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Linearization around the steady state (Asset pricing application)

Economics Asked on October 5, 2021

I am working through a linearization example from Colacito and Croce (2011). In the paper the following expression is derived:

begin{align}
& (v^{i}_{c,t})^{theta}=E_{t}[delta e^{Delta c^{i}_{t+1}theta(1-frac{1}{psi})}(1+v^{i}_{c,t+1})^{theta}]
end{align}

Then linearizing around steady state for v: $v^{i}_{ss}=frac{delta}{1-delta}$, they get:

begin{equation}
v^{i}_{c,t}=E_{t}[delta + frac{delta}{1-delta}(1-frac{1}{psi})Delta c^{i}_{t+1} + delta v^{i}_{c,t+1}]
end{equation}

I am struggling to see how you derive the second equation from the first equation by simply linearizing around v. Can someone try to derive this and see how to get there?

Many thanks!

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