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ECON 90034 Economics of Finance Tutorial 2 Solutions (Week 3) Consumption and investment decisions under certainty Svetlana Danilkina Suppose an individual has $7 millions today and can invest them in the following projects: Project Investment Outlay Rate of Return A 1,000,000 8% B 1,000,000 20% C 2,000,000 4% D 3,000,000 30% Graph the production possibility (opportunity) set in a C0, C1 framework

(a)

In order to graph the production opportunity set, we need to order the investments by their rates of return and sum up the total investment required to undertake the first through i-th project. This is shown in the table Project

One plus the rate of return

Outlay for the i-th project

D B A C

1.30 1.20 1.08 1.04

3m 1m 1m 2m

Sum of the outlays (investment) 3m 4m 5m 7m

Sum of the income (available for C1) 3*1.3 = 3.9m 3.9 + 1*1.2 = 5.1m 5.1 + 1*1.08 = 6.18m 6.18 + 2*1.04 = 8.26m

The production possibility (opportunity) set plots the relationship between resources utilised today for real (physical) investments (i.e. consumption foregone along the C0 axis) and the extra consumption provided for the next period; or the relationship between consumption today C0 and consumption tomorrow C1 achievable through real investment. For example, if only project D is undertaken then $3 million in current consumption is sacrificed to receive 1.3 × ($3 million) = $3.9 million in consumption tomorrow. That corresponds to the consumption today C0 = 7 - 3 = 4m and consumption tomorrow C1 = 3.9m. If both projects D and B are undertaken then $3 million plus $1 million = $4 million in current consumption would be sacrificed to receive 1.3 × ($3 million) + 1.2 × ($1 million) = $3.9m + $1.2m = $5.1m in consumption tomorrow. That corresponds to the consumption today C0 = 7 – 3 – 1 = 3m and consumption tomorrow C1 = 5.1m. If we aggregate all the investment opportunities then $7 million would be foregone and the production opportunity set would look like

ECON 90034 Tutorial 2. Consumption and investment decisions under certainty

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C1

Slope = - 1.04 Slope = - 1.08

8.26 6.18 5.1

Production possibility (opportunity) set

Slope = - 1.2

3.9

Slope = - 1.3

IA

IC

IB

2 (b)

3

4

ID

7

C0 , millions

If the market rate of return is 10%, draw the capital market line (CML) and identify the optimal investment (production) point. Write down the formula for CML.

To find the capital market line draw a line with a slope of – (1 + r) = –1.1 such that it is tangent to the production possibility set. From the graph you can see where the tangency point is. This is the optimal production point (P0, P1). Hence, the optimal production decision is to invest in projects B and D. Both of these projects have higher rate of return than the capital market rate of return of 10%, so you should invest in both of them. The corresponding present value of final wealth is P 5.1 = 7.64 million . W0* = P0 + 1 = 3+ 1+ r 1.1 The other two projects – A and C have lower rate of return than the capital market rate of return of 10%, so you should invest in neither.

C1 optimal production point

8.4

Slope = -1.08

8.26

the capital market line

P1 = 5.1 3.9

Slope = -1.1 Slope = -1.2

2

IB

P0 = 3

C0 , millions 4

ID

7

W0* =7.64

ECON 90034 Tutorial 2. Consumption and investment decisions under certainty

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The corresponding formula for the capital market line is

C0 +

(c)

C1 P C = W 0* , where W 0* = P0 + 1 = 7.64 => C0 + 1 = 7.64 . 1+ r 1+ r 1.1

Alex has utility function UA(C0,C1) = C0*C1 that represents his preferences about consumption today and in the future. Calculate and show the following points on the graph: initial wealth W0; optimal production point (P0, P1); optimal consumption point (C0*, C1*); the present value of the final wealth W0*. Is he a lender or a borrower?

Alex can choose any point on the CML as his consumption point. The best point is the one that gives him the highest utility. This is a point on CML where one of his indifference curves is tangent to the CML. First, we need to calculate his marginal rate of substitution: 𝑀𝑅𝑆 =

𝐶 𝑀𝑈0 �𝑀𝑈 = 1. 𝐶0 1

To calculate his optimal consumption point, we need to use two equations: formula for the CML and the tangency condition (MRS=1+r). We need to solve them together to find C0 and C1. tangency condition is MRS = C1 /C0 = 1 + r => C1 /C0 = 1.1 C CML: C0 + 1 = 7.64 1.1 From the tangency condition we have C1 = 1.1C0; substituting into CML we obtain 2C0 = 7.64 => C0 = 3.82. Then C1 = 1.1C0 = 1.1*3.82 = 4.20. As Alex’ consumption point is to the right of the production point, he is a borrower – he is borrowing 3.82-3 = 0.82 on the capital market.

C1

optimal production point for lender and borrower

8.4

the capital market line optimal consumption point for borrower (Alex)

P1 = 5.1

Slope = -1.1

C1*=4.20 borrow 0.82

P0 = 3

C0 , millions

C0*=3.82 W0=7 W0* =7.64

Real investment ECON 90034 Tutorial 2. Consumption and investment decisions under certainty

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(d)

Repeat part (c) for Nina, who has utility function UN(C0 , C1) = C0 *(C1)3. Is she a lender or a borrower?

To calculate Nina’s optimal consumption point, we need to calculate her marginal rate of substitution: (𝐶1 )3 𝐶1 𝑀𝑈0 𝑀𝑅𝑆 = = �𝑀𝑈 = 2 1 3𝐶0 (𝐶1 ) 3𝐶0 and then use two equations: formula for the CML and the tangency condition (MRS = 1 + r). tangency condition is MRS = C1 /3C0 = 1 + r => C1 /3C0 = 1.1 C CML: C0 + 1 = 7.64 1.1 From the tangency condition we have C1 = 1.1*3*C0; substituting into CML we obtain 4C0 = 7.64 => C0 = 1.91. Then C1 = 3.3C0 = 3.3*1.91 = 6.30. As Nina’s consumption point is to the left of the production point, she is a lender – she is lending 3 - 1.91 = 1.09 on the capital market.

optimal consumption point for lender (Nina)

C1

optimal production point for lender and borrower

8.4 C1*=6.30

the capital market line

P1 = 5.1 Slope = -1.1

lend 1.09

C0*=1.91

C0 , millions

P0 = 3

W0=7 W0* =7.64

Real investment

(e)

Use your solution to (c) and (d) to graphically demonstrate the Fisher separation theorem for the case where one individual ends up lending in financial markets and another one borrowing.

ECON 90034 Tutorial 2. Consumption and investment decisions under certainty

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optimal consumption point for lender

C1

optimal production point for lender and borrower

8.4 C1 *

the capital market line

lender

optimal consumption point for borrower

P1 = 5.1

Slope = -1.1

C1* borrower

C0 , millions C0 *

P0 = 3

lender

C 0 * W0=7 W0* =7.64 borrower

optimal consumption point for lender

C1

optimal production point for lender and borrower

8.4 C1 *

the capital market line

lender

optimal consumption point for borrower

P1 = 5.1

Slope = -1.1

C1* borrower

lend

C0 * lender

borrow

P0 = 3

C0 , millions C 0 * W0=7 W0* =7.64 borrower

Real investment

Both borrower and lender make the same real investment in the amount of 4 million: 3m in project D and 1m in project B. The lender lends the amount of P0 - C0*lender on the capital market; the borrower borrows the amount of C0*borrower - P0 on the capital market. Though they consume different amounts due to the difference in their preferences, they make the same real investment decision. This is the Fisher separation theorem.

ECON 90034 Tutorial 2. Consumption and investment decisions under certainty

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(f)

Graphically analyse the effect of an exogenous decrease in the interest rate from 10% to 6% on (i)

the investment in real (physical) assets

The decrease in interest rates shifts the capital market line as shown on the graph. The new CML has slope of -1.06. It is tangent to the production possibility set at point M. This is new production point: P0 = 2m, P1 = 6.18m. Investment in real assets increases from 4m to 5m. It is now worthwhile to invest in three real projects: D, B and A (that have higher rate of return than the capital market rate of return of 6%)

C1

optimal production point for r = 6% optimal production point for r = 10%

8.4 P 1 = 6.18 for 6% P 1 = 5.1 for 10%

the capital market line for r = 10%

M the capital market line for r = 6%

L Real investment for 6%

P0 = 2 for 6%

P0 = 3 for 10%

W0=7

W 0* =7.64 for 10%

C0 , millions

Real investment for 10%

(ii)

the present value of final wealth of borrowers and lenders

The present value of final wealth for lenders and borrowers increases from 7.64 to 7.83: P 6.18 = 7.83million W0* = P0 + 1 = 2 + 1+ r 1.06

(iii)

the utility of borrowers and lenders.

Borrowers originally chose points to the right of the point L – the production point for 10% capital market rate. After the rate cut, their utility has increased unambiguously (because they can borrow at the lower interest rate). ECON 90034 Tutorial 2. Consumption and investment decisions under certainty

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C1

optimal production point for lender and borrower for r = 6% optimal production point for lender and borrower for r = 10%

8.4

the capital market line for r = 10%

P 1 = 6.18 for 6% P 1 = 5.1 for 10%

M L

optimal consumption points for borrower the capital market line for r = 6%

C0 , millions P0 = 2 for 6%

P0 = 3 for 10%

W0=7

W 0* =7.64 W0 * =7.83 for 10% for 6%

The case for a lender is ambiguous. Some lenders become borrowers after the rate cut. Some of them (though not all) experience an increase in utility:

optimal consumption points for a lender who now is a borrower for r = 6%

C1

optimal consumption points for lender for r = 10%

optimal production point for r = 10%

8.4 P 1 = 6.18 for 6% P 1 = 5.1 for 10%

the capital market line for r = 10%

M L

the capital market line for r = 6%

optimal production point for r = 6%

C0 , millions P0 = 2 for 6%

P0 = 3 for 10%

W0=7

W 0* =7.64 for 10%

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The lenders who stay lenders suffer a decline in utility (as they get lower return on their money on the capital market):

optimal consumption points for lender for r = 6% optimal consumption points for lender for r = 10%

C1

optimal production point for lender and borrower for r = 6%

8.4 P 1 = 6.18 for 6% P 1 = 5.1 for 10%

optimal production point for lender and borrower for r = 10%

the capital market line for r = 10%

M L

the capital market line for r = 6%

C0 , millions P0 = 2 for 6%

P0 = 3 for 10%

W0=7

W 0* =7.64 for 10%

ECON 90034 Tutorial 2. Consumption and investment decisions under certainty

W0* =7.83 for 6%

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