CHAPTER 5-高宏代写
时间:2023-03-24
CONSUMPTION-SAVINGS
FRAMEWORK
CHAPTER 3
2BASICS
Introduction
Consumption-Savings Framework – provides foundation for
Goods-market demand function (again…but w/different interpretation)
Financial-market supply function
Two time periods
Important: all analysis will be conducted from the perspective of the very
beginning of period 1…
…so a “future” (period 2) for which to save
Dynamic models are the staple of modern macroeconomic analysis
An explicit accounting of time
Two periods are sufficient to illustrate the basic principles
Soon will extend beyond two periods (Chapter 8)
3BASICS
Introduction
Timeline of events
Notation
c1: consumption in period 1
c2: consumption in period 2
P1: nominal price of consumption in period 1
P2: nominal price of consumption in period 2
Y1: nominal income in period 1 (“falls from the sky”)
Y2: nominal income in period 2 (“falls from the sky”)
A0: nominal wealth at the beginning of period 1/end of period 0
A1: nominal wealth at the beginning of period 2/end of period 1
A2: nominal wealth at the beginning of period 3/end of period 2
i: nominal interest rate between periods
r: real interest rate between periods
π2: net inflation rate between period 1 and period 2
y1: real income in period 1 ( = Y1/P1)
y2: real income in period 2 ( = Y2/P2)
2 1 2
2
1 1
1
P P P
P P
Period 1 Period 2
A0 A2Economic events during
period 1: income,
consumption, savings
A1 Economic events during
period 2: income,
consumption, savings
Start of the
world
End of the
worldStart of economic
planning horizon
End of economic
planning horizon
4STOCKS VS. FLOWS
Macro Fundamentals
Stock variables, aka accumulation variables
Quantity variables whose natural measurement occurs at a particular
moment in time
Checking account balance
Credit card indebtedness
Mortgage loan payoff
Flow variables
Quantity variables whose natural measurement occurs over the course
of a given interval of time
Income
Consumption
Savings
The two broad categories of income
Labor income
Asset income (generated by interest rate(s) on (components of)
wealth)
Economic
examples
Economic
examples
Interpret A in our framework as net
wealth ( = total assets – total debts)
All income is a
FLOW
regardless of
source
5BASICS
Macro Fundamentals
Building blocks of consumption-savings framework
Utility
Describes the benefits of engaging in financial market (and other)
activities
Budget constraint
Describes the costs of engaging in financial market (and other)
activities
Utility and budgets two DISTINCT concepts
As in basic consumer analysis (Chapter 1)
Only after describing utility and budgets separately do we bring
the two together to obtain predictions from the framework
6UTILITY
Model Structure
Preferences u(c1, c2) with all the “usual properties”
Lifetime utility function
Strictly increasing in c1
Strictly increasing in c2
Diminishing marginal utility in c1
Diminishing marginal utility in c2
Plotted as indifference curves
Utility side of consumption-savings
framework identical to Chapter 1
framework
c1
u(c1,c2)
c2
u(c1,c2)
c2
c1
7BUDGET CONSTRAINT(S)
Model Structure
Suppose again Y “falls from the sky”
Y1 in period 1, Y2 in period 2
Need two budget constraints to describe economic opportunities
and possibilities
One for each period
Period-1 budget constraint
Period-2 budget constraint
1 1 1 1 0(1 )Pc A Y i A
Total expenditure in period 1:
period-1 consumption +
wealth to carry into period 2
Total income in period 1:
period-1 Y + income from
wealth carried into period 1
(inclusive of interest)
2 2 2 2 1(1 )Pc A Y i A
Total expenditure in period 2:
period-2 consumption +
wealth to carry into period 3
Total income in period 2:
period-2 Y + income from
wealth carried into period 2
(inclusive of interest)
1 1 1 0 1 0Pc A A Y iA
2 2 2 1 2 1Pc A A Y iA
Savings during
period 1 (a flow)
Savings during
period 2 (a flow)
Asset income
during period 1 (a
flow)
Asset income
during period 2 (a
flow)
DEFINITION: A consumer’s savings
during a given period is the change in
his wealth during that period
can rewrite as
can rewrite as
8BUDGET CONSTRAINT(S)
Model Structure
Adopt a lifetime view of the budget constraint(s)
All analysis conducted from perspective of beginning of period 1
Period-1 budget constraint
Period-2 budget constraint
1 1 1 1 0(1 )Pc A Y i A
2 2 2 2 1(1 )Pc A Y i A
9BUDGET CONSTRAINT(S)
Model Structure
Adopt a lifetime view of the budget constraint(s)
All analysis conducted from perspective of beginning of period 1
Period-1 budget constraint
Period-2 budget constraint
1 1 1 1 0(1 )Pc A Y i A
Asset position at end of
period 1/beginning of
period 2 the key link
- will think further about
this soon…2 2 2 2 1(1 )Pc A Y i A
Assume = 0 (no bankruptcies + strictly increasing utility)
10
BUDGET CONSTRAINT(S)
Model Structure
Adopt a lifetime view of the budget constraint(s)
All analysis conducted from perspective of beginning of period 1
Period-1 budget constraint
Period-2 budget constraint
Combine into lifetime budget constraint (LBC)
Solve period-2 budget constraint for A1…
…and substitute into period-1 budget constraint
1 1 1 1 0(1 )Pc A Y i A
Asset position at end of
period 1/beginning of
period 2 the key link
- will think further about
this soon…2 2 2 2 1(1 )Pc A Y i A
2 2 2
1 1 1 0(1 )
1 1
Pc Y
Pc Y i A
i i
Present discounted
value (PDV) of all
lifetime expenditure
Present discounted value (PDV)
of all lifetime income
Assume = 0 (no bankruptcies + strictly increasing utility)
11
BUDGET CONSTRAINT(S)
Model Structure
Adopt a lifetime view of the budget constraint(s)
All analysis conducted from perspective of beginning of period 1
Period-1 budget constraint
Period-2 budget constraint
Combine into lifetime budget constraint (LBC)
Solve period-2 budget constraint for A1…
…and substitute into period-1 budget constraint
1 1 1 1 0(1 )Pc A Y i A
Asset position at end of
period 1/beginning of
period 2 the key link
- will think further about
this soon…2 2 2 2 1(1 )Pc A Y i A
2 2 2
1 1 1 0(1 )
1 1
Pc Y
Pc Y i A
i i
Present discounted
value (PDV) of all
lifetime expenditure
Present discounted value (PDV)
of all lifetime income
For graphical simplicity, will often assume A0 = 0 (i.e., consumer begins planning horizon with zero net wealth).
Note this is a different assumption than A2 = 0.
Assume = 0 (no bankruptcies + strictly increasing utility)
12
LIFETIME BUDGET CONSTRAINT
Model Structure
Graphically
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
Solve for c2
c1
c2
13
LIFETIME BUDGET CONSTRAINT
Model Structure
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
subtract P1c1
2 1 1 2
1
2 2 2
1
1 1
c P Y Y
c
i P P i P
divide by P2
1 1 2
2 1
2 2 2
(1 ) (1 )P i i Y Y
c c
P P P
multiply by (1+i)
14
LIFETIME BUDGET CONSTRAINT
Model Structure
Graphically
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
Solve for c2
c1
c2
2
2 1 1
22 2
1(1 1) Yic c
P i
P
Y
P P
15
LIFETIME BUDGET CONSTRAINT
Model Structure
Graphically
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
Solve for c2
c1
c2
2
2 1 1
22 2
1(1 1) Yic c
P i
P
Y
P P
Rearrange further using
definition of inflation:
2 1
2
1 2 2
1
1
1
P P
P P
2
2 1 1
22 2
1
1
1 Yi
c c Y
P P
i
slope = -(1+i)/(1+π2)
The larger is (1+i)/(1+π2), the
steeper is the budget line
IMPORTANT: Changes in nominal
interest rates (Fed) and/or
inflation affect the slope of the LBC
16
CONSUMER OPTIMIZATION
Model Structure
Consumer’s decision problem: maximize lifetime utility subject to
lifetime budget constraint – bring together both cost side and
benefit side
Choose c1 and c2 subject to
Plot budget line
Superimpose indifference map
At the optimal choice
2 2 2
1 1 1
1 1
Pc Y
Pc Y
i i
c1
c2
slope = -(1+i)/(1+π2)
optimal choice
* *
1 1 2
* *
2 1 2 2
( , ) 1
( , ) 1
u c c i
u c c
CONSUMPTION-SAVINGS
OPTIMALITY CONDITION
- key result in modern macro
analysis
MRS (between
consumption in
consecutive time periods)
price ratio (across
consecutive time
periods)
17
LAGRANGE ANALYSIS
The Mathematics of the Consumption-Savings Model
Apply Lagrange tools to consumption-savings optimization
Objective function: u(c1,c2)
Constraint:
Step 1: Construct Lagrange function
Step 2: Compute first-order conditions with respect to c1, c2, λ
Step 3: Combine (1) and (2) (with focus on eliminating multiplier)
2 2 2
1 2 1 1 1( , ) 0
1 1
Y Pc
g c c Y Pc
i i
2 2 2
1 2 1 2 1 1 1( , , ) ( , )
1 1
Y Pc
L c c u c c Y Pc
i i
* *
1 1 2
* *
2 1 2 2
( , ) 1
( , ) 1
u c c i
u c c
CONSUMPTION-SAVINGS
OPTIMALITY CONDITION
MRS (between
consumption in
consecutive time periods)
price ratio (across
consecutive time
periods)
1 1 2 1( , ) 0u c c P
2
2 1 2( , ) 0
1
P
u c c
i
LBC(1) (2) (3)
18
SAVINGS AND ASSET POSITIONS
Macro Fundamentals
Definition: A consumer’s savings during a given time period is the
change in his wealth during that time period
Assets/wealth (whether positive or negative) are a means for
“transferring income over time”
19
SAVINGS AND ASSET POSITIONS
Macro Fundamentals
Definition: A consumer’s savings during a given time period is the
change in his wealth during that time period
Assets/wealth (whether positive or negative) are a means for
“transferring income over time”
c1
c2
slope = -(1+i)/(1+π2)
Y1/P1
Y2/P2
(continuing to assume A0 = 0)
optimal choice
Optimal c1 > Y1/P1 consumer
has negative wealth at end of
period 1 period-1 savings is
negative (due to A0 = 0)
20
SAVINGS AND ASSET POSITIONS
Macro Fundamentals
Definition: A consumer’s savings during a given time period is the
change in his wealth during that time period
Assets/wealth (whether positive or negative) are a means for
“transferring income over time”
c1
c2
slope = -(1+i)/(1+π2)
Y1/P1
Y2/P2
optimal choice
c1
c2
slope = -(1+i)/(1+π2)
Y1/P1
Y2/P2
optimal choice
OR
Consumer
dissaved
during period 1
Consumer
saved during
period 1
(continuing to assume A0 = 0)
Optimal c1 < Y1/P1 consumer has
positive wealth at end of period 1
period-1 savings is positive (due to
A0 = 0)
Optimal c1 > Y1/P1 consumer
has negative wealth at end of
period 1 period-1 savings is
negative (due to A0 = 0)