Input-Output Analysis Exercises Energy Management Class P9 João Rodrigues Instituto Superior Técnico Lisbon, Portugal joao.rodrigues@ist.utl.pt 8 and November 2
Review Input-Output Analysis is a tool primarily used to calculate the total indirect requirements associated with final demand. The economic system: intersectoral transactions Z, added value v, imports im, final demand y and total supply/demand x. x = Z + y and x = Z + v + im Leontief model: demand stimulates supply and there is a fixed recipe of production. A = technical coefficients and L = Leontief inverse. δx = Lδy where L = (I A) and A = Zˆx Leontief multiplier: if there is a fixed coefficient between total production and some input, b i = e i /x i, demand stimulates a certain amount of that input. B = multiplier. δe = B δy where B = b L and b = ˆx e Notation: = transpose and ˆ = diagonal matrix. João Rodrigues (IST) Energy Management P9 8 and November 2 2 / 3
Problem Problem Consider an economy with 3 sectors. a) Indicate which sector has the largest contribution to GDP. And to inter-industry purchases? b) Obtain a reasonable approximation to the Leontief inverse. c) Determine in which sector an unitary increase in final demand leads to the highest increase in total demand of the same sector. And total demand of any sector? d) Determine in which sector an unitary increase in final demand leads to the highest increase in imports of the same sector. And imports of any sector? João Rodrigues (IST) Energy Management P9 8 and November 2 3 / 3
Problem a) The economy is described as: Z = 5 3 6 2 3 2 5 2 5 y = 2 5 5 x = 6 57 33 im = 65 95 v = 84 4 422 All data except total demand and added value is obtained from the source data. Total demand is obtained from the balance of sector supplies, and added value from the balance of sector demands. The largest contribution to GDP comes from Sector C, which has the largest added value. The sector with highest share of inter-industry purchases is B, because it has the highest column sum of the matrix of intersectoral transactions, excluding the main diagonal. João Rodrigues (IST) Energy Management P9 8 and November 2 4 / 3
Problem b) We try to use the Taylor expansion L = I + A + A 2 +... A =.3.9..2.9.4.3.27.9 A 2 =.36.9.2.7.32.2.28.95.9 Because the entries of the matrix of technical coefficients are small, higher order terms can be neglected and: L I + A =.3.9..2.9.4.3.27.9 João Rodrigues (IST) Energy Management P9 8 and November 2 5 / 3
Problem c) We use the Leontief inverse: δx = Lδy. The effect of a unitary variation in the demand of sector A in the total demand of every sector is the first column of the Leontief inverse: δy = and δx = Lδy =.3.2.3 Therefore, the meaning of entry a ij is the marginal effect in total demand of sector i of an increase in demand of sector j. To know in which sector an increase in final demand has the largest marginal increase in total demand of that same sector we compare the entries on the main diagonal of the Leontief inverse. The answer is A. The effect on total demand of all sectors is obtained by summing over columns of the Leontief inverse. Now the answer is sector B. João Rodrigues (IST) Energy Management P9 8 and November 2 6 / 3
Problem c - continued) This question could have been obtained using multipliers: δe = B δy, B = b L and b = ˆx e. In this case, the production input is totat output itself, so e = x. The multiplier effect of a margin increase in final demand of A in the total production of A is obtained using direct coefficients and stimulus of: δy = b = On the other hand, if the focus is on the effect of a marginal increase of final demand of A in the total production of all sectors: δy = and b = João Rodrigues (IST) Energy Management P9 8 and November 2 7 / 3
Problem d) We use multipliers again, but now the production inputs are imports, e = im. The multiplier effect of a marginal increase in final demand of A in the imports of A is obtained using direct coefficients and stimulus of:.44 δy = and b = On the other hand, if the focus is on the effect of a marginal increase of final demand of A in the imports of all sectors: δy = and.44 b =..79 João Rodrigues (IST) Energy Management P9 8 and November 2 8 / 3
Problem d - continued) If we want to perform a single operation to calculate the three multipliers of final demand on imports of the same sector, we want to find: ˆB =.46..8 where B = ˆb L and b =.44..79 If we want to perform a single operation to calculate the three multipliers of final demand on imports of all sectors, we want to find:.422.44 B =. where B = b L and b =..85.79 It is always sector A that has a largest marginal effect on imports. But even though sector B has no direct imports, it has a positive effect on imports of sectors A and C. João Rodrigues (IST) Energy Management P9 8 and November 2 9 / 3
Problem Problem 2 Consider an economy with 2 sectors and no imports of which the following data is available: A = [.2.3.4.5 ] L = [.79.7.43 2.86 ] x = [ 5 a) Describe the full IO system. b) Determine the required demand stimulus (in %) that will lead to an added value increase of 3% in sector and 4% in sector 2. c) The energy intensity of the economy is total direct energy consumption per GDP, or EI = e + e 2 v + v2. Considering that direct energy requirements per unit of total output are constant, determine under which conditions the energy intensity decreases, when the demand stimulus is applied. João Rodrigues (IST) Energy Management P9 8 and November 2 / 3 ]
Problem 2 a) Using the matrix of technical coefficients and total output we obtain the matrix of intersectoral transactions. [ ] [ ] [ ] [ ] 2 45 35 4 Z = y = v = x = 4 75 35 3 5 b) We want to determine dy /y and dy 2 /y 2 for which dv /v =.3 and dv 2 /v 2 =.4. If the production structure is fixed, b = v x = v + dv x + dx and x + dx x = v + dv v so dx x = dv v Next we observe that: δy = (I A) δx implies ŷ δy = ŷ (I A) ˆxr João Rodrigues (IST) Energy Management P9 8 and November 2 / 3
Problem 2 b - continued) Performing substitutions we find [ ] ŷ.7 δy = given r = ˆx δx = ˆv δv =.5 [.3.4 c) If all variations are expressed in relative terms, maybe it is a good idea to examine energy intensity in relative terms too. ] EI = E V so dei CI = de E dv V where E = e + e 2 V = v + v 2 Expansion of de = de + de 2 and dv = dv + dv 2 leads to: dei ( e = EI E v ) ( e (r r 2 ) = V E 4 ) (.3.4) 7 So if e /E > 4/7 the energy intensity will decrease. João Rodrigues (IST) Energy Management P9 8 and November 2 2 / 3
Problem Problem 3 Consider an economy with 3 sectors of which the following data is available: 5?.24.32.67 5 Z = 2 2? L =.4.45.94 y = 6 2?.26.43.6 35 a) Determine the primary inputs (added value + imports) of each sector. b) For which sector a marginal increase in final demand leads to the highest marginal increase in total output of that sector? And total output of all sectors? c) Repeat the previous answer, for primary inputs instead of total output. d) The energy imports of each sector are 5, 2 and 5 toe. Determine for which sector a marginal increase in final demand leads to the highest increase in energy imports. No solutions this time! João Rodrigues (IST) Energy Management P9 8 and November 2 3 / 3