The national income higher i d ei private consumption

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Unformatted text preview: ter 6 IS­LM Model M 0 = kY − li Notice that the amount of money in the economy is fixed! The amount of money in the economy will tend to The increase when national income increases. increase But it will tend to decrease if higher interest But rate depresses the economic activity. rate Collecting the four equations together, this is the linear system that we Collecting get: get: Y C I G − −=0 ( −) − = a b C − 1 tY I ei d + = kY − = 0 li M 1 b(1 −t ) The matrix form representation will be: 0 k If you were to compute the If determinant of matrix A, what would be the most rational way to do this? be −1 −1 0 0 −1 0 1 0 0 Y G0 0 C − a = e I d −l i M 0 7 IS­LM Model Let us apply Cramer’s rule to solve this system. First, compute the determinant of the coefficient matrix. Its last row has First, two zeroes in it, so it makes sense to expand the determinant by the last row. row. − 1 A =− − k1 0 − 1 0 1 0 1 0 −l b(1 −t ) e 0 − 1 − 1 − 1 0 0 1 In fact, we are lucky since in both third-order determinants there is a In column or a row with two zeroes. That means we only have to compute two two-order determinants. two 1 −1 −1 = −1 −1 0 A = −k × e( −1) −1 b (1 − t ) − 1 3+3 = −1 + b (1 − t ) × ( −1) − l × ( − 1) 3+ 3 × ( b(1 − t ) − 1) A= ke − ( − ) + bl 1 t l 8 IS­LM Model To find the equilibrium level of national income, for example, we first To compute the auxiliary determinant by substituting the first column in matrix A by the right-hand side column vector: matrix G0 0 −a −1 0 0 d 0 1 e M0 A1 = −1 −1 0 0 −l A1 = e( −1) Notice that the fourth column contains Notice two zeroes, so let’s expand the determinant by the last column. determinant G0 3+4 −1 −a M0 −1 0 −1 0 −l ( −1) 0 −1 −1 G0 = −eM 0 ( −1) − l d +1 −a −1 0 = M 0 e − l [ d ( −1) +1( − G0 − a ) ] = M 0 e + l ( G0 + d + a ) G0 4 +4 −1 −a d −1 0 −1 0= 1 −1 = −1 9 IS­LM Model By Cramer’s rule in order to obtain the equilibrium value of national By income, we need to divide the auxiliary determinant A by the determinant 1 of the coefficient matrix: M 0 e +l (G0 +d +a ) Y= ke −bl (1 −t ) +l * We can rearrange this expression in order to express the equilibrium We level of national income as a linear combination of the two exogenous variables, money stock and the government expenditure. variables, e l a +d Y* = M0 + G0 + = ek + l [1 − b(1 − t ) ] ek + l [l − b(1 − t ) ] ek + l [l − b(1 − t ) ] = A × M 0 + B × G0 + C A is the Keynesian money multiplier, B is the government is expenditure multiplier, while C is the structural coefficient. expenditure 10 Leontief Input­Output Models Consider the following question: How much output should each one of the n industries in the economy produce so there is no shortage of steel anywhere in the economy? Let us formalize our economy with n industries by means of an Let input-coefficient matrix. a23...
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