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2. Considering the following example $y_t = c_t + i_t$, (1.1) $c_t = \alpha y_{t-1} + \epsilon_{ct}$ $0 < \alpha < 1$ (1.2) $i_t = \beta (c_t - c_{t-1}) + \epsilon_{it}$ $\beta > 0$ (1.3) where $y_t$, $c_t$, and $i_t$ denote real GDP, consumption, and investment in time period $t$, respectively. In this Keynesian model, $y_t$, $c_t$, and $i_t$ are endogenous variables. The previous period's GDP and consumption, $y_{t-1}$ and $c_{t-1}$, are called predetermined or lagged endogenous variables. The terms $\epsilon_{ct}$ and $\epsilon_{it}$ are zero mean random disturbances for consumption and investment, and the coefficients $\alpha$ and $\beta$ are parameters to be estimated. Please show: if you use the AC model with two periods, AKA: $y_t = \alpha y_{t-1} + \beta y_{t-2} + x_t$ $-\beta$ Then, $\alpha = a + b$, and $\beta = \frac{-b}{a+b}$

          2. Considering the following example
$y_t = c_t + i_t$,
(1.1)
$c_t = \alpha y_{t-1} + \epsilon_{ct}$ 
$0 < \alpha < 1$
(1.2)
$i_t = \beta (c_t - c_{t-1}) + \epsilon_{it}$
$\beta > 0$
(1.3)
where $y_t$, $c_t$, and $i_t$ denote real GDP, consumption, and investment in time
period $t$, respectively. In this Keynesian model, $y_t$, $c_t$, and $i_t$ are endogenous
variables. The previous period's GDP and consumption, $y_{t-1}$ and $c_{t-1}$, are
called predetermined or lagged endogenous variables. The terms $\epsilon_{ct}$ and $\epsilon_{it}$
are zero mean random disturbances for consumption and investment, and the
coefficients $\alpha$ and $\beta$ are parameters to be estimated.
Please show: if you use the AC model with two periods, AKA:
$y_t = \alpha y_{t-1} + \beta y_{t-2} + x_t$
$-\beta$
Then, $\alpha = a + b$, and $\beta = \frac{-b}{a+b}$

2. Considering the following example
yt = ct + it,
(1.1)
ct = α yt-1 + ϵct
0 < α < 1
(1.2)
it = β (ct - ct-1) + ϵit
β > 0
(1.3)
where yt, ct, and it denote real GDP, consumption, and investment in time
period t, respectively. In this Keynesian model, yt, ct, and it are endogenous
variables. The previous period's GDP and consumption, yt-1 and ct-1, are
called predetermined or lagged endogenous variables. The terms ϵct and ϵit
are zero mean random disturbances for consumption and investment, and the
coefficients α and β are parameters to be estimated.
Please show: if you use the AC model with two periods, AKA:
yt = α yt-1 + β yt-2 + xt
-β
Then, α = a + b, and β = (-b)/(a+b)

Added by Holly C.

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