Introduction to Probability · Problem 11.2.22

Markov Chains — Grinstead & Snell

Mathematics — Probability · absorbing-markov-chains

Problem

In the Leontief economic model, there are

n

industries

1, 2, \ldots, n

. The

i

th industry requires an amount

0 \leq q_{ij} \leq 1

of goods (in dollar value) from company

j

to produce 1 dollar's worth of goods. The outside demand on the industries, in dollar value, is given by the vector

\mathbf{d} = (d_1, d_2, \ldots, d_n)

. Let

\mathbf{Q}

be the matrix with entries

q_{ij}

. (a) Show that if the industries produce total amounts given by the vector

\mathbf{x} = (x_1, x_2, \ldots, x_n)

then the amounts of goods of each type that the industries will need just to meet their internal demands is given by the vector

\mathbf{xQ}

. (b) Show that in order to meet the outside demand

\mathbf{d}

and the internal demands the industries must produce total amounts given by a vector

\mathbf{x} = (x_1, x_2, \ldots, x_n)

which satisfies the equation

\mathbf{x} = \mathbf{xQ} + \mathbf{d}

. (c) Show that if

\mathbf{Q}

is the

Q

-matrix for an absorbing Markov chain, then it is possible to meet any outside demand

\mathbf{d}

. (d) Assume that the row sums of

\mathbf{Q}

are less than or equal to 1. Give an economic interpretation of this condition. Form a Markov chain by taking the states to be the industries and the transition probabilities to be the

q_{ij}

. Add one absorbing state 0. Define

q_{i0} = 1 - \sum_j q_{ij}

. Show that this chain will be absorbing if every company is either making a profit or ultimately depends upon a profit-making company. (e) Define

\mathbf{xc}

to be the gross national product. Find an expression for the gross national product in terms of the demand vector

\mathbf{d}

and the vector

\mathbf{t}

giving the expected time to absorption.

Related theory

Markov Chains Transition Matrices And Stationary Distributions

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