The Leontief Open Economy Production Model

Transcription

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 consumed or produced by each sector are measured in $ (value of goods consmbed or produced). We begin with a consumption matrix G: the 4 >2 column lists the amount of goods (in $) consumed Ðor demanded Ñ by sector W4 from sectors W" ß W# ß ÞÞÞß W8 to produce one unit ($) of its product.. Columns of G are consumption vectors by sectors W" â W â W 4 8 to produce 1 unit of product (measured in $) Æ Æ Æ W" Ä -"" â -"4 â -"8 Consumed from W3 Ä -3" â -34 â -38 œg W Ä - â - â - 8 8" Of course, each - 34!Þ -"" For example, Column " œ -3" lists the consumption (demands) of Sector 1 from - 8" Sectors W",..., W8 to produce one unit ($) of product. Of course, we could also think of this as a list of costs ($) for Sector 1 to produce one unit of product, and the sum of the entries gives the total cost of all products consumed (demanded) by Sector 1 to produce one unit of its product.

2 The same is true for each column: the entries in Column 4 represent the consumption (demands, or costs) of sector W from sectors W ß W ß ÞÞÞß W in order for W to produce unit ($) of product. 4 " # 8 4 one Each column sum represents the costs for that sector to produce one unit. If a column sum is ", we say that sector is profitable.) -"4 B4 -"4 If we weight Column 4 with a scalar B 4 0, then the vector B4-34 œ B4-34 lists - B the consumption (demand, cost) by sector W4 from other sectors to produce B4 units of its own product. Suppose B" ß B# ß ÞÞÞß B 8 (! ) are amounts to be produced by sectors W" ß W# ß ÞÞÞß W8. We B" B# call Bœ is called a production vector. Then what does the vector GB mean? B 8 It is a linear 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 "" " " " " 88 8 total demanded from sector W" by sectors W" ß ÞÞÞ ß W to produce B œ total demanded from sector W# by sectors W" ßÞÞÞ ß W8 to produce B total demanded from sector W,C sectors W ß ÞÞÞ ß W to produce B 8 " 8 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 B of products (in $) from each sector œ the list GB of total demands ($) 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.

3 Now we add another feature to the economic model: an open sector. It is nonproductive, that is, it produces nothing that sectors W" ß ÞÞÞß W8use. The open sector simply demands the prodcuts of sectors W" ß ÞÞÞß W8Þ (For example, these could be demands of the government, demands for foreign export, for individual consumption, for charitable groups,...). ".# Supppose. œ tells how many units (in $) the open sector demands from each. 8 sector Wß ÞÞÞß W8 Þ The vector. is called the final demand vector We want to know if it is possible to set a level of production B so that both the productive and open sectors are satisfied, with nothing left over. That is, we want to find an B so that Leontief Open Economy Production Model B œ GB. (***) Å Å Å Total production œ Demand from + Demand from productive sectors open sector to produce B ( called intermediate ( called final demand ) demand ) If M is the identity matrix, we can rewrite this equation as : MB œ GB. MB GB œ. or ÐM GÑ B œ. To take a simple example, suppose the economy has 4 productive sectors the consumption matrix is Consumption vectors for W" W# W$ 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

4 Then ÐM GÑ œ Þ*! Þ!& Þ$! Þ#! Þ"& Þ(& Þ!& Þ"! Þ$! Þ"! Þ*! Þ#& Þ"& Þ#! Þ"! Þ)! #&!!! "!!!! 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 ($) The equation (***) will be satisfied if )&&)! &!'#! B œ, that is, *'"'! "!$##! if W " produces units ($), W produces units ($), # ÞÞÞ etc. Notice: the row reduced echelon form shows that MGis invertible, so the solution for " B is actually unique: B œðmgñ. Þ ( We could have solved the matrix equation " ÐM GÑB œ. in the first place by trying to find M GÑ, but that's even more work. ) This theorem is too hard for us to prove, but we can give an intuitive argument for it. 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 a # # 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: Sum Column 1 œ Ð+ -ÑB Ð,.ÑD Ÿ αb αd œ α ÐB DÑ Sum Column 2 œ Ð+ -ÑC Ð,.ÑA Ÿ αc αa œ α ÐC AÑ 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 7 the elements in G are nonnegative, this means that every individual entry - in the matrix G 7 satisfies! Ÿ - Ÿ α 7 Þ Since α 7 Ä! as 7 Ä (because 7!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 7" 3) Since G Ä! as 7 Ä, it follows that for large 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 suggests that MGreally is invertible, and, since ÐMGG ÞÞÞG Ñhas nonnegative entries, it also suggests that all the entries in the exact matrix ÐM GÑ " will be nonnegative. 4) If ÐM GÑ is 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.

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 has 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.. But then, to produce G. ß the sectors require an additional GÐG. ÑÑ in 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, 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 U.S. economy in 1958, with data for 81 sectors grouped here (for manageability) into ( 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 It turns out that MGis invertible ( as it should be, according to the Theorem stated)

8 (%!!! &'!!! "!&!! 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 of ÐM GÑ " Suppose G is an 8 8 consumption matrix with entries! and column sums each less than "Þ Let production vector that a final demand Ð Ñ B œ satisfies.? B œ production vector that satisfies a different final demand?. Ðfor the moment, just think of? B and? d as fancy names for two new production/final demand vectors having no connection to B ß. Ñ Then by (**) B œ G B.? B œ G? B?.

9 Adding the equations and rearranging, we get B? B œ GÐB? B ÑÐ.?. Ñ So production level ÐB? B Ñfulfils 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, increase 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 œ col 1 of ÐM GÑ " "! " So 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 W " requires a change? B in the production vector required to satisfy the new demand, and? B œ the first column of ÐM GÑ ". 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.

10 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