P <- matrix(c(0, 0, 1, 0, 1, 0, 0, 1, 0), ncol = 3, byrow = TRUE)
P## [,1] [,2] [,3]
## [1,] 0 0 1
## [2,] 0 1 0
## [3,] 0 1 0As you can see if the first row is State i, then it is not only easier to get into state j ( second row) in two steps, it is actually impossible without first going to state k( third row ), which would then take you to state j with 100% certainty within two steps.
P <- matrix(c(1, 0, 0, 0, 1, 0, 0, 0, 1), ncol = 3, byrow = TRUE)
P## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1In this Probability Matrix we can see that this process is not ergodic because our initial state has every effect on our eventual process of transition. Across both states, starting in either state compeletely removes all probabilty of moving to another state. Because your initial condition perpetually dictates your path, this is a non ergodic process. By staying in the initial state for perpetuity you do not cycle between any other states therefore have no long run cycle in this non ergodic process.
P <- matrix(c(0.2, 0.2, 0.6, 0, 0, 1, 0, 0.1, 0.9), ncol = 3, byrow = TRUE)
P## [,1] [,2] [,3]
## [1,] 0.2 0.2 0.6
## [2,] 0.0 0.0 1.0
## [3,] 0.0 0.1 0.9P %^% 1000## [,1] [,2] [,3]
## [1,] 0 0.09091 0.9091
## [2,] 0 0.09091 0.9091
## [3,] 0 0.09091 0.9091With this Probability Matrix we can see that the second and third states are absorbing states in that once you enter into them your probability of returning to the first state is 0. Therefore the the long run cycle of this Markov Matrix is only a subset of the sencond and third states.
P1 <- matrix(c(0.4, 0.6, 0, 0, 1, 0, 0, 0.1, 0.9), ncol = 3, byrow = TRUE)
P1## [,1] [,2] [,3]
## [1,] 0.4 0.6 0.0
## [2,] 0.0 1.0 0.0
## [3,] 0.0 0.1 0.9P2 <- matrix(c(0.5, 0.5, 0, 0, 0, 1, 0, 0.5, 0.5), ncol = 3, byrow = TRUE)
P2## [,1] [,2] [,3]
## [1,] 0.5 0.5 0.0
## [2,] 0.0 0.0 1.0
## [3,] 0.0 0.5 0.5P1_2 <- P1 %*% P2
P1_2## [,1] [,2] [,3]
## [1,] 0.2 0.20 0.60
## [2,] 0.0 0.00 1.00
## [3,] 0.0 0.45 0.55We have just shown that even two Markov Matrices with differing amounts of states and probabilities, in order to matrix multiply we are always going to be multiplying n numbers that sum to one with another m numbers that sum to one and summing them therefore we will ALWAYS be left with a Markov Matrix in which the sum of the rows equals 1.
\(Yaseen = (Aij) | Jurgen = (Bij)\)
\(\sum_{J=1}^{N}Aij = \sum_{J=1}^{N}Bij = 1\)
\((Yaseen * Jurgen)_{ij} = \sum_{k = 1}^{N} Aik *Bkj\)
\(= (A_{i1}B_{j1}+ \cdots + A_{iN}B_{jN})\)
\((Yaseen * Jurgen)_{ij} = \sum_{k = 1}^{N} Aik *Bkj\)
\(= A_{11}B_{11} +\cdots + A_{in}B_{n1} + \cdots A_{in}B_{nn}\)
\(= A_{i1}(B_{11} +\cdots +B_{1n}) +\cdots A_{in}(B_{n1} + \cdots B_{nn})\)
\(= A_{i1}+\cdots A_{in} = 1\)
Y <- matrix(c(2, 5, 10), ncol = 1)
P <- matrix(c(0.3, 0.3, 0.4, 0.3, 0.3, 0.4, 0.3, 0.3, 0.4), ncol = 3, byrow = TRUE)
B <- 0.952
Gamma <- 2
# Utility in our example is just a function of consumption, and Y is the
# aggregate total of consumption because it is the entire endownment of the
# economy.
U_Y <- (Y^1 - Gamma)/(1 - Gamma)
MU <- Y^(-Gamma)
Kernel <- MU %*% (1/t(MU))
Q <- P * B * Kernel
eigen(Kernel)## $values
## [1] 3.000000000000000000 0.000000000000000222 0.000000000000000000
##
## $vectors
## [,1] [,2] [,3]
## [1,] 0.98667 -0.99662 0.92914
## [2,] 0.15787 0.07973 -0.36573
## [3,] 0.03947 0.01993 0.05427This can be further proved in the NxN case by the fact that all Markov Matrices A are such that A is an NxN Matrix.
We Know that NxN Matrices are such that the Trace(A) = Y1 +…+Yn where Yi is an eigenvalue of A.
And that the deterimnat of an NxN matrix is such that det(A) = Y1…YN where Yi is an eigenvalue of A.
Every asset pricing kernel will have a trace of NBeta, because we know the diagonal elements will have identical marginal utility of consumption when staying in the same state. Therefore we know that the sum of any Asset pricing kernel must be such that Y1+….+Yn = NBeta.
After placing any NxN Asset Pricing kernel into RREF we will always observe that we have at least One Free Variable and therefore will Violate the Full Rank Assumption.
Because our matrix A does not have Full Rank we know that Our Matrix is a Non Invertible Matrix.
Non Invertible Matrices are such that thier determinant is equal to 0.
Therefore we know that any asset Pricing Kernel has Eigenvalues that sum to N*Beta.
And the product of those same eigenvalues equals 0.
In order for N numbers to sum to a nonzero number and to produce 0 when multiplied we know that one of them must also be 0.
Therefore we have just proven that every Asset pricing kernel has at least one Eigenvalue of 0.
Y<-matrix(c(300000000000000,300000000000001,300000000000002), ncol = 1)
Y## [,1]
## [1,] 300000000000000
## [2,] 300000000000001
## [3,] 300000000000002B<- 0.952
Gamma<- 2
MU<- Y^(-Gamma)
Kernel<- MU%*%(1/t(MU))
Kernel## [,1] [,2] [,3]
## [1,] 1 1 1
## [2,] 1 1 1
## [3,] 1 1 1Q<- B*Kernel
Q## [,1] [,2] [,3]
## [1,] 0.952 0.952 0.952
## [2,] 0.952 0.952 0.952
## [3,] 0.952 0.952 0.952RF <- 1/sum(Q[1,])
RF2 <- 1/(sum(Q[2,]))
RF3 <- 1/sum(Q[3,])
#THREE RISK FREE RATES ACROSS THE STATES
RF## [1] 0.3501RF2## [1] 0.3501RF3## [1] 0.3501As we can see from the example above when there is very little difference in Endowment between the three states, the ratios of Marginal Utililities between non identical states will all be approximately equal based the almost negligible differnces in marginal utility across the three states. Therefore we can observe an asset pricing kernel that has almost identical Risk Free Rates across all three states with non Identical Endonments.
Y<-matrix(c(1,20,30000), ncol = 1)
Y## [,1]
## [1,] 1
## [2,] 20
## [3,] 30000P<- matrix(c(1,0,0, 0,1,0, 0,0,1), ncol=3, byrow = TRUE)
P## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1B<- 0.952
Gamma<- 2
MU<- Y^(-Gamma)
Kernel<- MU%*%(1/t(MU))
Kernel## [,1] [,2] [,3]
## [1,] 1.000000000000 400.0000000000 900000000
## [2,] 0.002500000000 1.0000000000 2250000
## [3,] 0.000000001111 0.0000004444 1Q<- P*B*Kernel
Q## [,1] [,2] [,3]
## [1,] 0.952 0.000 0.000
## [2,] 0.000 0.952 0.000
## [3,] 0.000 0.000 0.952RF <- 1/sum(Q[1,])
RF2 <- 1/(sum(Q[2,]))
RF3 <- 1/sum(Q[3,])
#THREE RISK FREE RATES ACROSS THE STATES
RF## [1] 1.05RF2## [1] 1.05RF3## [1] 1.05THis is another example of a stable RF rate, however I have now included that the probabilitiiy of staying in the same state that we started in is 1 and every other probabiltity is zero. We are essentially showing that when there is no MacroEconomic uncertantity, after our initial condition, the Marginal utility ratios of non identical states will all go to zero. Although we have huge discrepancies between our endownment sizes across states, when you konw that you are going to stay in one state forever, the marginal utility is equalized between time = t and time = t +1, therefore the only paramter with which you discount assets is your Beta discount factor (patience level between money today and money at time = t+1). Therefore in each state the sum of the rows is just Beta* The probability of staying in the same state= Beta across the diagonols. The Rf Rate is just 1 / E[mv] = 1 / B across all states.
Y<-matrix(c(300000000000000,300000000000001,300000000000002), ncol = 1)
Y## [,1]
## [1,] 300000000000000
## [2,] 300000000000001
## [3,] 300000000000002B<- 0.952
Gamma<- 0
MU<- Y^(-Gamma)
Kernel<- MU%*%(1/t(MU))
Kernel## [,1] [,2] [,3]
## [1,] 1 1 1
## [2,] 1 1 1
## [3,] 1 1 1Q<- B*Kernel
Q## [,1] [,2] [,3]
## [1,] 0.952 0.952 0.952
## [2,] 0.952 0.952 0.952
## [3,] 0.952 0.952 0.952RF <- 1/sum(Q[1,])
RF2 <- 1/(sum(Q[2,]))
RF3 <- 1/sum(Q[3,])
#THREE RISK FREE RATES ACROSS THE STATES
RF## [1] 0.3501RF2## [1] 0.3501RF3## [1] 0.3501Lastly and most importantly is the case in which we have a differntiable utility function with Gamma = 0. We can see that when we take the marginal utility of any consumption or any parameter in our utility function it all goes to one when we take the derivative. Therefore every row will sum to N* Beta. Likewise, every state will have a risk free rate = 1 / N * B. When the marginal value of consumption in all states is equal to the same value, you are indiffernt between returns in different states, and therefore will have a stable RF rate across three rates. It is important to note that in this exmaple we did not include probabilities, nor did we include Marginal utilities that are so similar it would be notationally intensive to view the difference between them.
# Initial Endowments of 3 Individuals
Y1 <- matrix(c(10, 5, 0, 8), ncol = 1)
Y1## [,1]
## [1,] 10
## [2,] 5
## [3,] 0
## [4,] 8Y2 <- matrix(c(4, 12, 4, 18), ncol = 1)
Y2## [,1]
## [1,] 4
## [2,] 12
## [3,] 4
## [4,] 18Y3 <- matrix(c(3, 5, 12, 16), ncol = 1)
Y3## [,1]
## [1,] 3
## [2,] 5
## [3,] 12
## [4,] 16Y <- Y1 + Y2 + Y3
Y## [,1]
## [1,] 17
## [2,] 22
## [3,] 16
## [4,] 42# Initial Markov Probability Matrix In which rows sum to one.
P <- matrix(c(0.25, 0, 0.5, 0.25, 0.1, 0, 0.2, 0.7, 0.55, 0, 0.3, 0.15, 0.4,
0.6, 0, 0), ncol = 4, byrow = TRUE)
P## [,1] [,2] [,3] [,4]
## [1,] 0.25 0.0 0.5 0.25
## [2,] 0.10 0.0 0.2 0.70
## [3,] 0.55 0.0 0.3 0.15
## [4,] 0.40 0.6 0.0 0.00# Long Run Invariant Distribution
P %^% 100## [,1] [,2] [,3] [,4]
## [1,] 0.3502 0.1354 0.2888 0.2256
## [2,] 0.3502 0.1354 0.2888 0.2256
## [3,] 0.3502 0.1354 0.2888 0.2256
## [4,] 0.3502 0.1354 0.2888 0.2256B <- 0.952
Gamma <- 2# Utility in our example is just a function of consumption, and Y is the
# aggregate total of consumption because it is the entire endownment of the
# economy. U_Y<- (Y^1-Gamma)/ (1-Gamma)
MU <- Y^(-Gamma)
MU## [,1]
## [1,] 0.0034602
## [2,] 0.0020661
## [3,] 0.0039062
## [4,] 0.0005669Kernel <- (1/MU) %*% (t(MU))
Kernel## [,1] [,2] [,3] [,4]
## [1,] 1.0000 0.5971 1.129 0.1638
## [2,] 1.6747 1.0000 1.891 0.2744
## [3,] 0.8858 0.5289 1.000 0.1451
## [4,] 6.1038 3.6446 6.891 1.0000Q <- B * P * Kernel
Q## [,1] [,2] [,3] [,4]
## [1,] 0.2380 0.000 0.5374 0.03899
## [2,] 0.1594 0.000 0.3600 0.18284
## [3,] 0.4638 0.000 0.2856 0.02072
## [4,] 2.3243 2.082 0.0000 0.00000PV <- ginv(diag(4) - Q)
# MV gives us the present discounted market value of the Economys entire
# productive Capacity.
MV <- PV %*% Y
MV## [,1]
## [1,] 306.0
## [2,] 495.2
## [3,] 272.8
## [4,] 1784.2M1 <- PV %*% Y1
M1## [,1]
## [1,] 92.64
## [2,] 141.54
## [3,] 75.17
## [4,] 517.97M2 <- PV %*% Y2
M2## [,1]
## [1,] 89.75
## [2,] 155.79
## [3,] 79.85
## [4,] 550.94M3 <- PV %*% Y3
M3## [,1]
## [1,] 123.6
## [2,] 197.9
## [3,] 117.8
## [4,] 715.3# M1-M3 gives us the prices of the Individuals initial Endownments of the
# Economy.Theta1 <- M1/MV
Theta2 <- M2/MV
Theta3 <- M3/MV
# C1_1 denotes Indiviual 1's consumption in each state given that we are
# starting in State = S1.
C1_1 <- Theta1[1] * Y
C1_1## [,1]
## [1,] 5.146
## [2,] 6.660
## [3,] 4.844
## [4,] 12.715C2_1 <- Theta2[1] * Y
C2_1## [,1]
## [1,] 4.986
## [2,] 6.453
## [3,] 4.693
## [4,] 12.319C3_1 <- Theta3[1] * Y
C3_1## [,1]
## [1,] 6.867
## [2,] 8.887
## [3,] 6.464
## [4,] 16.967# A1_1 denotes the amounts of the four possible Arrow securities that
# Individual 1 Bought/Sold given that we are starting in State = S1.
A1_1 <- PV %*% (C1_1 - Y1)
A1_1## [,1]
## [1,] -0.000000000000006217
## [2,] 8.386669457565833596
## [3,] 7.423302103893926862
## [4,] 22.174112242069220713A2_1 <- PV %*% (C2_1 - Y2)
A2_1## [,1]
## [1,] 0.0000000000000151
## [2,] -10.5359304624091301
## [3,] 0.1687066612484784
## [4,] -27.6151805179549612A3_1 <- PV %*% (C3_1 - Y3)
A3_1## [,1]
## [1,] 0.00000000000002665
## [2,] 2.14926100484334492
## [3,] -7.59200876514237777
## [4,] 5.44106827588596964# To ensure our equilibrium condition holds we sum the total of arrow
# security holdings and it should sum to O.
A <- A1_1 + A2_1 + A3_1
signif(A, digits = 1)## [,1]
## [1,] 0.00000000000004
## [2,] 0.00000000000005
## [3,] 0.00000000000003
## [4,] 0.00000000000020# C1_1 denotes Indiviual 1's consumption in each state given that we are
# starting in State = S2.
C1_2 <- Theta1[2] * Y
C1_2## [,1]
## [1,] 4.859
## [2,] 6.287
## [3,] 4.573
## [4,] 12.003C2_2 <- Theta2[2] * Y
C2_2## [,1]
## [1,] 5.348
## [2,] 6.921
## [3,] 5.033
## [4,] 13.212C3_2 <- Theta3[2] * Y
C3_2## [,1]
## [1,] 6.794
## [2,] 8.792
## [3,] 6.394
## [4,] 16.784# A1_2 denotes the amounts of the four possible Arrow securities that
# Individual 1 Bought/Sold given that we are starting in State = S2.
A1_2 <- PV %*% (C1_2 - Y1)
A1_2## [,1]
## [1,] -5.182085684293951999
## [2,] -0.000000000000007994
## [3,] 2.803132390428351162
## [4,] -8.041483493014190742A2_2 <- PV %*% (C2_2 - Y2)
A2_2## [,1]
## [1,] 6.51010448142943066
## [2,] -0.00000000000004974
## [3,] 5.97289234805747427
## [4,] 10.34379916472604322A3_2 <- PV %*% (C3_2 - Y3)
A3_2## [,1]
## [1,] -1.328018797135507967
## [2,] 0.000000000000006661
## [3,] -8.776024738485856957
## [4,] -2.302315671712055867# To ensure our equilibrium condition holds we sum the total of arrow
# security holdings and it should sum to O.
A <- A1_2 + A2_2 + A3_2
signif(A, digits = 1)## [,1]
## [1,] -0.00000000000003
## [2,] -0.00000000000005
## [3,] -0.00000000000003
## [4,] -0.00000000000020# C1_3 denotes Indiviual 1's consumption in each state given that we are
# starting in State = S3
C1_3 <- Theta1[3] * Y
C1_3## [,1]
## [1,] 4.684
## [2,] 6.061
## [3,] 4.408
## [4,] 11.572C2_3 <- Theta2[3] * Y
C2_3## [,1]
## [1,] 4.976
## [2,] 6.439
## [3,] 4.683
## [4,] 12.293C3_3 <- Theta3[3] * Y
C3_3## [,1]
## [1,] 7.341
## [2,] 9.500
## [3,] 6.909
## [4,] 18.135# A1_3 denotes the amounts of the four possible Arrow securities that
# Individual 1 Bought/Sold given that we are starting in State = S3.
A1_3 <- PV %*% (C1_3 - Y1)
A1_3## [,1]
## [1,] -8.32614166762352070
## [2,] -5.08832927409610569
## [3,] -0.00000000000001288
## [4,] -26.37377870161802917A2_3 <- PV %*% (C2_3 - Y2)
A2_3## [,1]
## [1,] -0.18922516451134097
## [2,] -10.84217181087237769
## [3,] -0.00000000000002709
## [4,] -28.71851060975733816A3_3 <- PV %*% (C3_3 - Y3)
A3_3## [,1]
## [1,] 8.51536683213478796
## [2,] 15.93050108496837680
## [3,] -0.00000000000002243
## [4,] 55.09228931137493390# To ensure our equilibrium condition holds we sum the total of arrow
# security holdings and it should sum to O.
A <- A1_3 + A2_3 + A3_3
signif(A, digits = 1)## [,1]
## [1,] -0.00000000000007
## [2,] -0.00000000000010
## [3,] -0.00000000000006
## [4,] -0.00000000000040# C1_1 denotes Indiviual 1's consumption in each state given that we are
# starting in State = S4.
C1_4 <- Theta1[4] * Y
C1_4## [,1]
## [1,] 4.935
## [2,] 6.387
## [3,] 4.645
## [4,] 12.193C2_4 <- Theta2[4] * Y
C2_4## [,1]
## [1,] 5.249
## [2,] 6.793
## [3,] 4.940
## [4,] 12.969C3_4 <- Theta3[4] * Y
C3_4## [,1]
## [1,] 6.816
## [2,] 8.820
## [3,] 6.415
## [4,] 16.839# A1_4 denotes the amounts of the four possible Arrow securities that
# Individual 1 Bought/Sold given that we are starting in State = S4.
A1_4 <- PV %*% (C1_4 - Y1)
A1_4## [,1]
## [1,] -3.80294171986601404
## [2,] 2.23200179786875541
## [3,] 4.03272980798377745
## [4,] -0.00000000000004974A2_4 <- PV %*% (C2_4 - Y2)
A2_4## [,1]
## [1,] 4.73610492031864183
## [2,] -2.87103472294870166
## [3,] 4.39125523449399680
## [4,] -0.00000000000008171A3_4 <- PV %*% (C3_4 - Y3)
A3_4## [,1]
## [1,] -0.93316320045266365
## [2,] 0.63903292507988896
## [3,] -8.42398504247781332
## [4,] -0.00000000000009681# Sum to Zero
A <- A1_4 + A2_4 + A3_4
signif(A, digits = 1)## [,1]
## [1,] -0.00000000000004
## [2,] -0.00000000000006
## [3,] -0.00000000000004
## [4,] -0.00000000000020In a recursive competitive equillibrium our holdings of Arrow securities based on the initial State is identical to the holdings for the next state, and across all states for thatt matter. It only depends on the initial given condition, afterwhich it remains constant in perpetuity, essentially defining your role in the insurance framework as a seller a buyer or as none, all as a result of your initial endowment across states and your position(relative wealth) in the first state that occurs.
We have shown different holdings among individuals of different amounts of arrow secutities given different intial condidions(starting state), therefore proving that the recursive competitive equillibrium is dependeant upon History (being the first condition/state).