Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Question
![Consider the Keynesian consumption function
Yt =B₁ + B₂x2t + εt
(
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<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[EtZ2t] = 0
Darive/ show that the coverance is this:
Cov(x2t, Et)
o² = Var(&t)
=
1
1- B₂
02
1- B₂
(
1
1- B₂ Var (2₂)
-Cov(Z2t, &t) +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc93dd3a-e660-4464-bc8c-e382a2c34aae%2Ff1103d49-53cb-41a3-a6b3-e0fc8733038b%2Fm96k4ur_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the Keynesian consumption function
Yt =B₁ + B₂x2t + εt
(
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<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[EtZ2t] = 0
Darive/ show that the coverance is this:
Cov(x2t, Et)
o² = Var(&t)
=
1
1- B₂
02
1- B₂
(
1
1- B₂ Var (2₂)
-Cov(Z2t, &t) +
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