U₁(C+1) U₁(C₁) Box 10.2 Calculating the Equilibrium Price Function Equation (10.4). implicitly defines the equilibrium price series. Can it be solved directly to produce the actual equilibrium prices {q(): j=1,2, N}? The answer is positive. First, we must specify parameter values and functional forms. In particular, we need to select values for ō and for the various output levels Y, to specify the probability transition matrix T and the form of the represen- tative agent's period utility function (a CRRA function of the form U(c)=(c)/(1-) is a natural choice). We may then proceed as fol- lows. Solve for the {q(): j=1,2, ..., N} as the solution to a system of linear equations. Note that Eq. (10.4). can be written as the fol- lowing system of linear equations (one for each of the N possible current states ): Modify the Box 10.2 example on page 291 as follows: • The payouts in the three states are (Y¹, Y², Y³) = (4,3,2) • All other assumptions remain the same. Hints: Solving the problem involves applying Eq. (10.4). To see how this applies I re-write Eq. (10.4) below. U'(Y₁)q₁ = ¿E₁[U' (Ÿ₁+1)(+1+Ỹ₁+1)] U' (Y₁)q₁ = 8E₁[U' (Ỹ₁+1)(Ÿ₁+1)] +8E₁[U' (Ỹ₁+1)(+1)] where Eq. (1) matches the system of equations shown in Box 10.2. (1) ⚫ You should be able to validate that for U(c) = ln(c) implies E₁[U'(Ỹ₁+1)(Ỹ₁+1)] = 8 ⚫ To understand the Transition Matrix, assume the current state 0₁ = 1.5. The LHS is 19(1.5)=9(1.5), which is the LHS of the first equation in the system. ⚫ Since we assume we are in ₁ the Transition matrix tells us there is .5 probability of state ₁ = 1.5 eventuating in the next period so that (.5)q(1.5) = 39(1.5), which is a RHS term in the system in Box 10.2 ⚫ Apply this set-up to all three states, which generates the system shown.
U₁(C+1) U₁(C₁) Box 10.2 Calculating the Equilibrium Price Function Equation (10.4). implicitly defines the equilibrium price series. Can it be solved directly to produce the actual equilibrium prices {q(): j=1,2, N}? The answer is positive. First, we must specify parameter values and functional forms. In particular, we need to select values for ō and for the various output levels Y, to specify the probability transition matrix T and the form of the represen- tative agent's period utility function (a CRRA function of the form U(c)=(c)/(1-) is a natural choice). We may then proceed as fol- lows. Solve for the {q(): j=1,2, ..., N} as the solution to a system of linear equations. Note that Eq. (10.4). can be written as the fol- lowing system of linear equations (one for each of the N possible current states ): Modify the Box 10.2 example on page 291 as follows: • The payouts in the three states are (Y¹, Y², Y³) = (4,3,2) • All other assumptions remain the same. Hints: Solving the problem involves applying Eq. (10.4). To see how this applies I re-write Eq. (10.4) below. U'(Y₁)q₁ = ¿E₁[U' (Ÿ₁+1)(+1+Ỹ₁+1)] U' (Y₁)q₁ = 8E₁[U' (Ỹ₁+1)(Ÿ₁+1)] +8E₁[U' (Ỹ₁+1)(+1)] where Eq. (1) matches the system of equations shown in Box 10.2. (1) ⚫ You should be able to validate that for U(c) = ln(c) implies E₁[U'(Ỹ₁+1)(Ỹ₁+1)] = 8 ⚫ To understand the Transition Matrix, assume the current state 0₁ = 1.5. The LHS is 19(1.5)=9(1.5), which is the LHS of the first equation in the system. ⚫ Since we assume we are in ₁ the Transition matrix tells us there is .5 probability of state ₁ = 1.5 eventuating in the next period so that (.5)q(1.5) = 39(1.5), which is a RHS term in the system in Box 10.2 ⚫ Apply this set-up to all three states, which generates the system shown.
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
Related questions
Question
Transcribed Image Text:U₁(C+1)
U₁(C₁)
Box 10.2
Calculating the Equilibrium Price Function
Equation (10.4). implicitly defines the equilibrium price series. Can
it be solved directly to produce the actual equilibrium prices
{q(): j=1,2, N}? The answer is positive. First, we must specify
parameter values and functional forms. In particular, we need to
select values for ō and for the various output levels Y, to specify
the probability transition matrix T and the form of the represen-
tative agent's period utility function (a CRRA function of the form
U(c)=(c)/(1-) is a natural choice). We may then proceed as fol-
lows.
Solve for the {q(): j=1,2, ..., N} as the solution to a system of
linear equations. Note that Eq. (10.4). can be written as the fol-
lowing system of linear equations (one for each of the N possible
current states ):
![Modify the Box 10.2 example on page 291 as follows:
• The payouts in the three states are (Y¹, Y², Y³) = (4,3,2)
• All other assumptions remain the same.
Hints:
Solving the problem involves applying Eq. (10.4). To see how this applies I re-write
Eq. (10.4) below.
U'(Y₁)q₁ = ¿E₁[U' (Ÿ₁+1)(+1+Ỹ₁+1)]
U' (Y₁)q₁ = 8E₁[U' (Ỹ₁+1)(Ÿ₁+1)] +8E₁[U' (Ỹ₁+1)(+1)]
where Eq. (1) matches the system of equations shown in Box 10.2.
(1)
⚫ You should be able to validate that for U(c) = ln(c) implies E₁[U'(Ỹ₁+1)(Ỹ₁+1)] = 8
⚫ To understand the Transition Matrix, assume the current state 0₁ = 1.5. The LHS is
19(1.5)=9(1.5), which is the LHS of the first equation in the system.
⚫ Since we assume we are in ₁ the Transition matrix tells us there is .5 probability of
state ₁ = 1.5 eventuating in the next period so that (.5)q(1.5) = 39(1.5), which
is a RHS term in the system in Box 10.2
⚫ Apply this set-up to all three states, which generates the system shown.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1b2e65f2-c472-45bf-885c-00dffce84016%2F9989d0ce-b04c-4d46-868e-85a40cbee858%2Fqnrezni_processed.png&w=3840&q=75)
Transcribed Image Text:Modify the Box 10.2 example on page 291 as follows:
• The payouts in the three states are (Y¹, Y², Y³) = (4,3,2)
• All other assumptions remain the same.
Hints:
Solving the problem involves applying Eq. (10.4). To see how this applies I re-write
Eq. (10.4) below.
U'(Y₁)q₁ = ¿E₁[U' (Ÿ₁+1)(+1+Ỹ₁+1)]
U' (Y₁)q₁ = 8E₁[U' (Ỹ₁+1)(Ÿ₁+1)] +8E₁[U' (Ỹ₁+1)(+1)]
where Eq. (1) matches the system of equations shown in Box 10.2.
(1)
⚫ You should be able to validate that for U(c) = ln(c) implies E₁[U'(Ỹ₁+1)(Ỹ₁+1)] = 8
⚫ To understand the Transition Matrix, assume the current state 0₁ = 1.5. The LHS is
19(1.5)=9(1.5), which is the LHS of the first equation in the system.
⚫ Since we assume we are in ₁ the Transition matrix tells us there is .5 probability of
state ₁ = 1.5 eventuating in the next period so that (.5)q(1.5) = 39(1.5), which
is a RHS term in the system in Box 10.2
⚫ Apply this set-up to all three states, which generates the system shown.
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