linearized around its steady state, plotted for t going from - ∞ to + ∞. The Euler line is the locus y ˙ = 0 the Phillips line is the locus π ˙ = 0. The variable y is output π is inflation y n is the natural level of output. The Euler equation is given by equation ( 4) and the Phillips curve is given by equation ( 1). Phase Diagrams of the Euler-Phillips System in the NK and WUNK Models As a result, with wealth in the utility function, the steady-state Euler equation describes output as a decreasing function of the real interest rate-as in the traditional IS curve but through a different mechanism. They keep consumption constant only if the hedonic returns on wealth fall enough to offset the increase in financial returns: this requires output to decline. When the real interest rate r h is higher, people have a financial incentive to save more and postpone consumption. The steady-state Euler equation imposes that the total rate of return on wealth equals the time discount rate, so it now involves output y. The total rate of return becomes r h + u ' ( 0 ) y, where the hedonic returns are measured by u ' ( 0 ) y. With wealth in the utility function, the returns on wealth are not only financial but also hedonic. It imposes that the financial rate of return on wealth equals the time discount rate-otherwise, households would not keep their consumption constant. (8)The standard steady-state Euler equation boils down to r h = δ.
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