Consumer Behavior: Income and Substitution Effects
The Consumer’s Reaction to a Change in Income
Engel Curve or Engel’s Law
The Consumer’s Reaction to a Change in Price
The Consumer’s Demand Function
Cobb-Douglas Utility Function
The Slutsky Substitution Effect
The Hicks substitution effect
Homework 51)a) the IS curve ln Yt= ln Y(t+1) – (1Ɵ)rtso th.docxadampcarr67227
Homework 5
1)
a) the IS curve: ln Yt= ln Y(t+1) – (1/Ɵ)rt
so the slope is: drt/dyt (is) = -Ɵ/Yt. That means that an increase in Ɵ will result in a steeper curve.
LM curve: Mt/Pt = Yt^(Ɵ/v) (1+rt / rt)^(1/v)
Ln(Mt/Pt) = (Ɵ/v) ln Yt +(1/v)ln(1+rt) – (1/v)ln rt.
0 = (Ɵ/v)(1/Yt)dYt + (1/v)(1/(1+rt)) drt – (1/v)(1/rt)drt.
The slope is: drt/dyt (LM) = (Ɵrt(1+rt))/Yt. That means that an increase in Ɵ will result in a steeper curve.
b) the curve IS is not affected by the value of V. while curve LM shifts upwards, since a decrease in v will result in an increase for the demand for real money.
c) IS is not affected byΓ(.)
optimal money holdings: BΓ’(Mt/Pt) = (it/(1+it)) U’(Ct)
B(Mt/Pt)^(-v) = (it/1+it) Yt^-Ɵ
Mt/Pt= B^(1/v) Yt^(Ɵ/v) (1+rt/rt)^(1/v)
So this means that the LM curve will shift downwards.
2)
a) AC= (PC/)+(αYP/2)i
AC/ = -(PC/^2) + (αYP/2)I = 0
C/^2 = αYi/2
So *=(2C/αYi)^(1/2)
b) average real money holdings:M/P= αY/2
M/P = (αY/2) (2C/αYi)^(1/2)
M/P= (αCY/2i)^(1/2)
Ln(m/p) = (1/2)(lnα+lnY+lnC-ln2-lni)
(1/(M/P))((M/P)/i) = -(1/2)(1/i)
Elasticity of real money with respect to i: ((M/P)/)(i/(M/P)) = -1/2
The elasticity with respect to Y : ((M/P)/Y)(Y/(M/P)) = ½
Average real money holdings increase in Y, and decrease in i.
4)
a)when p is at a level that generates maximum output, LS meets LD.
b) when p is above the level that generates maximum output, will cause unemployment.
7)
a)
b)i)
ii)
iii)
13)
a) the asset has an expected rate of return r. capital gain/loss plus dividends per unit time = rvp. There is no dividends per unit time while searching for the palm tree, and there is b probability per unit time of capital gain of (vc-vp)-c. the difference in the price of the asset is(vc-vp) and –c is what the asset pays, so at the end we have rvp=b(vc-vp-c)
b) there is probability aL that a person will find another person with a coconut and trade with that person and gain u̅. the difference in the price of the asset is (vp-vc). So we end up with
rvp=al(vp-vc+u̅).
c) vp=(rvc/aL)+vc-u̅.
r((rvc/aL )+vc-u̅)= b(vc-(rvc/aL)-vc+u̅-c)
vc(r(r+aL+b))/aL = u̅(r+b)-bc
the value of being in state C: vc= (aL(u̅(r+b)-bc)) / r(r+aL+b)
the value of being in state p: vp= ((u̅(r+b)-bc)/(r+aL+b)) + (aL(u̅(r+b)-bc)/r(r+aL+b)) - u̅
so finally
vc-vp = (bc+u̅aL)/(r+aL+b).
e) vc-vp ≥c
vc-vp = (bc+u̅a(b/a))/(r+a(b/a)+b) = (bc+bu̅)/(r+2b)
(bc+bu̅)/(r+2b) ≥ c
That means that
Bc+bu̅≥c and c(r+2b-b) ≤ bu̅
So finally we have
c≤ bu̅ / (r+b).
f) it is a steady-state equilibrium for no one who finds a tree to climb it for any value of c>0.
Yes there are values of c which there is more than one steady-state equilibrium for 0<c< bu̅/(r+b)
Yes, L = b/a has a higher welfare than L=0. When L=0 people don’t gain any utility since they don’t climb a tree and don’t have a chance to trade with other people and gain a coconut.
0 1 2 3 4 5 -3 -2.2000000000000002 -1.8 -1.8 -2.2000000000000002 -3
0 1 2 3 4 5 7 6.5 5.5 3.5 1
0 1 2 3 4 -2 -2.5 -3.5 -5.5 -8
LD.
Consumer Behavior: Income and Substitution Effects
The Consumer’s Reaction to a Change in Income
Engel Curve or Engel’s Law
The Consumer’s Reaction to a Change in Price
The Consumer’s Demand Function
Cobb-Douglas Utility Function
The Slutsky Substitution Effect
The Hicks substitution effect
Homework 51)a) the IS curve ln Yt= ln Y(t+1) – (1Ɵ)rtso th.docxadampcarr67227
Homework 5
1)
a) the IS curve: ln Yt= ln Y(t+1) – (1/Ɵ)rt
so the slope is: drt/dyt (is) = -Ɵ/Yt. That means that an increase in Ɵ will result in a steeper curve.
LM curve: Mt/Pt = Yt^(Ɵ/v) (1+rt / rt)^(1/v)
Ln(Mt/Pt) = (Ɵ/v) ln Yt +(1/v)ln(1+rt) – (1/v)ln rt.
0 = (Ɵ/v)(1/Yt)dYt + (1/v)(1/(1+rt)) drt – (1/v)(1/rt)drt.
The slope is: drt/dyt (LM) = (Ɵrt(1+rt))/Yt. That means that an increase in Ɵ will result in a steeper curve.
b) the curve IS is not affected by the value of V. while curve LM shifts upwards, since a decrease in v will result in an increase for the demand for real money.
c) IS is not affected byΓ(.)
optimal money holdings: BΓ’(Mt/Pt) = (it/(1+it)) U’(Ct)
B(Mt/Pt)^(-v) = (it/1+it) Yt^-Ɵ
Mt/Pt= B^(1/v) Yt^(Ɵ/v) (1+rt/rt)^(1/v)
So this means that the LM curve will shift downwards.
2)
a) AC= (PC/)+(αYP/2)i
AC/ = -(PC/^2) + (αYP/2)I = 0
C/^2 = αYi/2
So *=(2C/αYi)^(1/2)
b) average real money holdings:M/P= αY/2
M/P = (αY/2) (2C/αYi)^(1/2)
M/P= (αCY/2i)^(1/2)
Ln(m/p) = (1/2)(lnα+lnY+lnC-ln2-lni)
(1/(M/P))((M/P)/i) = -(1/2)(1/i)
Elasticity of real money with respect to i: ((M/P)/)(i/(M/P)) = -1/2
The elasticity with respect to Y : ((M/P)/Y)(Y/(M/P)) = ½
Average real money holdings increase in Y, and decrease in i.
4)
a)when p is at a level that generates maximum output, LS meets LD.
b) when p is above the level that generates maximum output, will cause unemployment.
7)
a)
b)i)
ii)
iii)
13)
a) the asset has an expected rate of return r. capital gain/loss plus dividends per unit time = rvp. There is no dividends per unit time while searching for the palm tree, and there is b probability per unit time of capital gain of (vc-vp)-c. the difference in the price of the asset is(vc-vp) and –c is what the asset pays, so at the end we have rvp=b(vc-vp-c)
b) there is probability aL that a person will find another person with a coconut and trade with that person and gain u̅. the difference in the price of the asset is (vp-vc). So we end up with
rvp=al(vp-vc+u̅).
c) vp=(rvc/aL)+vc-u̅.
r((rvc/aL )+vc-u̅)= b(vc-(rvc/aL)-vc+u̅-c)
vc(r(r+aL+b))/aL = u̅(r+b)-bc
the value of being in state C: vc= (aL(u̅(r+b)-bc)) / r(r+aL+b)
the value of being in state p: vp= ((u̅(r+b)-bc)/(r+aL+b)) + (aL(u̅(r+b)-bc)/r(r+aL+b)) - u̅
so finally
vc-vp = (bc+u̅aL)/(r+aL+b).
e) vc-vp ≥c
vc-vp = (bc+u̅a(b/a))/(r+a(b/a)+b) = (bc+bu̅)/(r+2b)
(bc+bu̅)/(r+2b) ≥ c
That means that
Bc+bu̅≥c and c(r+2b-b) ≤ bu̅
So finally we have
c≤ bu̅ / (r+b).
f) it is a steady-state equilibrium for no one who finds a tree to climb it for any value of c>0.
Yes there are values of c which there is more than one steady-state equilibrium for 0<c< bu̅/(r+b)
Yes, L = b/a has a higher welfare than L=0. When L=0 people don’t gain any utility since they don’t climb a tree and don’t have a chance to trade with other people and gain a coconut.
0 1 2 3 4 5 -3 -2.2000000000000002 -1.8 -1.8 -2.2000000000000002 -3
0 1 2 3 4 5 7 6.5 5.5 3.5 1
0 1 2 3 4 -2 -2.5 -3.5 -5.5 -8
LD.
Advanced macroeconomics, 4th edition. Romer.
Chapter12.
12.1. The stability of fiscal policy. (Blinder and Solow, 1973.) By definition, the budget deficit equals the rate of change of the amount of debt outstanding: δ(t) ≡ D ̇(t). Define d(t) to be the ratio of debt to output: d(t) = D(t)/Y(t). Assume that Y(t) grows at a constant rate g > 0.
(a) Suppose that the deficit-to-output ratio is constant: δ(t)/Y(t) = a, where a > 0.
̇
(i) Find an expression for d(t) in terms of a, g, and d(t). ̇
(ii) Sketch d(t) as a function of d(t). Is this system stable?
(b) Suppose that the ratio of the primary deficit to output is constant and equal to a > 0. Thus the total deficit at t, δ(t), is given by δ(t) = aY(t) + r(t)D(t), where r(t) is the interest rate at t. Assume that r is an increasing function of the debt-to-output ratio: r(t) = r(d(t)), where r′(•) > 0, r′′(•) > 0, limd→−∞ r(d) < g, limd→∞ r(d) > g.
̇
(i) Find an expression for d(t) in terms of a, g, and d(t). ̇
(ii) Sketch d(t) as a function of d(t). In the case where a is sufficiently small that d ̇ is negative for some values of d, what are the stability properties of the system? What about the case where a is sufficiently large that d ̇ is positive for all values of d ?
12.2. Precautionary saving, non-lump-sum taxation, and Ricardian equivalence.
(Leland, 1968, and Barsky, Mankiw, and Zeldes, 1986.) Consider an individual who lives for two periods. The individual has no initial wealth and earns labor incomes of amounts Y1 and Y2 in the two periods. Y1 is known, but Y2 is random; assume for simplicity that E[Y2] = Y1. The government taxes income at rate τ1 in period 1 and τ2 in period 2. The individual can borrow and lend at a fixed interest rate, which for simplicity is assumed to be zero. Thus second-period consumption is C2 = (1 − τ1)Y1 − C1 + (1 − τ2)Y2. The individualchoosesC1 tomaximizeexpectedlifetimeutility,U(C1)+E[U(C2)].
(a) Find the first-order condition for C1.
(b) Show that E[C2] = C1 if Y2 is not random or if utility is quadratic.
(c) Show that if U ′′′(•) > 0 and Y2 is random, E[C2] > C1.
(d) Suppose that the government marginally lowers τ1 and raises τ2 by the same amount, so that its expected total revenue, τ1Y1 + τ2E[Y2], is un- changed. Implicitly differentiate the first-order condition in part (a) to find an expression for how C1 responds to this change.
(e) Show that C1 is unaffected by this change if Y2 is not random or if utility is quadratic.
(f) Show that C1 increases in response to this change if U ′′′(•) > 0 and Y2 is random.
12.3
Consider the Barro tax-smoothing model. Suppose that output, Y, and the real interest rate, r, are constant, and that the level of government debt out- standing at time 0 is zero. Suppose that there will be a temporary war from time 0 to time τ. Thus G(t) equals GH for 0 ≤ t ≤ τ, and equals GL there- after,whereGH >GL.Whatarethepathsoftaxes,T(t),andgovernmentdebt outstanding, D(t)?
12.4
Consider the Barro tax-smoothing model. Supp.
Problem Set 4Due Tuesday, April 12ECON 434 Internati.docxsleeperharwell
Problem Set 4
Due: Tuesday, April 12
ECON 434: International Finance and Macroeconomics
Penn State: Spring, 2016
1. Government Budget Constraints. This problem looks at the feasibility constraints facing the
government and the sustainability of current-account balances. This will generalize some of the results
we obtained in class.
__ Consider an economy that lasts for N periods. In period t, the government purchases Gt dollars
worth of goods, and it collects Tt dollars worth of taxes. The government can also purchase B
g
t bonds
in period t; if it holds B
g
t bonds in period t, then it receives (1 + r)B
g
t bonds in period t + 1. (For
simplicity, assume that the interest rate r is constant.) If B
g
t < 0, then it means the government is in
debt.
(a) What is the government's period-t budget constraint? (Your answer should contain Gt, Tt, B
g
t ,
and r.)
(b) How do you compute the primary �scal de�cit in period t? How do you compute the secondary
�scal de�cit in period t?
(c) Combine the period budget constraints for t = 1, . . . ,N to show that:
B
g
0 +
N∑
t=1
Tt
(1 + r)
t
=
N∑
t=1
Gt
(1 + r)
t
+
B
g
N
(1 + r)
N
. (1)
(d) Suppose that N is a �xed, �nite number. What condition does B
g
N have to satisfy, and why?
(e) Individuals who pay taxes have �nite lives, but institutions, such as governments, can live a long
time, possibly forever. This leads to the possibility that a government could, in principle, keep
rolling over its debt, even as generations of citizens come and go.1 Mathematically, we model this
by letting N →∞. What condition does B
g
N
(1+r)N
have to satisfy as N →∞? Explain your answer
in words.
(f) Suppose that the government starts out in debt, with B
g
0 < 0. Is it possible for the government
to run primary de�cits forever? Why or why not?
(g) Now, suppose that the government starts out with positive assets B
g
0 > 0. We'll look at the case of
an in�nitely-lived government (N = ∞), and we'll assess whether it's possible for the government
to make purchases while never taxing its citizens (i.e., Tt = 0, for t = 1,2, . . .).
__ Consider the following plan. Before making purchases in period t, the government has
(1 + r)B
g
t−1 dollars at its disposal from the interest it earned on its previous period's assets.
The government decides to take a fraction δ of this money to spend on period-t government
purchases Gt; the remaining fraction 1−δ is used to buy more bonds.
i. Provide an expression for B
g
t in terms of B
g
t−1, and provide an expression for Gt in terms of
B
g
t−1. (Both expressions will depend on δ and r.)
ii. Provide an expression for B
g
t in terms of B
g
0 and t, and provide an expression for Gt in terms
of B
g
0 and t. (Both expressions will depend on δ and r.)
1For an example, see �The Case of the Undying Debt� by François Velde: https://www.chicagofed.org/publications/working-
papers/2009/wp-12.
1
iii. In each period t, does the government run a primary surplus or de�cit.
This slide set is a work in progress and is embedded in my Principles of Finance course that I teach to computer scientists and engineers.
http://financefortechies.weebly.com/
1- Consider the model of liquidity and financial intermediation presen.docxfitzlikv
1. Consider the model of liquidity and financial intermediation presented in class. The economy has an infinite horizon and is populated by a large number of individuals who each live for three periods. In period t there are N t individuals born where N t = n N t 1 and n 1 . Individuals receive utility from consuming in middle age and when old. That is, individuals have utility function U ( c 2 , t + 1 , c 3 , t + 2 ) = ln c 2 , t + 1 + ln c 3 , t + 2 where ( 0 , 1 ) . For simplicity you can assume that n = 1 and = 1 . Individuals have one unit of time that is supplied inelastically as labor, but possess different productivities denoted by [ 0 , 1 ] . An individual with productivity produces units of the consumption good when young. Assume that the productivity of the young generation is uniformly distributed over the interval [ 0 , 1 ] in every period. There is a risk-neutral bank takes deposits and commits to a one-period return denoted by r d , so that one unit of the consumption good deposited with the bank yields the gross return r d units of the consumption good in t + 1 . When an individual withdraws from the bank, they pay a fixed cost > 0 , measured in units of the consumption good. This fixed cost has many interpretations including the annual fee on a deposit account or the time required to go to a bank (or bank machine) to withdraw (in fact during COVID-19 you could also consider the risk associated with making an in-person withdrawal as a component of this fixed cost). Individuals have three stores of value available; fiat money ( m ) , deposits ( d ) , and capital ( k ) . There is a supply of fiat money available in period t denoted by M t , where M t = z M t 1 for all t where z 1 . Capital has a two period return X > n 2 and a one-period return x = X 1/2 , however, unmatured capital cannot be sold, traded or borrowed against. For parts (a) - (d) you can assume that the real rate of return on fiat money in a stationary equilibrium is n = 1 , and the return on deposits satisfies r d ( n , x ) , where x = X 1/2 . Further for parts (a) - (e) you can assume that the money supply is constant so that z = 1 and M t = M for all t . Please answer the following questions and assume a stationary equilibrium (i.e. c 2 , t ( ) = c 2 ( ) for all t and c 3 , t ( ) = c 3 ( ) for all t ) . (a) Write down the utility maximization problem of a type worker including all budget constraints. Explain how individuals in the economy will use money, deposits and capital. (b) Write down the Lagrangian and solve for the optimal choices of capital, deposits, real money and consumption for any type individual. If it is easier, you an solve the problem by considering a type w worker and doing the following: (i) Assume the bank does not exist so that individuals do not make deposits and can only use fiat money and capital. Solve for the equilibrium levels of consumption ( c 2 , t + 1 and c 3 , t + 2 ) , capital ( k t ) , and individual real money demand ( t m t.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
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Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
2. Intertemporal Choice
Persons often receive income in
“lumps”; e.g. monthly salary.
How is a lump of income spread over
the following month (saving now for
consumption later)?
Or how is consumption financed by
borrowing now against income to be
received at the end of the month?
3. Present and Future Values
Begin with some simple financial
arithmetic.
Take just two periods; 1 and 2.
Let r denote the interest rate per
period.
4. Future Value
E.g., if r = 0.1 then $100 saved at the
start of period 1 becomes $110 at the
start of period 2.
The value next period of $1 saved
now is the future value of that dollar.
5. Future Value
Given an interest rate r the future
value one period from now of $1 is
FV = 1 + r .
Given an interest rate r the future
value one period from now of $m is
FV = m(1 + r ).
6. Present Value
Suppose you can pay now to obtain
$1 at the start of next period.
What is the most you should pay?
$1?
No. If you kept your $1 now and
saved it then at the start of next
period you would have $(1+r) > $1,
so paying $1 now for $1 next period
is a bad deal.
7. Present Value
Q: How much money would have to be
saved now, in the present, to obtain
$1 at the start of the next period?
A: $m saved now becomes $m(1+r) at
the start of next period, so we want
the value of m for which
m(1+r) = 1
That is, m = 1/(1+r),
the present-value of $1 obtained at the
start of next period.
8. Present Value
The present value of $1 available at
the start of the next period is
1
PV =
.
1+r
And the present value of $m
available at the start of the next
period is
m
PV =
1+r
.
9. Present Value
E.g., if r = 0.1 then the most you
should pay now for $1 available next
period is
1
PV =
1+ 0⋅1
= $0 ⋅ 91.
And if r = 0.2 then the most you
should pay now for $1 available next
period is
1
PV =
1+ 0⋅ 2
= $0 ⋅ 83.
10. The Intertemporal Choice Problem
Let m1 and m2 be incomes received in
periods 1 and 2.
Let c1 and c2 be consumptions in
periods 1 and 2.
Let p1 and p2 be the prices of
consumption in periods 1 and 2.
11. The Intertemporal Choice Problem
The intertemporal choice problem:
Given incomes m1 and m2, and given
consumption prices p1 and p2, what is
the most preferred intertemporal
consumption bundle (c1, c2)?
For an answer we need to know:
– the intertemporal budget constraint
– intertemporal consumption
preferences.
12. The Intertemporal Budget Constraint
To start, let’s ignore price effects by
supposing that
p1 = p2 = $1.
13. The Intertemporal Budget Constraint
Suppose that the consumer chooses
not to save or to borrow.
Q: What will be consumed in period 1?
A: c1 = m1.
Q: What will be consumed in period 2?
A: c2 = m2.
15. The Intertemporal Budget Constraint
c2
So (c1, c2) = (m1, m2) is the
consumption bundle if the
consumer chooses neither to
save nor to borrow.
m2
0
0
m1
c1
16. The Intertemporal Budget Constraint
Now suppose that the consumer
spends nothing on consumption in
period 1; that is, c1 = 0 and the
consumer saves
s 1 = m 1.
The interest rate is r.
What now will be period 2’s
consumption level?
17. The Intertemporal Budget Constraint
Period 2 income is m2.
Savings plus interest from period 1
sum to
(1 + r )m1.
So total income available in period 2 is
m2 + (1 + r )m1.
So period 2 consumption expenditure
is
18. The Intertemporal Budget Constraint
Period 2 income is m2.
Savings plus interest from period 1
sum to
(1 + r )m1.
So total income available in period 2 is
m2 + (1 + r )m1.
So period 2 consumption expenditure
c 2 = m2 + ( 1 + r )m1
is
19. The Intertemporal Budget Constraint
m2 +
c2
(1 + r )m1
the future-value of the income
endowment
m2
0
0
m1
c1
20. The Intertemporal Budget Constraint
c2
( c1 , c 2 ) = ( 0, m2 + ( 1 + r )m1 )
m +
2
(1 + r )m1
is the consumption bundle when all
period 1 income is saved.
m2
0
0
m1
c1
21. The Intertemporal Budget Constraint
Now suppose that the consumer
spends everything possible on
consumption in period 1, so c2 = 0.
What is the most that the consumer
can borrow in period 1 against her
period 2 income of $m2?
Let b1 denote the amount borrowed
in period 1.
22. The Intertemporal Budget Constraint
Only $m2 will be available in period 2
to pay back $b1 borrowed in period 1.
So
b1(1 + r ) = m2.
That is,
b1 = m2 / (1 + r ).
So the largest possible period 1
consumption level is
23. The Intertemporal Budget Constraint
Only $m2 will be available in period 2
to pay back $b1 borrowed in period 1.
So
b1(1 + r ) = m2.
That is,
b1 = m2 / (1 + r ).
So the largest possible period 1
consumption level is m
c1 = m1 +
2
1+r
24. The Intertemporal Budget Constraint
c2
( c1 , c 2 ) = ( 0, m2 + ( 1 + r )m1 )
m +
2
(1 + r )m1
is the consumption bundle when all
period 1 income is saved.
m2
the present-value of
the income endowment
0
0
m1
m2 c1
m1 +
1+r
25. The Intertemporal Budget Constraint
c2
( c1 , c 2 ) = ( 0, m2 + ( 1 + r )m1 )
m +
2
(1 + r )m1
m2
0
0
is the consumption bundle when
period 1 saving is as large as possible.
m + m2 ,0
( c1 , c 2 ) = 1
1+r
is the consumption bundle
when period 1 borrowing
is as big as possible.
m2 c1
m1
m1 +
1+r
26. The Intertemporal Budget Constraint
Suppose that c1 units are consumed
in period 1. This costs $c1 and
leaves m1- c1 saved. Period 2
consumption will then be
c 2 = m2 + (1 + r )(m1 − c1 )
27. The Intertemporal Budget Constraint
Suppose that c1 units are consumed
in period 1. This costs $c1 and
leaves m1- c1 saved. Period 2
consumption will then be
c 2 = m2 + (1 + r )(m1 − c1 )
which is
slope
intercept
c 2 = −(1 + r )c1 + m2 + (1 + r )m1 .
28. The Intertemporal Budget Constraint
c2
( c1 , c 2 ) = ( 0, m2 + ( 1 + r )m1 )
m +
2
(1 + r )m1
m2
0
0
is the consumption bundle when
period 1 saving is as large as possible.
m + m2 ,0
( c1 , c 2 ) = 1
1+r
is the consumption bundle
when period 1 borrowing
is as big as possible.
m2 c1
m1
m1 +
1+r
29. The Intertemporal Budget Constraint
m2 +
c2
c 2 = −(1 + r )c1 + m2 + (1 + r )m1 .
(1 + r)m1
slope = -(1+r)
m2
0
0
m1
m2 c1
m1 +
1+r
30. The Intertemporal Budget Constraint
m2 +
c2
c 2 = −(1 + r )c1 + m2 + (1 + r )m1 .
(1 + r)m1
m2
0
0
Sa
vi
slope = -(1+r)
ng
Bo
rro
wi
m1
ng
m2 c1
m1 +
1+r
31. The Intertemporal Budget Constraint
(1 + r )c1 + c 2 = (1 + r )m1 + m2
is the “future-valued” form of the budget
constraint since all terms are in period 2
values. This is equivalent to
c2
m2
c1 +
= m1 +
1+r
1+r
which is the “present-valued” form of the
constraint since all terms are in period 1
values.
32. The Intertemporal Budget Constraint
Now let’s add prices p1 and p2 for
consumption in periods 1 and 2.
How does this affect the budget
constraint?
33. Intertemporal Choice
Given her endowment (m1,m2) and
prices p1, p2 what intertemporal
consumption bundle (c1*,c2*) will be
chosen by the consumer?
Maximum possible expenditure in
m2 + (1 + r )m1
period 2 is
so maximum possible consumption
m2 + (1 + r )m1
in period 2 is c 2 =
.
p2
34. Intertemporal Choice
Similarly, maximum possible
expenditure in period 1 is
m2
m1 +
1+r
so maximum possible consumption
in period 1 is
m1 + m2 / ( 1 + r )
c1 =
.
p1
35. Intertemporal Choice
Finally, if c1 units are consumed in
period 1 then the consumer spends
p1c1 in period 1, leaving m1 - p1c1 saved
for period 1. Available income in
period 2 will then be
m2 + (1 + r )(m1 − p1c1 )
so
p 2c 2 = m2 + (1 + r )(m1 − p1c1 ).
36. Intertemporal Choice
p 2c 2 = m2 + ( 1 + r )( m1 − p1c1 )
rearranged is
( 1 + r )p1c1 + p 2c 2 = (1 + r )m1 + m2 .
This is the “future-valued” form of the
budget constraint since all terms are
expressed in period 2 values. Equivalent
to it is the “present-valued” form
p2
m2
p1c1 +
c 2 = m1 +
1 +r
1 +r
where all terms are expressed in period 1
values.
39. The Intertemporal Budget Constraint
c2
( 1 + r )m1 + m2
p2
m2/p2
0
0
m1/p1
c1
m1 + m2 / ( 1 + r )
p1
40. The Intertemporal Budget Constraint
c2 (1 + r )p1c1 + p 2c 2 = (1 + r )m1 + m2
( 1 + r )m1 + m2
p2
p1
Slope = − (1 + r )
p2
m2/p2
0
0
m1/p1
c1
m1 + m2 / ( 1 + r )
p1
41. The Intertemporal Budget Constraint
c2 (1 + r )p1c1 + p 2c 2 = (1 + r )m1 + m2
( 1 + r )m1 + m2
p2
Sa
vi
Bo
rro
wi
m2/p2
0
ng
p1
Slope = − (1 + r )
p2
0
m1/p1
ng
c1
m1 + m2 / ( 1 + r )
p1
42. Price Inflation
Define the inflation rate by π where
p1 (1 + π ) = p 2 .
For example,
π = 0.2 means 20% inflation, and
π = 1.0 means 100% inflation.
43. Price Inflation
We lose nothing by setting p1=1 so
that p2 = 1+ π .
Then we can rewrite the budget
constraint p
m
p1c1 +
as
2 c =m +
2
1
2
1 +r
1 +r
1+π
m2
c1 +
c 2 = m1 +
1 +r
1 +r
44. Price Inflation
1+π
m2
c1 +
c 2 = m1 +
1 +r
1 +r
rearranges to
1+r
m1 + m
c2 = −
c1 + (1 + π)
2
1+ π
1 + r
so the slope of the intertemporal budget
constraint is
1+r
−
.
1+ π
45. Price Inflation
When there was no price inflation
(p1=p2=1) the slope of the budget
constraint was -(1+r).
Now, with price inflation, the slope of
the budget constraint is -(1+r)/(1+ π ).
This can be written as
1+r
− (1 + ρ ) = −
1+ π
ρ is known as the real interest rate.
46. Real Interest Rate
1+r
− (1 + ρ ) = −
1+ π
gives
r −π
ρ=
.
1+ π
For low inflation rates (π ≈ 0), ρ ≈ r - π .
For higher inflation rates this
approximation becomes poor.
47. Real Interest Rate
r
0.30 0.30 0.30 0.30 0.30
π
0.0
0.05 0.10 0.20 1.00
r - π 0.30 0.25 0.20 0.10 -0.70
ρ
0.30 0.24 0.18 0.08 -0.35
48. Comparative Statics
The slope of the budget constraint is
1+ r
− (1 + ρ) = −
.
1+ π
The constraint becomes flatter if the
interest rate r falls or the inflation
rate π rises (both decrease the real
rate of interest).
52. Comparative Statics
1+r
slope = − (1 + ρ ) = −
1+ π
c2
The consumer saves. An
increase in the inflation
rate or a decrease in
the interest rate
“flattens” the
budget
constraint.
m2/p2
0
0
m1/p1
c1
53. Comparative Statics
1+r
slope = − (1 + ρ ) = −
1+ π
c2
If the consumer saves then
saving and welfare are
reduced by a lower
interest rate or a
higher inflation
rate.
m2/p2
0
0
m1/p1
c1
57. Comparative Statics
1+r
slope = − (1 + ρ ) = −
1+ π
c2
The consumer borrows. A
fall in the inflation rate or
a rise in the interest rate
“flattens” the
budget constraint.
m2/p2
0
0
m1/p1
c1
58. Comparative Statics
1+r
slope = − (1 + ρ ) = −
1+ π
c2
If the consumer borrows then
borrowing and welfare are
increased by a lower
interest rate or a
higher inflation
rate.
m2/p2
0
0
m1/p1
c1
59. Valuing Securities
A financial security is a financial
instrument that promises to deliver
an income stream.
E.g.; a security that pays
$m1 at the end of year 1,
$m2 at the end of year 2, and
$m3 at the end of year 3.
What is the most that should be paid
now for this security?
60. Valuing Securities
The security is equivalent to the sum
of three securities;
– the first pays only $m1 at the end of
year 1,
– the second pays only $m2 at the
end of year 2, and
– the third pays only $m3 at the end
of year 3.
61. Valuing Securities
The PV of $m1 paid 1 year from now is
m1 / (1 + r )
The PV of $m2 paid 2 years from now is
2
m2 / ( 1 + r )
The PV of $m3 paid 3 years from now is
3
m3 / ( 1 + r )
The PV of the security is therefore
m1 / (1 + r ) + m2 / (1 + r ) 2 + m3 / (1 + r ) 3 .
62. Valuing Bonds
A bond is a special type of security
that pays a fixed amount $x for T
years (its maturity date) and then
pays its face value $F.
What is the most that should now be
paid for such a bond?
64. Valuing Bonds
Suppose you win a State lottery. The
prize is $1,000,000 but it is paid over
10 years in equal installments of
$100,000 each. What is the prize
actually worth?
68. Valuing Consols
x
x
x
PV =
+
+
+
1 + r (1 + r ) 2 (1 + r ) 3
1
x
x
=
+
+
x +
1+r
1 + r (1 + r ) 2
1
=
[ x + PV] .
1+r
Solving for PV gives
x
PV = .
r
69. Valuing Consols
E.g. if r = 0.1 now and forever then the
most that should be paid now for a
console that provides $1000 per year is
x $1000
PV = =
= $10,000.
r
0⋅1