The Leontief Open Economy Production Model
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1 The Leontief Open Economy Production Model We will look at the idea for Leontief's model of an open economy a term which we will explain below. It starts by looking very similar to an example of a closed economy that we saw near the beginning of this course. Here, however, we start with a matrix whose columns represent consumption (rather than production as in the earlier example). In the case of simple examples, we can solve the equations that we will set up. However, proofs of the deeper and more interesting theorems about what happens in general are beyond the scope of this course. Assume that an economy has 8 sectors W" ß ÞÞÞß W8. Amounts of consumption or production in each sector are measured in common units, say $. We begin with a consumption matrix G: the 4 >2 column lists the amounts (in $) demanded Ð œ needed or consumed Ñ by sector W4 from sectors W" ß W# ß ÞÞÞß W8 in order for W 4 to produce one unit ($) of output. Consumption vectors for W" â W4 Æ â W8 W Consumed from W W Of course, entry each -!Þ 34 " 3 8 Ä Ä Ä - â - â - - â - â - - â - â - "" "4 "8 3" " œg -"" For example, Column " œ -3" lists the demands in $) by Sector 1 from - 8" Sectors W",..., W8 to produce one unit ($) of product. We could also think of this a list of costs for Sector 1 to produce one unit of product; the sum of the entries in Column " gives the total cost of all products consumed by Sector 1 to produce one unit of its product.
2 In the same way: the entries in Column 4represent the demands of sector W 4 (in $) from sectors W" ß W# ß ÞÞÞß W8 in order for W4 to produce one unit ($) of product. The column sum >2 represents the costs to Sector 4in producing one unit. (If the 4 column sum is ", we say that sector W 4 is profitable.) -"4 B" -"4 If we weight Column 4 with a scalar B 4 0, then the vector B4-34 œ B" -34 lists - B - 84 " 84 the demands (costs) of sector W4 from other sectors to produce B4 units ($) of its own product. If B" ß B# ß ÞÞÞß B 8 (! ) are amounts to be produced by sectors W" ß W# ß ÞÞÞß W8Þ units, then B" B# Bœ is called a production vector. What does the vector GB mean? It is a linear B 8 combination of the columns of G with the B 3 's as weights: -"" -"# -"8 - B ÞÞÞ- B GB œ B" -31 B# -3# ÞÞÞB8-38 œ -3" B" ÞÞÞ - B B ÞÞÞ- B 81 8# 88 total demanded by Sectors W ß ÞÞÞ ß W œ total demanded by Sectors W ß ÞÞÞ ß W total demanded by Sectors W ß ÞÞÞ ß W from W " 8 " from W " 8 3 from W " 8 8 "" " " " " 88 8 to produce B to produce B to produce B In other words, GB lists the demands that must be supplied from each sector (to the others) to deliver the production vector B Þ The equation GB œ B asks for an equilibrium value an production vector B for which everything is in balance where the list of amounts B ($) produced by each sector œ the list of total demands GB ($) from the sectors. For equilibrium, the production from each sector must be not too large, not too small, but just right. This is a closed economy everything is just exchanged among the sectors. Now we add another feature to the economic model: an open sector. It is nonproductive; it produces nothing that sectors W ß ÞÞÞß W use. The open sector simply demands goods " 8
3 produced by sectors W" ß ÞÞÞß W8Þ (For example, these could be demands of the government, demands for foreign export, for charitable groups,...).. A final demand vector. œ. from each sector Wß ÞÞÞß W8Þ " # 8 tells how many units ($) the open sector demands We want to know if it is possible to set a level of production B so that both the productive and open sectors are satisfied, with nothing left over. That is, we want to find an B so that Leontief Open Economy Production Model B œ GB. (***) Å Å Å Total production œ Demand from + Demand from productive sectors open sector to produce B ( intermediate ( called final demand ) demand ) We can rewrite this equation as: ÐM GÑ B œ. To take a simple example, suppose the economy has 4 productive sectors the consumption matrix is Consumption vectors for W" W# W$ W% Æ Æ Æ Æ Þ"! Þ!& Þ$! Þ#! Þ"& Þ#& Þ!& Þ"! Gœ Þ$! Þ"! Þ"! Þ#& Þ"& Þ#! Þ"! Þ#! ÐThe column sums in G are all " so each sector is profitable. ) W ß ÞÞÞW " % and Then ÐM GÑ œ Þ*! Þ!& Þ$! Þ#! Þ"& Þ(& Þ!& Þ"! Þ$! Þ"! Þ*! Þ#& Þ"& Þ#! Þ"! Þ)!
4 #&!!! "!!!! If the demand from the open sector is. œ, then we want to solve $!!!! &!!!! Þ*! Þ!& Þ$! Þ#! #&!!! Þ"& Þ(& Þ!& Þ"! "!!!! ÐM GÑB œ B œ œ. Þ$! Þ"! Þ*! Þ#& $!!!! Þ"& Þ#! Þ"! Þ)! &!!!! Row reducing the augmented matrix gives Þ*! Þ!& Þ$! Þ#! #&!!! "!!! )&&)! Þ"& Þ(& Þ!& Þ"! "!!!!! "!! &!'#! µ Þ$! Þ"! Þ*! Þ#& $!!!!!! "! *'"'! Þ"& Þ#! Þ"! Þ)! &!!!!!!! " "!$##! These calculations were carried out to many decimal places, but the displayed numbers are rounded, at the end, to the nearest unit ($) The equation (***) will be satisfied if W " produces units ($), W produces units ($), ÞÞÞ etc. # The row reduced echelon form shows that MGis invertible, so the solution for B is " actually unique: B œðmgñ. Þ ( We could have solved by first trying to find MGÑ ", but that's even more work.) Here's a theorem that's too hard for us to prove: Theorem ( using the same notation as above) If G and. have nonnegative entries and the column sums in G are all "( that is, if every sector is profitable), then ÐM GÑ must be invertible and therefore there is a unique production vector satisfying the " equation ÐM GÑB œ., namely B œ ÐM GÑ.. Moreover, this B is economically feasible in the sense that all its entries are!þ
5 Here is an informal, imprecise discussion that might give you some confidence that the theorem is actually true. 1) Suppose G is any 8 8 matrix with entries all 0 and all column sums "Þ 7 Then G Ä! as 7 Ä. Å (the 8 8zero matrixñ This statement just means that each entry - in the matrix G 7 can be made as close to! as we like by choosing 7large enough. To keep things simple, we will only look at the case when G œ +,, -. a # # matrix. But the idea is similar when Gis 8 8. Let α œ the larger of the two column sums in G. By definition of α, +- Ÿ α and,.ÿα Ðone of the two actually equals α. ) By our assumptions above about G, we see that! Ÿ α ". An observation: suppose Bß Cß Dß A!Þ What happens to column sums when any # # matrix B C G is multiplied by on the left? B C +, B C +B,D +C,A G œ œ, -. -B.D -C.A so the column sums satisfy: Column 1: Ð+ -ÑB Ð,.ÑD Ÿ αb αd œ α ÐB DÑ Column 2: Ð+ -ÑC Ð,.ÑA Ÿ αc αa œ α ÐC AÑ Å In other words: for each column in the product G B C, column sum in product Ÿ α B C Ðold column sum in Ñ Use this fact, letting B C œgœ +, -. Þ Then B C # +, +, G œ G œ and # # sum for column " of G Ÿ α (old column 1 sum in G) œ αð+-ñ Ÿ α α œ α sum for column 2 of G Ÿ α (old column 2 sum in G) œ αð,.ñ Ÿ α α œ α. # #
6 B C Repeating the calculation again using, we see that in the matrix œg# G $ œ G G # +, œ G Ÿ -. #, the sum of each column is α $. 7 7 Continuing in this way, we get that each column sum in G is Ÿ α Þ Since all the elements in G 7 are nonnegative, this means that every individual entry in the matrix -in the matrix G 7 satisfies!ÿ-ÿα 7 Þ Since α 7 Ä! as 7Ä 7 (because!ÿα "ß ) we get that each entry -in G approaches! as 7Ä Þ 7" 2) MG œðmgñðmgg ÞÞÞG Ñ Why? Just multiply it out to verify: ÐMGÑÐMGG ÞÞÞG Ñ œ ÐM GÑM ÐMGÑG ÐMGÑG ÞÞÞ ÐMGÑG œm GMMGG MG G ÞÞÞMG G œmggg G ÞÞ. ÞÞÞG G G œmg 7" # # # $ 7 7" # 7 7" 7 3) Since G Ä! as 7 Ä, it follows that for large 7 7" G! and therefore M ÐMGÑÐMGG ÞÞÞG Ñ Å so ÐM G G ÞÞÞ G Ñ is approximately an inverse for ÐM GÑ, that is " ÐMGG ÞÞÞG Ñ ÐMGÑ This approximation suggest= that MGreally is invertible, and, since ÐMGG ÞÞÞG Ñhas nonnegative entries, it also suggests that all the entries in the exact ÐM GÑ " will be nonnegative. 4) If ÐM GÑ invertible and ÐM GÑ has all nonnegative elements, then of " " course there is a unique solution B œ ÐM GÑ. and ÐM GÑ. has all nonnegative entries. is "
7 Having done all this, we can also write: " ÐMGÑ ÐMGG ÞÞÞG Ñ, so " Bœ ÐMGÑ. ÐMGG ÞÞÞG Ñ. or B. G. G. ÞÞÞG. Does that make any sense? Think about it like this: we ask the sectors to meet the final demand., and they set out to do it. But producing the goods listed in. creates additional demands: each sector needs demands from the others to be able to produce anything at all. In fact, the sectors must supply each other with the products listed in G. so that they can produce., so they also need to produce these products. But then, to produce G. ß the sectors require GÐG. ÑÑin products and so on and so on... To make it all work, the needed production B is an infinite sum: B œ. G. G. ÞÞÞG. ÞÞÞ 7 But because G Ä! as 7Ä, so we approximate B. G. G. ÞÞÞG. for large 7. The column vectors in ÐM GÑ " actually have an economic interpretation: see the remarks following the next example Example Here is an example with a slightly more realistic set of data involving the Leontief Open Economy Production Model. The data is from #13, p. 137 in the textbook. The consumption matrix G is based on input-output data for the US economy in 1958, with data for 81 sectors grouped here (for manageability) into 7 larger sectors: (1) nonmetal household and personal products (2) final metal products (such as autos) (3) basic metal products and mining (4) basic nonmetal products and agriculture (5) energy (6) services and (7) entertainment and miscellaneous products. Units are in millions of dollars. (From Wassily W. Leontief, The Structure of the US Economy, Scientific American, April 1965, pp ) G œ
8 It turns out that MGis invertible ( as it should be, according to the Theorem stated (%!!! &'!!! "!&!! above). For a final demand vector. œ #&!!!, the solution to BœGB. is easy "(&!! "*'!!! &!!! with computer software such as Matlab. Rounded to 4 places, Matlab gives ÐM GÑ " œ so (%!!! &'!!! "!&!! " " B œðmgñ. œðmgñ #&!!! œ "(&!! "*'!!! &!!! ( entries rounded to whole numbers) An economic interpretation for the columns vectors of ÐM GÑ " Suppose G is an 8 8 consumption matrix with entries! and column sums each less than "Þ Let B œ production vector that satisfies a final demand.? B œ another production vector that satisfies a different final demand?. Ðfor the moment, think of? B,? d just as fancy names for two new production/final demand vectors having no connection to B ß. Ñ
9 Then B œ G B.? B œ G? B?. Adding the equations and rearranging, we get B? B œ GÐB? B ÑÐ.?. Ñ So production level ÐB? B Ñ fulfils the final demand Ð.?. Ñ At this point you can think of?. as we would in calculus as a change in the final demand., and? B is the corresponding change made necessary in the production vector B to satisfy the new final demand. "! Suppose we let?. œ, that is, change the final demand from sector W"! by 1 unit. Since? B œg? B?., we have ÐMGÑ? B œ?., or " "? B œðmgñ?. œðmgñ "!! col 1 col 2 col 8 of of of œ" "! " ÞÞÞ! " ÐM GÑ ÐM GÑ ÐM GÑ col 1 of œ " ÐM GÑ "! " This says that the? B corresponding to?. œ is the first column of ÐM GÑ Þ! In other words: a change of one unit in the final demand from Sector " requires a change?b in the production vector required to satisfy the new demand, and:?b œ the first column of ÐM GÑ ".
10 Similarly, you can show that a change of one unit in the demand of Sector W # causes a " change in the production vector?b œthe second column of ÐMGÑ à etc. The column vectors in ÐM GÑ " are not just accidental vectors that pop up in computing the inverse; they have economic meaning. "! So, in the preceding example: a change of?. œ ß that is a change of 1 unit! Ðœ " million dollars Ñin the demand for goods from sector W" forces the necessary production vector Bœ to change to B? B, where => "?B œ" column of ÐMGÑ œ
The Leontief Open Economy Production Model
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