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Prof.
 Marek Weretka's
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Econ 301 Intermediate Microeconomics 
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Problem Set 5
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Problem 1 (Intertemporal Choice)
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Gerald is a CEO in Brainies Consulting Inc.
 His income in the first year is 
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\begin_inset Formula ${\small m}_{{\small1}}{\small=\$200}$
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and in the second 
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\begin_inset Formula ${\small m}_{{\small2}}{\small=\$200}$
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.
 Assume that the interest rate is 
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\begin_inset Formula ${\small r=100\%}$
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: (His time horizon is limited to these two years.)
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a) Find 
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and 
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of Gerald's income.
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b) Show on the graph 
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Gerald's budget set.
 Mark 
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, 
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and the slope of his budget line
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c) Explain what borrowing/lending strategy gives Gerald each of the two
 "extreme" consumption points.
 (how much does he borrow/lend in the first period, how much does he pay
 back/ receive in the second period?)
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d) Euppose his utility function is
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\begin_inset Formula 
\[
{\small U(C}_{1}{\small;C}_{2}{\small)\ =\ \ln C}_{{\small1}}+\ln{\small C}_{{\small2}}
\]

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Fi
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nd analytically Gerald's optimal choice and show it on the graph.
 Does the optimal consumption involve saving or borrowing?
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Problem 2 (Intertemporal Choice)
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The preferences of a manager and a sportsman are given by 
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\begin_inset Formula 
\[
{\small U(C}_{1}{\small;C}_{2}{\small)\ =\ \ln C}_{{\small1}}+\frac{1}{1+\delta}\ln{\small C}_{{\small2}}
\]

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Suppose 
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\begin_inset Formula $r=\delta=100\%$
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.
 
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a) Give economic interpretation to coefficient 
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b) Derive the optimal consumption plan of a manager 
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if his income in two periods is given by 
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\begin_inset Formula ${\small m=(0,3000)}.$
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Find optimal level of savings S in the first period.
 
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c) Derive optimal consumption plan of a sportsman, 
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\begin_inset Formula $\left(C_{1},C_{1}\right)$
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if his income in two periods is given by 
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\begin_inset Formula ${\small m=(1500,0)}.$
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Find optimal level of savings in the first period.
 
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d) Are the manager and the sportsman smoothing their consumption profiles?
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e) Show, using one of the secrets of happiness, that the optimal consumption
 of the manager is decreasing over time when 
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\begin_inset Formula $r<\delta.$
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Problem 3 (Annuity and Perpetuity)
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a) Derive PV formula for perpetuity with payment 
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and interest equal to 
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\begin_inset Formula $r$
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, 
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b) Derive PV formula for annuity that pays 
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maturity is 
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and interest rate is 
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Problem 4 (Present Value, use a calculator)
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a) Buying or renting: You are moving in to Madison (you are going to stay
 here forever!) and you would like to find an apartment.
 You can either buy it or rent it.
 The monthly rent is 
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\begin_inset Formula ${\small\$500}$
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and the monthly interest rate is 
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\begin_inset Formula ${\small0.1\%}$
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.
 Alternatively, you can purchase the apartment, paying 
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\begin_inset Formula ${\small\$600,000}$
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: How are you going to finance your new accommodation? Argue using the 
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\begin_inset Formula ${\small PV}$
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formula.
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b) You take a loan to buy a car that costs 
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\begin_inset Formula ${\small\$4000}$
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.
 What is your monthly payment if you want to pay back the loan after 3 years,
 (36 payments) and the monthly interest rate is 0.5%: (Hint: Loan with constant
 payments is like annuity!)
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c) You are hired by Merrill Lynch --a financial management and advisory
 firm -- to help asses the value of a T-bond (T-bond is a bond issued by
 the Department of the Treasury) with the face value 
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\begin_inset Formula ${\small F=\$1000}$
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; coupon 
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\begin_inset Formula ${\small c=\$100}$
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(paid annually till 
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\begin_inset Formula ${\small T-1}$
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(included) and face value 
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paid in 
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) and time to maturity equal to 
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\begin_inset Formula ${\small T=10}$
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years.
 The interest rate is equal to 
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\begin_inset Formula ${\small r=10\%}$
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.
 Find the PV of such bond.
 Is it a good or bad deal to buy such bond for $900?: Explain
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d) You want to receive 
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\begin_inset Formula ${\small\$40,000}$
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per year when retired (61 and 80).
 How much do you have to save between 21-60 years.
 (Interest rate is 5%) 
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e) You save 
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\begin_inset Formula ${\small\$20,000}$
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per year when working (21 and 60).
 How much will you consume per year when you retire (age 61--80) (Interest
 rate is 5%) 
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Problem 5 (Life cycle problems, use Excel program)
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a) You are going to earn 
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\begin_inset Formula ${\small m}_{t}={\small\$200,000}$
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when working (age 21 and 60), and then you are going to live for the next
 20 years with 
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\begin_inset Formula ${\small m}_{t}={\small\$0}$
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.
 Find the constant level of consumption 
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\begin_inset Formula ${\small C}$
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during years (21-80) that can be financed by your income (Interest rate
 is 5%).
 Find the level of savings 
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\begin_inset Formula ${\small S}_{t}{\small=m}_{t}{\small-C}$
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for periods 
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\begin_inset Formula ${\small t<60}$
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and for 
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b) Consider the problem in a).
 Assume that you inherit 
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\begin_inset Formula ${\small\$1000,000}$
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(one million $) apart from your regular income, at the age of 
\size default

\begin_inset Formula ${\small20}$
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.
 Find 
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 and 
\begin_inset Formula ${\small S}_{t}{\small?}$
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c) Consider the problem in b).
 Assume that you also leave your offspring a bequest of 
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\begin_inset Formula ${\small\$1000,000}$
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(one million $).
 Find 
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and 
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d) Optional: Accumulated wealth in period 
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is given by 
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\begin_inset Formula ${\small W}_{t}{\small=W}_{t-1}{\small(1+r)+s}_{t}$
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\begin_inset space \space{}
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that is it coincides with wealth from the previous period plus interest
 and savings in 
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\begin_inset Formula $t$
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.The agent's initial wealth is 
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\begin_inset Formula ${\small W}_{20}{\small=0}.$
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Using Excel sheet, plot the behavior of wealth 
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\begin_inset Formula ${\small W}_{t}$
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over your life time for points a-c.
 When the agent's wealth attains it's maximum? What can you say about the
 wealth level at the end of life: 
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\begin_inset Formula $W_{80}?$
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Questionnaire Econ 301
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This is an anonymous questionnaire.
 
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Answer the questions but do not sign it.
 The questionnaire will be collected on Tuesday 
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Which year are you in and what is your major?
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I.
 Problem sets:
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1.
 How much time (on average) do you spend solving PS?
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2.
 Are PS: supereasy, easy, OK, difficult, very difficult.
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3.
 Are PS helpful in understanding the material?
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4.
 How difficult math do you need to solve problems? supereasy, easy, OK,
 difficult, very difficult.
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II.
 Class
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1.
 Do you find solving problem in class useful?
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2.
 Do you prefer more intuitive or formal exposition of the material?
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3.
 How about the pace of the course: too slow, slow, OK, fast, too fast.
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4.
 What fraction of the theory presented in the class can you follow? 20%,
 40%, 60%, 80%, 100%
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5.
 How would you improve the course?
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III.
 Midterm
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1.
 Do you think the midterm was fair ?
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2.
 The midterm's difficulty level: Was it supereasy, easy, OK, difficult,
 very difficult?
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3.
 The contents: where you surprised by any of the questions?
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4.
 Length: Was the midterm very short, short, OK, long , too long?
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IV.
 TA sections
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1.
 Do you find them helpful in understanding the material?
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2.
 Do you find them helpful in solving your assignments?
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