The Leontief Model - Section 2.6 (a) Consider an economy with three sectors: Manufacturing, Agriculture and Services. For each unit of output,...
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2. The Leontief Model - Section 2.6
(a) Consider an economy with three sectors: Manufacturing, Agriculture and Services. For each
unit of output, Manufacturing requires 0.2 units from other companies in that sector, 0.3
units from Agriculture and 0.1 units from Services. For each unit of output, Agriculture
uses 0.2 units from Manufacturing and 0.1 units from Agriculture, but needs no input from
Services. Finally, for each unit of output the Services sector consumes no products from
Manufacturing, 0.3 units of Agricultural products and 0.2 units of Service products. Construct
the consumption matrix for this economy.
(b) Find the production level x for the economy described above that will satisfy a final demand
of 40 units of Manufacturing, 60 units of Agricultural products and 80 units of Services.

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(c) Suppose demand increases by 1 unit of Manufacturing products compared to the demand
d given in part (b). Denote by Ad the vector representing the change in demand, namely i} i. Compute the production vector Ax that satisfies a final demand equal to Ad. How does
this vector relate to the matrix (I — 0)”? ii. Compute the production vector 3cm that satisfies the new demand dnew. iii. How are x, xnew and Ax related to each other? iv. For bonus points: explain why the result you found above regarding x,xnew and Ax
is always true using the production equation and the identity dnew = d + Ad. (d) On page 137 of the text book it is explained that, if the column sums of a matrix C are all
strictly less than 1I then the inverse of the matrix I — C can be approximated by 1
Ad = [ll] . The new demand vector is therefore given by dHEW = d + Ad. (I—C)_1mI+C+02+Ca+...+Cm, where m is a sufficiently large positive integer. For the consumption matrix in part (a), how
large must m be taken so that the right hand side of the formula displayed above approximates
(I — (30—1 with error less than 1101? This means that every entry of the matrix sum I + C +
02 + 03 + . . . + 0&quot;” must be within 0.01 of the corresponding entry of (I — (II—1.

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