Consider the Keynesian consumption function Yt = B₁ + B₂x2t + ε where yt is per capita consumption, and x2+ is per capita income. The coefficient B₂ is interpreted causally as the marginal propensity to consume, and we expect 0

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Consider the Keynesian consumption function
Yt = B₁ + B₂x2t + &t
where yt is per capita consumption, and x2+ is per capita income.
The coefficient ₂ is interpreted causally as the marginal propensity
to consume, and we expect 0<B₂ < 1.
At the same time,
X2t = Yt + Z2t
where z2t denotes per capita investment.
Assume for now that investment is exogenous and
Cov(&t, Z2t) = E[εtZ2t] = 0
Darive/ show that the coverance is this:
Cov(x2t, Et)
o² = Var(&t)
=
=
1
1- B₂
0²
1- B₂
-Cov(Z2t, Et) +
(
(
1
- Var(&t)
1- B₂
Transcribed Image Text:Consider the Keynesian consumption function Yt = B₁ + B₂x2t + &t where yt is per capita consumption, and x2+ is per capita income. The coefficient ₂ is interpreted causally as the marginal propensity to consume, and we expect 0<B₂ < 1. At the same time, X2t = Yt + Z2t where z2t denotes per capita investment. Assume for now that investment is exogenous and Cov(&t, Z2t) = E[εtZ2t] = 0 Darive/ show that the coverance is this: Cov(x2t, Et) o² = Var(&t) = = 1 1- B₂ 0² 1- B₂ -Cov(Z2t, Et) + ( ( 1 - Var(&t) 1- B₂
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