Department of Economics ECO 204 2016-2017 …...ECO 204, 2016-2017 (AJAZ), Test 2` This test is copyright material and cannot be used for commercial purposes or posted anywhere without

ECO 204, 2016-2017 (AJAZ), Test 2`

This test is copyright material and cannot be used for commercial purposes or posted anywhere without prior permission. Report violations to [email protected]

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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)

∎ Department of Economics ∎ ECO 204 ∎ Microeconomic Theory for Commerce ∎ 2016-2017 (Ajaz)

Test 2 Solutions

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EXAM DETAILS

Duration: 2 hours

Total number of questions: 4

Total number of pages: 21 (including title page)

Total number of points: 105 + 5 Bonus Points

Please answer all questions. To earn credit you must show all calculations.

This is a closed note and closed book exam.

You may use a non-programmable calculator. Sharing is not allowed.

. KEEP YOUR ANSWERS AS BRIEF AS POSSIBLE AND SHOW ALL NECESSARY CALCULATIONS

GOOD LUCK!

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QUESTION 1 [TOTAL 20 Points] (a) [5 Points] The following table contains some data on Delta Airline’s and Boeing’s “monthly stock price = 𝑃”, “monthly returns = 𝑅𝐸𝑇”, and “monthly returns without dividends = 𝑅𝐸𝑇𝑋”:

DATE DELTA P DELTA RET DELTA RETX BOEING P BOEING RET BOEING RETX

10/30/2015 $ 50.84 0.133051 0.133051 $ 148.07 0.130737 0.130737

11/30/2015 $ 46.46 -0.083497 -0.086153 $ 145.45 -0.011549 -0.017694

12/31/2015 $ 50.69 0.091046 0.091046 $ 144.59 -0.005913 -0.005913

Source: CRSP through CHASS @ U of Toronto

Calculate Delta and Boeing’s dividend per share (if any) on 11/30/2015 and 12/31/2015. State all necessary assumptions, show all essential steps, and state the final answer up to two decimal places. Answer From the definition of returns:

Returns = Capital Gains + Dividend Yield

𝑅𝐸𝑇 = 𝑅𝐸𝑇𝑋 + Dividend Yield

Dividend Yield = 𝑅𝐸𝑇 − 𝑅𝐸𝑇𝑋

Dividend at time 𝑡

Price at time 𝑡 − 1= 𝑅𝐸𝑇𝑡 − 𝑅𝐸𝑇𝑋𝑡

Dividend Payment at time 𝑡 = (𝑅𝐸𝑇𝑡 − 𝑅𝐸𝑇𝑋𝑡) × Price at time 𝑡 − 1

Or:

Dividend Payment at time 𝑡 = (Dividend Yield at time 𝑡) × Price at time 𝑡 − 1 To see if there was a dividend payment on 11/30/2015 we would calculate:

Dividend Payment on 11/30/2015 = (𝑅𝐸𝑇11/30/2015 − 𝑅𝐸𝑇𝑋11/30/2015) × Price on 10/30/2015

Using the same procedure to see if a dividend was issued on 12/31/2015 we obtain:

DATE DELTA P DELTA RET DELTA RETX DELTA DIV

YIELD DELTA DIV BOEING P BOEING RET

BOEING RETX

BOEING DIV YIELD

BOEING DIV

10/30/2015 $ 50.84 0.133051 0.133051 0 $ - $ 148.07 0.130737 0.130737 0 $ -

11/30/2015 $ 46.46 -0.083497 -0.086153 0.002656 $ 0.14 $ 145.45 -0.011549 -0.017694 0.006145 $ 0.91

12/31/2015 $ 50.69 0.091046 0.091046 0 $ - $ 144.59 -0.005913 -0.005913 0 $ -

On 11/30/2015, Delta issued a dividend of $0.14/share while Boeing issued a dividend of $0.91/share. Neither company issued a dividend on 12/30/2015.


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(b) [5 Points] An investor has collected data from 6/29/2007 through 12/31/2015 on the monthly returns of Delta Airlines, Boeing, and the S&P 500 Index, as well as data on the monthly rates of US T-Bills issued on the first day of the next month (notice the missing value below):

DATE DELTA RET BOEING RET SP500 RET Monthly rate of T- Bill on 1st day of next month

6/29/2007 0.034121 -0.04404 -0.017816 0.0041333

7/31/2007 -0.095432 0.075603 -0.031982 0.0036000

… … … … …

… … … … …

11/30/2015 -0.083497 -0.011549 0.000505 0.0001917

12/31/2015 0.091046 -0.005913 -0.01753 Missing: Monthly rate of T-Bill issued on 1/1/2016

Source: CRSP through CHASS @ U of Toronto and FRED

The annualized rate of a one-month T-Bill issued on 1/1/2016 was 0.260%. What was the monthly rate of a T-Bill issued on 1/1/2016? Show all essential steps, state the final answer up to six decimal places, and use this value in parts below. Answer

Since 0.260% is in annualized terms, we divide by 12 to get the monthly rate in % terms: 0.260

12= 0.0217%. Next, we

divide by a 100 to express this in terms of basis points: 0.0217

100= 0.000217. Thus, we get:

DATE DELTA RET BOEING RET SP500 RET Monthly rate of T- Bill on 1st day of next month

6/29/2007 0.034121 -0.04404 -0.017816 0.0041333

7/31/2007 -0.095432 0.075603 -0.031982 0.0036000

… … … … …

… … … … …

11/30/2015 -0.083497 -0.011549 0.000505 0.0001917

12/31/2015 0.091046 -0.005913 -0.01753 0.000217


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(c) [3 Points] The following table gives the “summary statistics” of the data set in part (b):

DELTA RET BOEING RET SP500 RET Monthly rate on T- Bill on 1st day of next month

Mean 0.0201 0.00872 0.00390 0.0003842

Variance 0.0205 0.00614 0.00215 0.0000007

Source: CRSP through CHASS @ U of Toronto and FRED. Data range from 6/29/2007 through 12/31/2015

Suppose that on Jan 1st, 2016, an investor invests $1m in a risk free asset and a (i.e. one) risky asset. True or false: T-Bills are a “risk free asset” because of the four assets listed above, T-Bills has the smallest variance (and therefore “risk”)? Answer False. While T-Bills have the lowest variance (and thus lowest standard deviation, a measure of risk), the reason they

are risk free is because a T-Bill purchased on 1/1/2016 has zero risk provided the investor holds on to the T-Bill for one

month (its duration) and cashes the payment in US$.


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(d) [7 Points] The following table gives the “summary statistics” of the data set in part (b):

DELTA RET BOEING RET SP500 RET Monthly rate on T- Bill on 1st day of next month

Mean 0.0201 0.00872 0.00390 0.0003842

Variance 0.0205 0.00614 0.00215 0.0000007

Source: CRSP through CHASS @ U of Toronto and FRED. Data range from 6/29/2007 through 12/31/2015

Suppose that on Jan 1st, 2016, an investor invests $1m in a risk free asset and a (i.e. one) risky asset. Of the three risky assets (Delta, Boeing, S&P 500) which one should the investor select as the risky asset in her portfolio (remember in this part the portfolio consists of a risk free and a risky asset)? Provide a short explanation backed by graphs/numbers (if necessary). Answer The investor can select either Delta, or Boeing, or SP 500 (actually, an ETF tracking the SP 500) as the risky asset -- but which one should he pick? Let’s use the same logic we used to choose the optimal combination of two risky assets to construct a synthetic risky asset. The following graph shows the two combinations (“1” and “2”) of two risky assets (“A” and “B”):

𝑅 𝑡

𝐸 2

B

A

1

𝑅

𝐸 −

F

We argued that combination “2” was better than combination “1” because it has a higher Sharpe ratio (“price of risk”) –

based on this argument, we constructed the synthetic asset by looking for the combination that gave us the highest

price of risk. We can use the same logic here – pick the asset with the highest price of risk. The following table shows the

price of risk calculated by:

Price of Risk =Risk Premium of risky asset

Risk of risky asset=Return of risky asset − Risk free rate

Risk of risky asset


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DELTA BOEING SP500

Risk Premium = 0.01986 0.00850 0.00368

Risk = 0.14325 0.07838 0.04632

Price of Risk (Sharpe’s Ratio) = 0.13866 0.10848 0.07943

Delta has the highest price of risk. Thus, if the investor wishes to invest $1m (for one month in January 2016) in a risk

free asset and one risky asset, that risky asset should be Delta stocks.


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QUESTION 2 [TOTAL 10 Points] (a) [5 Points] Suppose an investor has a “mean-variance utility function” (with utility parameter 𝑐 > 0). Does this investor have “monotone” (aka “more is better”) preferences? What about “convex” (aka “taste for variety”) preferences? Answer The mean-variance utility function is:

𝑈 = 𝐸[ ] − 𝑐 2 Here 𝐸[ ] = Expected returns, 2 = Variance of returns (risk squared), 𝑐 > 0 is a parameter. Now, the marginal utilities are:

𝜕𝑈

𝜕 = −𝑐 < 0 (for a risky asset and risk aversion (i. e. 𝑐 > 0))

𝜕𝑈

𝜕𝐸[ ]= 1 > 0

“Risk” is a bad good and, as such, the investor does not have monotone preferences. Now, the indifference curves look like (one needs to show that the indifference look like this by, amongst other things, computing 𝑀𝑅𝑆 = 𝑐 and noting that the slope steepens with risk):

“Risk”

“Expected Return”

The investor does have convex preferences. For example, notice that the investor prefers “bundle C” to all combinations of bundles “A” and “B”:


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“Risk”

“Expected Return” A

B

C


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(b) [5 Points] Consider an investor with a mean-variance utility function with the parameter 𝑐 = 0.5. Suppose that initially, the investor has purchased an asset with a (historical) average return of 0.872% and risk of 0.07838 standard deviations. Now suppose that the risk of this asset increases by 1%. What must happen to this asset’s expected return in order for this investor to continue holding this asset? In general, will the “extra returns required to compensate the investor for higher risk” rise, fall, or stay the same with the level of risk? State all necessary assumption, show all essential steps, and state the final answer up to six decimal places. Answer The mean-variance utility function is:

𝑈 = 𝐸[ ] − 𝑐 2 = 𝐸[ ] − 0.5 2 Initially:

𝐸[ ] = estimated by historical average returns = 0.00872

Risk = 0.07838

𝑈 = 𝐸[ ] − 𝑐 2 = 0.00872 − 0.5 (0.07838 )2 = 0.005647 Now, the new level of risk is 1.01(0.07838) = 0.079162 standard deviations. Following the greater risk, for the investor to continue holding on to the asset, the investor must be indifferent to the initial situation. For this to happen we see that the asset has to offer higher returns:

𝑈 after higher risk = New Expected Returns − 0.5 (1.01 ∗ 0.07838 )2 = Initial 𝑈 = 0.005647

New Expected Returns = 0.005647 + 0.5 (1.01 ∗ 0.07838 )2 = 0.00878 This makes intuitive sense: the investor demands a higher return to absorb the greater risk. To see the relationship between the “extra returns required to compensate the investor for higher risk” and the “level of risk” we note from the previous part that:

𝑀𝑅𝑆 = 𝑐

That is, the slope of the indifference curves is proportional to the level risk. As the asset becomes riskier, the investor demands ever higher compensating returns.


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QUESTION 3 [TOTAL 45 Points + 5 Bonus Points] Consider a financial portfolio with a fraction (1 − 𝛽) consisting of a risk free asset and a fraction 𝛽 consisting of a risky asset. (a) [10 Points] [This part is independent of other parts in this question] Show that there is a linear relationship between the portfolio’s expected return and the portfolio risk. What are the intercept and slope of this linear function? Please show all steps. Answer The expected return of the portfolio is:

𝐸[ 𝑝] = 𝐸[(1 − 𝛽) + 𝛽 ] = (1 − 𝛽)𝐸[ ] + 𝛽𝐸[ ] = + 𝛽(𝐸[ ] − )

The variance of portfolio returns is:

Var[ 𝑝] = 𝑝2 = Var[(1 − 𝛽) + 𝛽 ]

Var[ 𝑝] = 𝑝2 = (1 − 𝛽)2 Var[ ]⏟

0

+ 𝛽2 Var[ ]⏟ 𝜎𝑟

+ 2𝛽(1 − 𝛽) cov[ , ]⏟ 0

Here we used the fact that a risk free asset’s returns have zero variance and zero covariance with the risky asset. Next:

Var[ 𝑝] = 𝑝2 = 𝛽2

2 ⇒ 𝛽 = 𝑝 =

Portfolio Risk

Risky asset Risk

Thus:

𝐸[ 𝑝] = + 𝛽(𝐸[ ] − ) = + 𝑝

(𝐸[ ] − ) = +

(𝐸[ ] − )

𝑝

We see that there is a linear relationship between portfolio returns and risk:

𝐸[ 𝑝] = +(𝐸[ ] − )

𝑝 = Intercept + Slope 𝑝

Here the intercept is the risk free rate and the slope is the risky asset’s price of risk.


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(b) [5 Points] Suppose an investor with a “mean-variance utility function” (with 𝑐 > 0) constructs a financial portfolio with a fraction (1 − 𝛽) consisting of a risk free asset and a fraction 𝛽 consisting of a risky asset. Derive the expression for the optimal fraction of the portfolio invested in the risk asset, i.e. 𝛽. Show all steps. Answer The investor chooses 𝛽 to:

max𝛽𝑈 = 𝐸[ 𝑝] − 𝑐 𝑝

2 = + 𝛽{𝐸[ ] − } − 𝑐𝛽2

2

𝑑𝑈

𝑑𝛽= − + 𝐸[ ] − 2𝑐𝛽

2 = 0

Fraction ofPortfolioin RiskyAsset

= 𝛽∗ =(𝐸[ ] − )

2𝑐 2 =

Risk Premium of Risky Asset

2 𝑐 (Risky Asset Risk)2


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(c) [10 Points] [This part is independent of other parts in this question] Consider the investor in part (b) and suppose that her optimal portfolio is a leveraged portfolio where she has borrowed 10% of the value of her own funds at the risk free rate and invested her funds and the borrowed funds in a risky asset whose price of risk is 0.108483. What is the risk of this risky asset if her utility parameter 𝑐 = 0.629139? Show all essential steps and express the final answer up to six decimal places Answer The investor chooses 𝛽 to:


2 = + 𝛽{𝐸[ ] − } − 𝑐𝛽2

2

At the optimal solution:


= 𝛽∗ =(𝐸[ ] − )

2𝑐 2 =



In this case, she has borrowed 10% of the value of her own funds at the risk free rate and invested everything in a risky asset, so that:


= 𝛽∗ =(𝐸[ ] − )

2𝑐 2 =


2 𝑐 (Risky Asset Risk)2= 1.1

Notice that:

𝛽∗ =(𝐸[ ] − )

2𝑐 2 =

1

2𝑐

(𝐸[ ] − )

=

1

2𝑐 (Risky asset′s price of risk) = 1.1

Thus:

= 1

2(1.1)𝑐(Risky asset′s price of risk) =

1

2(1.1)(0.629139)(0.108483) ≈ 0.078378


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(d) [10 Points] [This part is independent of other parts in this question] Consider the investor in part (b) and suppose that she has constructed the optimal portfolio. True or false: all else equal, at the optimal solution, which has the greater impact on optimal utility: a small increase in the risky asset’s return or the same increase in the risk free asset’s return? Answer The investor chooses 𝛽 to:


2 = + 𝛽{𝐸[ ] − } − 𝑐𝛽2

2



= 𝛽∗ =(𝐸[ ] − )

2𝑐 2 =



We’re being asked for the impact on optimal utility from a small increase in the risky asset return vs. an equivalent increase in the risk free rate. We could substitute 𝛽∗ into the expression for utility, derive 𝑈∗, and then compare the marginal utility of higher returns on the risky asset vs. the risk free asset. However, this would be complicated. A faster approach is to use the envelope theorem. The original utility function is:

𝑈 = + 𝛽{𝐸[ ] − } − 𝑐𝛽2

2

To see the impact on optimal utility from a small increase in the risky asset’s returns -- by the envelope theorem – we differentiate utility with respect to 𝐸[ ]:

𝑑𝑈

𝑑𝐸[ ]= 𝛽

Next we evaluate this derivative at the optimal values:

𝑑𝑈∗

𝑑𝐸[ ]= 𝛽∗

To see the impact on optimal utility from a small increase in the risky free asset’s returns we -- by the envelope theorem – differentiate utility with respect to :

𝑑𝑈

𝑑 = 1 − 𝛽

Next we evaluate this derivative at the optimal values:

𝑑𝑈∗

𝑑 = 1 − 𝛽∗

Which is better: a small increase in the risky asset return or an equivalent increase in the risk free rate? That is, which expression is larger:

𝑑𝑈∗

𝑑𝐸[ ]= 𝛽∗ 𝑣 .

𝑑𝑈∗

𝑑 = 1 − 𝛽∗

The answer is simple: if at the initial optimal solution, over 50% of the portfolio is in the risk asset then a small increase in the risky asset’s returns will have a larger impact on optimal utility than the equivalent increase in the risk free asset’s returns.


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(e) [5 Points] [This part is independent of other parts in this question] Suppose you wish to create a “synthetic” asset consisting of two risky assets “A” and “B”. True or false: regardless of their attitude towards risk, each investor will create a synthetic asset that is identical to that of every other investor. Give a short explanation. Answer True. One creates a synthetic asset by maximizing the price of risk (aka Sharpe’s ratio) which is independent of utility:

Price of Risk = (𝐸[ ] − )

=Risk Premium of Synthetic Asset

Risk of Synthetic Asset


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** Please use the following information for all remaining parts ** Consider the following information based on data from 6/29/2007 through 12/31/2015 (source CRSP at CHASS @ U Toronto and FRED):

Covariance Table

DELTA RET BOEING RET SP500 RET

DELTA RET 0.02052 0.00364 0.00167

BOEING RET 0.00364 0.00614 0.00254

SP500 RET 0.00167 0.00254 0.00215

DELTA RET BOEING RET SP500 RET

Mean 0.0201 0.00872 0.00390

Variance 0.0205 0.00614 0.00215

Std. Dev. 0.1433 0.07838 0.04632

The one month rate on a T-Bill issued on 1/1/2016 was 0.000216667. An investor wishes to create a financial portfolio (for the month of January) consisting of T-Bills and a synthetic asset consisting of Delta stocks, Boeing stocks, and stocks of a S&P 500 tracking ETF. Label Delta, Boeing, and the SP 500 tracking ETF assets “1”, “2”, and “3” respectively. The optimal synthetic asset consists of:

𝑤1 = Fraction of Delta Shares in Synthetic Asset 0.54

𝑤2 = Fraction of Boeing Shares in Synthetic Asset 0.68

𝑤3 = Fraction of S&P 500 Tracking ETF in Synthetic Asset -0.22

The synthetic asset’s price of risk is 0.101792. Suppose that the investor’s mean-variance utility function parameter 𝑐 = 1. (f) [10 Points + Bonus 5 Points] Interpret the negative weight on the S&P 500 tracking ETF. What fraction of the portfolio consisting of the risk free asset and the synthetic asset consists of the synthetic asset above? Bonus 5 points if you derive the synthetic asset’s risk by using the figures in the covariance table. Answer The investor chooses 𝛽 to:


2 = + 𝛽{𝐸[ ] − } − 𝑐𝛽2

2

Where the “risky” asset in fact is the synthetic asset consisting of Delta, Boeing, and a SP 500 tracking ETF stocks. At the optimal solution:


= 𝛽∗ =(𝐸[ ] − )

2𝑐 2 =



We need to calculate the synthetic risky asset’s risk premium and risk. First the risk premium:

Risk Premium = Returns − Risk Free Rate In turn,


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Returns = 𝑤1 1 +𝑤2 2 +𝑤3 3 = 0.54(0.0201) + 0.68(0.00872) − 0.22(0.00390) = 0.015926

Risk Premium = Returns − Risk Free Rate = 0.015926 − 0.000216667 = 0.01570933 Next, the synthetic asset’s risk – the easy way is as follows:

Price of Risk =Risk Premium of Synthetic Asset

Risk of Synthetic Asset=

0.01570933

Risk of Synthetic Asset= 0.101792

Risk of Synthetic Asset = 0.01570933

0.101792= 0.154327

_________

Bonus answer:

Var(Synthetic Asset Returns) = 𝑤12 1

2 +𝑤22 2

2 + 𝑤32 3

2 + 2𝑤1𝑤2cov12 + 2𝑤1𝑤3cov13 + 2𝑤2𝑤3cov23

Var(Synthetic Asset Returns)= 0.542(0.025) + 0.682(0.00614) + (−0.22)2(0.00215) + 2(0.54)(0.68)(0.00364)+ 2(0.54)(−0.22)(0.00167) + 2(0.68)(−0.22)(0.00254) = 0.011750

Synthetic Asset Risk = √Var(Synthetic Asset Returns) = √0.011750 = 0.108397

_________ Thus:


= 𝛽∗ =Risk Premium of Risky Asset

2 𝑐 (Risky Asset Risk)2=

0.01570933

2 × 1 × 0.011750 ≈ 67%


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QUESTION 4 [TOTAL 30 Points] Consider an agent who lives for exactly two periods 𝑡 = 0,1 in an economy with a single good (corn). At the beginning of

each period, the agent is endowed with “real” incomes 𝑌0 and 𝑌1 (in units of corn) respectively. Let 𝐶0 and 𝐶1 denote the

amounts of corn consumed in 𝑡 = 0 and 𝑡 = 1 respectively. At 𝑡 = 0, each agent can save or borrow corn at the real

interest rate > 0.

(a) [15 Points] Suppose an agent’s preferences are represented by a general differentiable utility function

𝑈(𝐶0, 𝐶1). Show that if the real interest rises then net savers at 𝑡 = 0 are better off whereas net borrowers are worse

off. State all assumptions.

Answer

The inter-temporal UMP with a 𝐹𝑉 Budget Constraint is:

max𝐶0, 𝐶1

𝑈(𝐶0, 𝐶1) . 𝑡. 𝐶0(1 + ) + 𝐶1 = 𝑌0(1 + ) + 𝑌1 ⏟ Real 𝐹𝑉 Budget Constraint

, 𝐶0 ≥ 0, 𝐶1 ≥ 0

max𝐶0,𝐶1

𝑈(𝐶0, 𝐶1) . 𝑡. 𝐶0(1 + ) + 𝐶1 = 𝑌0(1 + ) + 𝑌1⏟ Total FV Income

𝐹𝑉𝑌

, −𝐶0 ≤ 0, −𝐶1 ≤ 0

max𝐶0,𝐶1

𝑈(𝐶0, 𝐶1) . 𝑡. 𝐶0(1 + ) + 𝐶1 − 𝐹𝑉𝑌 = 0,−𝐶0 ≤ 0, −𝐶1 ≤ 0

max

𝐶0, 𝐶1, 𝜆1, 𝜆 , 𝜆3L = 𝑈(𝐶0, 𝐶1) − 𝜆1[𝐶0(1 + ) + 𝐶1 − 𝐹𝑉𝑌] − 𝜆2[−𝐶0] − 𝜆3[−𝐶1]

max

𝐶0, 𝐶1, 𝜆1, 𝜆 , 𝜆3L = 𝑈(𝐶0, 𝐶1) − 𝜆1[𝐶0(1 + ) + 𝐶1 − 𝐹𝑉𝑌] + 𝜆2𝐶0 + 𝜆3𝐶1

The FOCs and KT conditions are:

𝜕L

𝜕𝐶𝑜=𝜕𝑈

𝜕𝐶0⏟𝑀𝑈0

− 𝜆1(1 + ) + 𝜆2 = 0

𝜕L

𝜕𝐶1=𝜕𝑈

𝜕𝐶1⏟𝑀𝑈1

− 𝜆1 + 𝜆3 = 0

𝜕L

𝜕𝜆1= −[𝐶0(1 + ) + 𝐶1 − 𝐹𝑉𝑌] = 0 ⇒ 𝐶0(1 + ) + 𝐶1 = 𝐹𝑉𝑌

𝜆2 ≥ 0, 𝐶0 ≥ 0, 𝜆2𝐶0 = 0

𝜆3 ≥ 0, 𝐶1 ≥ 0, 𝜆3𝐶1 = 0

Noting that 𝜆2, 𝜆3 ≥ 0 , assuming that > −1 and that corn is a “good” good we have:

𝜆1 =𝑀𝑈0 + 𝜆21 +

> 0

𝜆1 = 𝑀𝑈1 + 𝜆3 > 0


ECO 204, 2016-2017 (AJAZ), Test 2`


Page 18 of 20



L ∗ = 𝑈∗ − 𝜆1∗ [𝐶0

∗(1 + ) + 𝐶1∗ − 𝐹𝑉𝑌⏟

0

] + 𝜆2∗𝐶0

∗⏟0

+ 𝜆3∗𝐶1

∗⏟0

= 𝑈∗

Thus to evaluate the change in 𝑈∗ due to a change in a parameter (in this case ) we should use the envelope theorem on the Lagrange equation.

𝑑L

𝑑 = −𝜆1(𝐶0 − 𝑌0)

Thus:

𝑑L ∗

𝑑 =𝑑𝑈∗

𝑑 = −𝜆1

∗(𝐶0∗ − 𝑌0

∗) = 𝜆1∗𝑆0∗

We showed that under some conditions 𝜆1

∗ > 0. Thus, when increases, utility will increase if 𝑆0∗, net savings at 𝑡 =

0, are positive and vice versa if net savings are negative at 𝑡 = 0.


ECO 204, 2016-2017 (AJAZ), Test 2`


Page 19 of 20


(b) [15 Points] Suppose that an agent’s preferences over are represented by the utility function 𝑈 = min(𝛼𝐶0, 𝛽𝐶1) where 𝛼, 𝛽 > 0. Show that if the real interest rises then net savers at 𝑡 = 0 are better off whereas net borrowers at 𝑡 = 0 are worse off. Answer

The inter-temporal UMP with 𝐹𝑉 Budget Constraint is:

max𝐶0, 𝐶1

𝑈 = min(𝛼𝐶0, 𝛽𝐶1) . 𝑡. 𝐶0(1 + ) + 𝐶1 = 𝑌0(1 + ) + 𝑌1⏟ 𝐹𝑉𝑌

However since the utility function can’t be differentiated we can’t setup and solve this UMP by the KT method. Instead, we note that the optimal choice must be on the “corners” line and the budget line:

𝛼𝐶0 = 𝛽𝐶1

𝐶0(1 + ) + 𝐶1 = 𝑌0(1 + ) + 𝑌1⏟ 𝐹𝑉𝑌

Solving these yields:

𝐶0∗ = 𝛽

𝐹𝑉𝑌

[𝛽(1 + ) + 𝛼]

𝐶1∗ = 𝛼

𝐹𝑉𝑌

[𝛽(1 + ) + 𝛼]

An agent will save at 𝑡 = 0 whenever:

When is 𝑆0 = 𝑌0 − 𝐶0∗ > 0?

𝑆0 = 𝑌0 − 𝛽𝐹𝑉𝑌

[𝛽(1 + ) + 𝛼]> 0

𝑆0 = 𝑌0 − 𝛽𝑌0(1 + ) + 𝑌1[𝛽(1 + ) + 𝛼]

> 0

⇒𝛼

𝛽>𝑌1𝑌0

Now what is the impact on utility when increases?

𝑈∗ = min(𝛼𝐶0∗, 𝛽𝐶1

∗) =𝛼𝛽 𝐹𝑉𝑌

[𝛽(1 + ) + 𝛼]= 𝛼𝛽 [𝑌0(1 + ) + 𝑌1]

[𝛽(1 + ) + 𝛼]

𝑑𝑈∗

𝑑 =

𝛼𝛽 𝑌0

[𝛽(1 + ) + 𝛼]− 𝐹𝑉𝑌

𝛼𝛽2

[𝛽(1 + ) + 𝛼]2

𝑑𝑈∗

𝑑 =

𝛼𝛽

[𝛽(1 + ) + 𝛼][𝑌0 −

𝛽 𝐹𝑉𝑌

𝛽(1 + ) + 𝛼] =

𝛼𝛽

[𝛽(1 + ) + 𝛼]2[𝛼𝑌0 − 𝛽𝑌1]


ECO 204, 2016-2017 (AJAZ), Test 2`


Page 20 of 20


When increases, utility will increase so long as 𝛼𝑌0 − 𝛽𝑌1 > 0 or whenever 𝛼

𝛽>𝑌1

𝑌0. This is the same conclusion we

reached in part (a): when increases, utility will increase if 𝑆0∗, net savings at 𝑡 = 0, are positive and vice versa if net

savings are negative at 𝑡 = 0.


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