Slides 4
For the Final Exam
Homework A
• Budget line stays the same due to unchanged
𝑄 𝑚𝑎𝑥
• Indifference curve stays the same due to
unchanged preference
• Optimum stays the same
• P doubles
• Monetary policy has no effect on optimum
since income and price both double
Intertemporal Budget Constraint (IBC)
• Idea: no one borrows or saves forever
• Before she dies, 𝐶2 = 𝑌2 + (1 + 𝑟)(𝑌1 −𝐶1 )
• Dividing (1 + 𝑟) on both sides gives
𝐶1 +
𝐶2
1+𝑟
=
𝑌2
𝑌1 +
1+𝑟
(IBC)
So what really matters is permanent income, not
current income.
Exercise: what happens if 𝑟 rises?
Preference
• Intertemporal (Total) Utility: 𝑈 𝐶1 , 𝐶2
• Instantaneous (Each Period) Utility:
𝑢 𝐶1 ), 𝑢(𝐶2
• Two properties: 𝑢′ > 0, 𝑢′′ < 0
• For example, 𝑢 𝑐 = log(𝑐)
• 𝑈 𝐶1 , 𝐶2 = 𝑢 𝐶1 ) + 𝛽𝑢(𝐶2 = log 𝐶1
+ 𝛽log(𝐶2 )
Dynamic Optimization
• max 𝑈 𝐶1 , 𝐶2
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐶1 +
𝐶2
1+𝑟
= 𝑌1 +
• Constraint optimization
• Can be transformed to an unconstraint
optimization problem by substituting 𝐶1 = 𝑌1
𝑌2
𝐶2
+
−
into the utility function
1+𝑟
1+𝑟
𝑌2
1+𝑟
• Then take derivative of 𝑈 with respect to 𝐶2 and
let
𝑑𝑈
𝑑𝐶2
=0
Calculus
• Consider 𝑓(𝑥, 𝑦) where 𝑥 𝑎𝑛𝑑 𝑦 are both
functions of 𝑧
• Then
𝑑𝑓
𝑑𝑧
=
𝜕𝑓 𝑑𝑥 𝜕𝑓 𝑑𝑦
+
𝜕𝑥 𝑑𝑧 𝜕𝑦 𝑑𝑧
• For our problem
𝑓 = 𝑈, 𝑥 = 𝐶1 , 𝑦 = 𝐶2 , 𝑧 = 𝐶2
Calculus Two
𝑑𝑈
=
𝑑𝐶2
𝜕𝑈 𝑑𝐶1 𝜕𝑈 𝑑𝐶2
+
𝜕𝐶1 𝑑𝐶2 𝜕𝐶2 𝑑𝐶2
=
𝑑𝑢 𝑑𝐶1 𝛽𝑑𝑢 𝑑𝐶2
+
𝜕𝐶1 𝑑𝐶2 𝑑𝐶2 𝑑𝐶2
=
𝑑𝑢
𝜕𝐶1
−
1
1+𝑟
𝛽𝑑𝑢
+
1
𝑑𝐶2
=0
Euler Equation (First Order Condition)
𝑑𝑢/𝑑𝐶1
= 𝛽(1 + 𝑟)
𝑑𝑢/𝑑𝐶2
This looks similar to
𝑀𝑈𝐴 𝑃𝐴
=
𝑀𝑈𝐵 𝑃𝐵
marginal rate of substitution equals relative
price at optimum.
Interpretation
• 𝑑𝑢/𝑑𝐶1 = 𝛽(1 + 𝑟) 𝑑𝑢/𝑑𝐶2
• So at optimum a person is indifferent between
two options: (I) consume one more unit now;
(II) save that one unit and consume later. At
optimum the two options yield the same
change in utility.
Log Utility Function
• For log utility function 𝑢 𝑐 = log 𝑐 the
Euler equation is
𝐶2
= 𝛽(1 + 𝑟)
𝐶1
or
𝐶2 = 𝛽(1 + 𝑟) 𝐶1
This shows the dynamic path of consumption
Solution of Dynamic Optimization
• Plugging the Euler Equation 𝐶2 = 𝛽(1 + 𝑟)𝐶1 into
the IBC gives
𝛽(1 + 𝑟)𝐶1
𝑌2
𝐶1 +
= 𝑌1 +
1+𝑟
1+𝑟
and we have the solution
𝑊
𝐶1 =
1+𝛽
where 𝑊 = 𝑌1 +
𝑌2
1+𝑟
denotes permanent income.
Exercise
• What if 𝛽 falls?
• What if 𝑟 rises?
• So whether a person saves or borrows in the
first period does not matter. Kind of counterintuitive
Ricardian Equivalence
• Consider the effect of temporary tax cut. The
permanent after-tax income is
𝑌2 − 𝑇2
𝑊 = 𝑌1 − 𝑇1 +
1+𝑟
• Temporary tax cut has no effect on 𝐶1 as long
as the present value of tax stays the same (so
𝑊 stays the same), i.e., as long as a current
tax cut will be followed by a future tax hike.
Smooth Consumption
• Let 𝛽 1 + 𝑟 = 1, then Euler equation implies
that
• 𝐶2 = 𝛽 1 + 𝑟 𝐶1
→ 𝐶2 = 𝐶1
So the optimum is smooth consumption. The
reason is diminishing marginal utility.
Random Walk
• Let’s add randomness 𝑒 to Euler equation
𝐶2 = 𝐶1 + 𝑒
This shows that consumption follows random
walk (RW).
• RW has the property that its change is
unpredictable, i.e.,
𝐸 𝐶2 − 𝐶1 𝐶1 = 𝐸 𝑒 𝐶1 = 0
New-Keynesian Theory I
• Use micro foundation
• IBC assumes no borrowing constraint. If
borrowing constraint (market imperfection)
exists, then for poor people the budget
constraint is
𝐶1 ≤ 𝑌1 (IBC2)
• Now current income matters again. Tax cut
will be very effective in stimulating
expenditure.
New-Keynesian Theory II
• Consider doubling money supply
• But assume prices of some markets are
flexible, others sticky (market imperfection)
• Then 𝑃𝐴 , 𝑃𝐵 𝑎𝑛𝑑 𝑌 will change
disproportionally.
• Then optimum will change since budget line
shifts.
• Monetary policy can be very effective.
Let’s Summarize
• http://www.youtube.com/watch?v=GTQnarz
mTOc
• http://www.youtube.com/watch?v=3u2qRXb4
xCU
History of Macroeconomics
• Classical model (Adam Smith, David Ricardo,
Thomas Malthus)
• Markets clear by themselves
• Key assumption: prices are flexible
• Policies (and government) are not needed
• Policies can create inefficiency, e.g., tax,
minimum wage, tariff
What’s Wrong?
Great Depression
• Widespread and sustained unemployment
(surplus in labor market)
• Signal for market failure
• Classical model cannot explain
• Here comes Keynesian theory which focuses
on insufficient demand (after stock market
crash)
Model of Sticky Prices
• Keynes believes that one reason for
unemployment is sticky nominal wage
• Keynes believes more spending is needed to
raise the price level and lower real wage
• Can you draw a graph?
World War II
• Keynes believes WWII cut short of great
depreciation
• Government expenditures rises
• Through multiplier effect, income rises more
than the increase in G
• So broken window can be good. In long run
we are all dead.
Phillips Curve
• Keynes ignores the role of expectation
• Therefore he believes in a negative relation
between inflation and unemployment, i.e., a
fixed Phillips curve
• Can you draw a graph?
• Then here comes stagflation in 1970s
What’s Wrong?
Stagflation
• In 1970 both unemployment rate and inflation
rate rose
• can be explained by expectation-augmented
Phillips curve
• Lucas critique: econometrics cannot be used
to estimate Phillips curve because we cannot
assume fixed expectation
Micro Foundation
• Hayek and Lucas emphasize the
microeconomic foundation for
macroeconomics, Keynes does not.
• The basic model is a two-period utility
maximization problem
New Classical Theory
•
•
•
•
•
Permanent income hypothesis
Random walk hypothesis for consumption
Ricardian equivalence
Smooth consumption
The new classical theory indicates very small
multiplier effect.
Real Business Cycle Theory
• is built upon the new classical theory
• Emphasizes that markets clear by themselves,
economy can be self-correcting
• Unemployment (intertemporal labor
substitution) is voluntary
• Fluctuation in income (business cycle) is a
Pareto Efficient. Government had better do
nothing.
Dynamic Stochastic General
Equilibrium (DSGE) models
• The simple two-period model can be
generalized
• Multiple periods
• Random variables
• General equilibrium
• To much for this course
New Keynesian Theory
•
•
•
•
•
Keynesian school is fighting back
Micro foundation is used
Market can still fail due to various reasons
Borrowing constraints, sticky prices, etc
So policy is needed to fix market failure
Who are they?
• Classical approach: Friedrich Hayek*, Robert
Lucas*, Robert Barro, Edward Prescott*
• Keynesian approach: John Keynes, Paul
Krugman*, Larry Summers, Ben Bernanke,
Joseph Stiglitz*, Gregory Mankiw
• Extreme Keynesian approach: ?
Yeah. It’s Me!
Marxian Economics
• People are greedy
• Income gap and social instability are inevitable
• Widespread market failure (e.g., no health
insurance market for the poor, great
depression)
• Government needs to do everything (planned
economy)
• Fatal drawback: incentives are ignored
(Most) People are Thin
• in Planned Economy
• http://www.youtube.com/watch?v=UMLtkp4
AFkc
Truth?
•
•
•
•
Somewhere in middle.
Macroeconomics is still a growing baby.
There are many unsettled issues
For example, introduce game theory to
Macro?