Question

Linear Algebra

9. Consider the following linear, dynamic AS/AD model of real aggregate income (y) and the price of goods (p). Note that eo (autonomous expenditure) and yo^p(potential GDP) are both exogenously given and assumed to be strictly positive.

\begin{aligned} &\begin{aligned} -y_{t+1} &=0.5 y_{t}+e_{t}-p_{t} \\ -& p_{t+1}=p_{t}+(21 / 400)\left(y_{t}-y_{t}^{p}\right) \end{aligned}\\ &\boldsymbol{e}_{\boldsymbol{t}+\mathbf{1}}=\boldsymbol{e}_{\boldsymbol{t}}=\boldsymbol{e}_{\boldsymbol{0}}\\ &y_{t+1}^{p}=y_{t}^{p}=y_{0}^{p} \end{aligned}

\text { a. Letting } x_{t}=\left[\begin{array}{llll} y_{t} & p_{t} & e_{t} & y_{t}^{p} \end{array}\right]^{T}, \text { write down the } 4 \text { by } 4 \text { matrix } A, \text { for which } A x_{t}=x_{t+1}

b. Find all of A's eigenvalues.

c. What do the eigenvalues of A tell you about the steady-state equilibrium for this model?

d. Find the steady-state equilibrium vector, x*, expressed as a function of its exogenous determinants (eo and y).

e. All else equal, how would an increase in autonomous expenditure affect the steady-state values of y and p? Explain.

f. Suppose that at time 0, xo= [26 6 20 32]T. What would y and p be at time 1? At time 2?


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