This solution for R is the reduced-form equation for R. (Compare your answer to question 4 in the exercises to section 7.1.) It expresses the endogenous variable R as a function of exogenous variables and parameters only.
Example 9.19 Consider the following closed economy IS-LM model:
Solve for the equilibrium level of income in terms of government spending, G. Ai what level of public spending does the government balance its budget .'
These can be written in the form
Now, |A| = 0.4<-10) - 1 (I ) = -5 and to find Y we need |A| | which is
Income is just a simple linear function of government spending. In this case the government expenditure multiplier is 2—each dollar increase in government spending increases income by 2 dollars. Tax revenue is
This tells us that, while an increase in G generates more income (through the multiplier), each dollar increase in government spending only generates an extra
50 cents in tax revenue. The government deficit is G—T, and the budget is balanced when this is zero, or when G = T. This implies that
The "Open" Leontief Input-Output Model
This model looks at the economy as a number of interrelated industrial sectors. The industries are interrelated because an industry's output, in general, is used as an input into some other industries' production processes as well as possibly finding its way into final demand by consumers. Therefore, in general, each industry is potentially the producer of an intermediate good that may also be used in final consumption. The problem is to find the production level for each industry that is just sufficient to supply the demands from industry and consumers alike.
To model such a system, we start by expressing all outputs and demands in money terms. Since prices are assumed to be fixed, we can always recover the implied physical quantities by dividing through by the appropriate price per unit. There are assumed to be n goods produced by n industries, with the money value of output of the /th industry given by x,. The production vector for the economy in money terms is therefore given by
The final demand by consumers in money terms for the output of industry i is fixed at dj. and so the final demand vector as a whole is ih d =
Finally we need to specify the input requirements of each industry. Denote by the amount of the money value of the output of industry ; needed to produce one unit of output in industry j. This is a fixed technological requirement and the full.
economywide array of input requirements is given by
Note that this is necessarily a square matrix, though, of course, some of the u-q may be zero, reflecting the fact that industry j may not use any of the output of industry i as input. We refer to A as the matrix of production coefficients. Noticc also that n,, may be positive for some or all industries, meaning that some (or all) industries require some of their own output to be used in their production process.
The total money value of the output of industry i required by all industries is a'Jxl ~ ai 1*1 + «'2*2 H-----1- <it><x"
where is the money value of the output of industry / required to produce the \ j units of output of industry j. In total, the demands made on the output of all industries can be expressed as a column array
Each row here is the total demand on the output of industry i made by the entire production sector. Of course, in general there will also be a final demand from the consumption sector for the output of industry i, d,. This economy demand for ihe output of industry i in the economy as a whole is
X/'if}X) +</, and for supply to equal demand in sector i. we must have n
If all demands in the economy are to be supplied, this must hold for all n industries, and so x = Ax + d (9.7)
We can now pose the problem set by this model. Given the matrix of production coefficients A (summarizing the input requirements of industry) and a linal demand vector d, what is the vector of outputs x, that will just satisfy equation (9,7)? We can rearrange equation (9.7) to give x — Ax = d or
(/ - A)x = d and so. if (/ - ,4)"1 exists, we may write the solution as x = (/ — /4)~'d
Theorem 9.17 If (/■ — A) 1 has only nonnegative entries, theri for any d > 0. there is a unique nonnegative solution for x.
We will use the input requirements matrix that was first presented in example 8.3. In this case the economy consists of three industries, an agricultural industry, a mining industry, and a manufacturing industry. To produce one unit of agricultural output, the agricultural sector requires $0.3 of its own output, $0.2 of mining output and $0.4 of manufacturing oiiipui To produce one unit of mining output, the mining sector requires $0.5 of agricultural output, S0.2 of its own output, and $0.2 of manufacturing output. To produce one unit of manufacturing output requires $0.3 of agricultural output, $0.5 of mining output, and $0.3 of its own output. Final demands by consumers amount to $20,000, $10,000, and $40,000 for goods 1.2. and 3. respectively. Find the equilibrium quantities of output for the three sectors.
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