THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL

Transcription

1 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL HANS G. EHRBAR Date: Econ 7004 Fall 2002, revised June

2 2 The latest version of this document here is available at ehrbar/fixco.pdf for screen reading, and a monochrome version to print on US letter paper. 1. Exams Those interested in the application of this theory to socialist planned economy might be interested in it that question 11 is formulates an exercise from [4, p. 54] in the present mathematical notation, and that section 9 discusses the updating of I A) 1. In the final exam 2002, questions 8, 9, 22, 23, 24, and 36 were assigned. For the mathematical part of the qualifier 2003, the following instructions were given: You should do Problems 8 and 10, then you have a choice of doing either 15 or 22, then everybody should do 29, and finally there is a choice of either 35 or 36. Many questions have several parts. I tried to formulate them in such a way that, if you cannot do one of the parts, it is possible for you to go on to the next part.

3 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 3 2. A Subsistence Economy Woods, in [22, chapter 2], discusses the following simple commodity economy: (1) (2) X 11 qr wheat X 21 t iron x 1 qr wheat X 12 qr wheat X 22 t iron x 2 t iron Our preferred notation will be to write the industries vertically instead of horizontally. This allows a smoother transition to the Leontief input-output tables, and to the matrix notation in which each process is written as a column. (3) From: \ Into: Agriculture Manufacturing Agriculture: X 11 qr wheat X 12 qr wheat Manufacturing: X 21 t iron X 22 t iron Output: x 1 qr wheat x 2 t iron This table shows what was produced in the economy, and what the inputs for these production processes were. The ith column shows the inputs into the ith industry, and the ith row indicates the uses for the outputs of the ith industry. In the above example, X 11 qr wheat and X 21 t iron were used as productive inputs into agriculture, and X 21 qr wheat and X 22 t iron were used in manufacturing. The output of agriculture was x 1 qr wheat and that of manufacturing x 2 t iron.

4 4 HANS G. EHRBAR Under the assumption of constant returns to scale this economy can also be written as (4) From: \ Into: Agriculture Manufacturing Agriculture: a 11 x 1 qr wheat a 12 x 2 qr wheat Manufacturing: a 21 x 1 t iron a 22 x 2 t iron Output: x 1 qr wheat x 2 t iron where a ij is the amount of commodity i needed to produce one unit of commodity a11 a12 j. Here A = [ a 21 a 22 ] is the matrix of input coefficients and x = [ x1 x 2 ] is the vector indicating the gross product. Using the matrices A and x, the coefficients in (4) can be written as (5) A diag(x) = Note that [ a11 a (6) Ax = 12 [ ] [ ] [ ] [ ] a11 a 12 x1 0 a11 x = 1 a 12 x 2 X11 X = 12. a 21 a 22 0 x 2 a 21 x 1 a 22 x 2 X 21 X 22 a 21 a 22 ] [ x1 x 2 ] = [ ] a11 x 1 + a 12 x 2 a 21 x 1 + a 22 x 2 is the vector of inputs necessary to produce the gross output x.

5 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 5 In chapter 2, Woods makes the assumption that the product is exactly used up in next period s production, therefore (7) Ax = x or, written out, (8) (9) a 11 x 1 + a 12 x 2 = x 1 a 21 x 1 + a 22 x 2 = x 2 What mathematical conditions must the four coefficients a 11, a 12, a 21, a 22 satisfy so that this has a positive solution x 1 > 0, x 2 > 0? Woods at first only looks at the case where a 12 > 0 and a 21 > 0, i.e., wheat is used to produce iron, and iron is used to produce wheat. Under this presupposition, the following three conditions are necessary and sufficient for the existence of a positive gross product vector which is exactly reproduced by this technology: (10) (11) (12) 1 a 11 > 0 1 a 22 > 0 (1 a 11 )(1 a 22 ) a 12 a 21 = 0

6 6 HANS G. EHRBAR Proof of necessity: collect the terms in (8) to get (13) a 12 x 2 = (1 a 11 )x 1 and do the same thing with (9) (14) a 21 x 1 = (1 a 22 )x 2 Since the lefthand side of (13) is positive, the righthand side must be positive too, which gives us (10). In the same way, we get (11) from (14). Now massage (13) to get (15) x 2 x 1 = 1 a 11 a 12 and do the same thing to (14) to get (16) a 21 1 a 22 = x 2 x 1. (15) and (16) together give (17) a 21 1 a 22 = 1 a 11 a 12

7 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 7 which, multiplied out, gives (18) but this is (12). a 21 a 12 = (1 a 11 )(1 a 22 ) Problem 1. The identity (12) means that the determinant of a certain matrix is zero. Which matrix? Why should that matrix have a zero determinant? Answer. It is the matrix I A. (7) can be written (I A)x = o. This can be the case with x o only if I A is singular, i.e., if its determinant is zero. Is there a price system in which the value of the outputs is at the same time the sum of the input costs, i.e., there is no profit? If we plug the prices p = [ p 1 p 2 ] into (4), we get for the two columns of that table the two equations (19) p 1 a 11 x 1 + p 2 a 21 x 1 = p 1 x 1 p 1 a 12 x 2 + p 2 a 22 x 2 = p 2 x 2 which can be simplified to (20) p 1 a 11 + p 2 a 21 = p 1 p 1 a 12 + p 2 a 22 = p 2. But this can be written in matrix form as [ ] [ ] a (21) p1 p 11 a 12 2 = [ ] p a 21 a 1 p 2, 22

8 8 HANS G. EHRBAR i.e. p A = p or equivalently A p = p. In other words, while the quantity vector which ensures exact reproduction is a right eigenvector of A, the price vector which allows this reproduction to be financed without profits is a left eigenvector of A (or, what is the same, a right eigenvector of A ). The existence of the price vector is guaranteed by the same conditions (10), (11), and (12) which guarantee the existence of the gross product vector, since these conditions are invariant if one goes from A to the transpose A. Problem 2. Find the prices for the following economy used in [19, section 1]: (22) (23) 280 qr wheat 12 t iron 400 qr wheat 120 qr wheat 8 t iron 20 t iron Answer. Prices must satisfy (24) (25) 280p p 2 = 400p 1 i.e. 120p p 2 = 0 120p 1 + 8p 2 = 20p 2 i.e. 120p 1 12p 2 = 0 The price of one ton of iron is therefore ten times the price of a quart of wheat. 3. Production with a Surplus Notation: x o means: each element x i 0. x o means: each element x i > 0.

9 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 9 (26) Definition: A is productive if a x o exists so that x Ax, i.e., after replacement of the inputs used up in the production of x a strictly positive net product y = x Ax (sometimes called final demand ) remains. (26) can also be written as (27) x = Ax + y Written out element by element this gives: (28) (29) y o x 1 = a 11 x 1 + a 12 x 2 + y 1 x 2 = a 21 x 1 + a 22 x 2 + y 2 In order to see geometrically what the condition of productivity means, define first of all N = I A. In terms of N, (27) can also be written as (30) Nx = y or, if you split N into its column vectors, [ ] [ ] x (31) n1 n 1 2 = y x 2

10 10 HANS G. EHRBAR i.e., (32) n 1 x 1 + n 2 x 2 = y. For A to be productive, it must be possible to express at least one strictly positive vector y o as a nonnegative linear combination of the column vectors of I A, or, in other words, the cone spanned by the column vectors of I A must include y. Now draw the two vectors n 1 and n 2 and connect their tips by a straight line segment. This straight line segment is the locus of all points of the form n 1 α + n 2 (1 α) for 0 α 1. Since nonnegative linear combinations whose coefficients add to 1 are called convex combinations, the straight line segment can be called the convex hull of n 1 and n 2. For the economy to be productive, this line segment must pass through the positive quadrant. A picture is in [2, p. 24], or below in Problem 3. Problem 3. [22, p. 23] Here are three A-matrices [ 1/2 ] 1/4 [ 1/2 ] 3/8 [ 1/2 ] 1/2 1/3 1/5 1/3 3/4 1/3 3/4 Below is the geometric representation of each of these economies. The first good is on the horizontal and the second on the vertical axis. The arrows indicating the coordinate axes have unit length:

11 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL Give the coordinates of the end points of the line segments in these three graphs. Explain what these graphs mean. Can these graphs help you decide which of these economies can produce a net product of y = [ 2 3 ]? Answer. If the line segment intersects the quarter-plane consisting of positive vectors then the economy is productive. The coordinates of the points are columns of I A. Starting from the left endpoint of the leftmost line segment going to the right endpoint of the rightmost line segment we have x -1/4 1/2-3/8 1/2-1/2 1/2 y 4/5-1/3 1/4-1/3 1/4-1/3 After this geometric illustration let us look at the mathematical conditions for the economy to be productive. Assume that y o and x o satisfy (27). Collect terms in (28) to get (33) (1 a 11 )x 1 = a 12 x 2 + y 1

12 12 HANS G. EHRBAR Since y 1 > 0 and a 12 x 2 0, we know that the rhs of this equation is positive. Therefore also the lhs (1 a 11 )x 1 > 0. In particular this means x 1 0, therefore x 1 > 0, and from this follows (34) 1 a 11 > 0. In the same way follows also (35) 1 a 22 > 0 but we will not need (35) for the rest of this proof here. Since 1 a 11 0 one can solve (33) for x 1 : (36) x 1 = a 12x 2 + y 1 1 a 11. If one plugs this into (29) one gets (37) x 2 = a 21 a 12 x 2 + y 1 1 a 11 + a 22 x 2 + y 2 or, multiplying through by 1 a 11 and collecting terms, ( (1 a11 )(1 a 22 ) a 12 a 21 ) x2 = a 21 y 1 + (1 a 11 )y 2 (38)

13 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 13 Since 1 a 11 > 0 and y 2 > 0, the righthand side in (38) is positive, therefore also the lefthand side. Therefore x 2 cannot be 0, therefore it must be positive, therefore also (39) (1 a 11 )(1 a 22 ) a 12 a 21 > 0. So, we have just shown that, if the system is productive, the three inequalities (34), (35), and (39) must hold, they are necessary conditions for productiveness. Regarding sufficiency one can say: if the economy satisfies (34), (35), and (39), (which we derived from its ability to produce a given positive net output vector), then it can also produce any arbitrary nonnegative net output vector. We have already done most of the work to show this. The gross product vector necessary to produce a given y is (40) (41) (1 a 22 )y 1 + a 12 y 2 x 1 = (1 a 11 )(1 a 22 ) a 12 a 21 a 21 y 1 + (1 a 11 )y 2 x 2 = (1 a 11 )(1 a 22 ) a 12 a 21 The second equation follows from (38), and the first by symmetry. These solutions are clearly nonnegative. To finish the proof, one has to plug these values into (27) and verify that these are indeed solutions (not done here).

14 14 HANS G. EHRBAR Problem 4. Show the following: if A is a 2 2 matrix whose off-diagonal elements are nonpositive, and which satisfies the three inequalities (34), (35), and (39), then I A has an inverse which consists of nonnegative elements only. Answer. The matrix inverse of a 2 2 matrix is, in general [ ] 1 [ ] a b 1 d b (42) = c d ad bc c a therefore (43) (I A) 1 = [ ] 1 1 a 22 a 12 (1 a 11 )(1 a 22 ) a 12 a 21 a 21 1 a The Price System Now let s see if this system allows a uniform positive profit rate. The wage goods are still folded into the A-matrix, therefore we will first assume that the wages are simply decided by the subsistence bundle, i.e., we are not yet showing wages or labor inputs separately. The price vector must satisfy (44) (45) p 1 = (1 + r)(p 1 a 11 + p 2 a 21 ) p 2 = (1 + r)(p 2 a 12 + p 2 a 22 )

15 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 15 or in matrix notation (46) p = (1 + r)p A This means: p has to be a left eigenvector of A. Here the Frobenius theorem comes in handy. 5. A Statement of the Frobenius Theorem The following is from [2, pp ]. We only need the assumptions that the square matrix A O and that A is indecomposable, i.e., every good enters, either directly or indirectly, as input into the production of every other good. Mathematically indecomposability can be expressed by one of the following three equivalent conditions: (47) (48) (49) For all i and j exists a t > 0 so that (A t ) ij > 0 I + A + A A n 1 O If x o and λx Ax for some λ > 0 then x o If A O is indecomposable, the Frobenius theorem states:

16 16 HANS G. EHRBAR (1) There is (up to a scalar) exactly one nonnegative price vector p o which gives equal rates of profit, i.e., which satisfies (50) p = (1 + r )p A or p A = r p This equilibrium price vector is not only nonnegative but its elements are 1 even strictly positive, i.e., p o. The eigenvalue 1+r is also called the dominant eigenvalue, notation frob(a). (2) For all λ > frob(a) follows (λi A) 1 O (3) The following inequalities hold for any vector q o and scalar µ: (51) (52) (53) (54) if q A µq then µ λ if q A µq then µ < λ if q A µq then µ λ if q A µq then µ > λ (4) If A A, then the associated Frobenius eigenvectors satisfy frob(a ) frob(a), (5) frob(a) = frob(a ).

17 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 17 (6) The Frobenius eigenvalue is the largest root of the characteristic equation det(a λi) = 0, it is a single root, and every other root has a smaller absolute value. Problem 5. Since frob(a) = frob(a ), conditions (51) until (54) are also true for right eigenvectors. Formulate these conditions for right eigenvectors. Answer. The following inequalities hold for any vector z o and scalar µ: (55) (56) (57) (58) if Az zµ then µ λ if Az zµ then µ < λ if Az zµ then µ λ if Az zµ then µ > λ Problem 6. Show: For any strictly positive price vector, which usually gives different rates of profit in the different industries, the smallest industrial rate of profit is smaller than r, and the biggest is bigger than r. This can be interpreted to mean that r is some kind of average rate of profit. Answer. r min is the minimum and r max the maximum industrial profit rate, i.e., (59) (1 + r min )p A p (1 + r max)p A.

18 18 HANS G. EHRBAR The left hand side inequality gives (60) from which follows, by (53), (61) p A r min p r min 1 + r, i.e., r min r. The right hand side inequality in (59) gives (62) from which follows, by (51), (63) which means r max r. p A r max p r max 1 + r From this follows immediately that r > 0 iff there is any price vector which gives positive profits everywhere. And from (4) follows that the wage-profit frontier is downward-sloping. If the matrix is 2 2, i.e., [ ] a11 a (64) A = 12 a 21 a 22

19 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 19 then the largest root of the characteristic equation is (65) (66) (67) frob(a) = a 11 + a 22 (a11 + a 22 ) + 2 (a 11 a 22 a 12 a 21 ) 2 4 = a 11 + a 22 + (a 11 + a 22 ) 2 4(a 11 a 22 a 12 a 21 ) 2 = a 11 + a 22 + (a 11 a 22 ) 2 + 4a 12 a 21 2 From (67) one can see that the term under the square root is positive for an indecomposable matrix, i.e., a matrix with a 12 > 0 and a 21 > The Fully Specified Model So far we have dealt with a simplified version of the model since we have not shown the labor inputs into the economy explicitly. We will do this now. Assume we have the production (68) (69) X 11 qr wheat X 21 t iron L 1 hours of labor x 1 qr wheat X 12 qr wheat X 22 t iron L 2 hours of labor x 2 t iron

20 20 HANS G. EHRBAR Under the assumption of constant returns to scale this can also be written as (70) (71) a 11 x 1 qr wheat a 21 x 1 t iron l 1 x 1 hours of labor x 1 qr wheat a 12 x 2 qr wheat a 22 x 2 t iron l 2 x 2 hours of labor x 2 t iron or in matrix terms (72) Ax l x x i.e., in order to produce the output bundle x one needs the input bundle Ax and the labor input l x. In order to be able to deal with alternative processes and joint production one also has to introduce the output matrix B and the intensity vector z. The above model is called an open economy because it has a non-produced input (labor). The model we had earlier was, by contrast, a closed economy. There are two ways to mathematically represent an open economy as a closed economy. You may either add the means of production which the laborers consume to the input requirements of the given products, or you may add a separate industry, the production of labor-power, which has the minimum means of subsistence as input requirements. Closing an economy in this way however means that one assumes that profits are paid on the labor input as well, contrary to the assumptions used by Sraffa.

21 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 21 Problem 7. Given an open economy with [ ] a b A = l = [ 1 c ] [ 0 (73) b = 0 0 w] where b is the means of subsistence the laborer has to consume for each hour of labor. w is here the real minimum wage. Write this economy as closed economy in the two ways indicated. Answer. If we do not want to show labor separately it becomes [ ] [ ] [ A c = A + bl a b 0 [1 ] a (74) = + c = 0 0 w w ] b cw and if we do want to show labor as a separate industry, it becomes [ ] [ ] a b 0 A b (75) A l = l = 0 0 w d 1 c d Here one can either set d = 0, or one can consider d the labor services directly purchased by households (hours of cleaning ladies or babysitters needed per hour of labor performed). Problem 8. [10, pp ] This exercise shows the connection between the matrix formalism for fixed-coefficient production functions, and the input-output tables published by many countries. The following equation describes a fictitious economy in

22 22 HANS G. EHRBAR some time period, perhaps a year: Agric: Manuf: 25 bushels wheat 14 yd cloth 80 years labor 100 bushels wheat 20 bushels wheat 6 yd cloth 180 years labor 50 yd cloth Written vertically instead of horizontally one gets (76) From: \ Into: Agriculture Manufacturing Agriculture: 25 bushels wheat 20 bushels wheat Manufacturing: 14 yards cloth 6 yards cloth Labor: 80 years labor 180 years labor Output: 100 bushels wheat 50 yards of cloth This table describes what was produced in the economy, and what inputs were used in these production processes. Each column is an industry, it shows the inputs into this industry, and each row indicates the uses for the outputs of the corresponding industry. 14 yards of cloth were used as productive input into agriculture, and 6 yards of cloth were used in manufacturing, and the output of manufacturing was 50 yards of cloth, etc. (i) Compute the A-matrix, the l-vector, and the gross product vector x for this economy. The matrix diag(x) is the diagonal matrix which has x as its diagonal vector. Show that the material inputs in the table (76) can be obtained as the matrix product A diag(x), and the labor inputs as the vector l diag(x). Remember, a ij is

23 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 23 the amount of commodity i needed to produce one unit of commodity j, and l i is the amount of labor needed to produce one unit of commodity j. Answer. (77) (78) (79) [ ] 1/4 2/5 A = 7/50 3/25 [ ] [ ] l 100 = 4/5 18/5 x = 50 [ ] [ ] [ ] 1/4 2/ A diag(x) = = 7/50 3/ [ ] [ ] l [ ] diag(x) = 4/5 18/5 = (ii) The net product can be obtained in two ways: either go to table (76) and subtract the wheat and cloth used in both sectors from the wheat and cloth produced, or by the matrix operation y = x Ax. Show that this gives the same results. Answer. From table (76) you can see: net output of wheat = the total output of 100 bushels wheat 25 bushels wheat used in agriculture 20 bushels wheat used in manufacturing = 55 bushels wheat; net output of cloth = 50 yards cloth 14 yards cloth 6 yards cloth = 30 yards cloth. Matrix algebra gives the same result: (80) y = x Ax = [ ] [ 1/4 2/5 7/50 3/25 ] [ ] = [ ] [ ] = [ ]

24 24 HANS G. EHRBAR (iii) Show that with the following prices and wages, the economy makes zero profits: (81) p = [ 2 5 ] w = 1 Answer. We have to show that equation (92) holds with r = 0, i.e., p A + wl = p : (82) [ ] [ ] 1/4 2/5 [ ] [ ] /5 18/5 = 2 5 7/50 3/25 (iv) Now we construct a fictitious household sector using the following conventions: (1) the household sector uses up the whole net product of the economy; (2) the household sector also uses up 40 man years in personal services (maids); (3) the household sector produces all the labor used in this economy, namely, 300 person

25 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 25 years. This gives the following economy (83) From: \ Into: Agriculture Manuf. Households Total Agriculture: 25 bsh wheat 20 bsh wheat 55 bsh wheat 100 bsh wheat Manuf.: 14 yd cloth 6 yd cloth 30 yd cloth 50 yd cloth Households: 80 yr labor 180 yr labor 40 yr labor 300 yr labor Output: 100 bsh wheat 50 yd cloth 300 yr labor In this table, the row sums are equal to the gross output vector. Construct A, x, and p for this economy. Show that Ax = x, i.e., this economy has a zero net product, and that p A = p, i.e., this economy has zero profits. Answer. The third element in the price vector is the wage w = 1: [ ] [ ] 1/4 2/5 11/ [ ] (84) A = 7/50 3/25 1/10 x = 50 p = /5 18/5 2/ (85) (86) Ax = [ 1/4 2/5 11/60 7/50 3/25 1/10 4/5 18/5 2/15 ] [ p A = [ ] [ 1/4 2/5 11/60 7/50 3/25 1/10 4/5 18/5 2/15 ] = [ ] ] = [ ]

26 26 HANS G. EHRBAR (v) Verify that the matrix A diag(x) gives the numbers in table (83) (except for the row sums and column sums). Answer. (87) [ 1/4 ] [ ] 2/5 11/ /50 3/25 1/ = 4/5 18/5 2/ [ 25 ] (vi) The following table describes the economy not in physical quantities but in monetary terms: (88) From: \ Into: Agriculture Manuf. Households Total Agriculture: $ 50 wheat $ 40 wheat $ 110 wheat $ 200 wheat Manuf.: $ 70 cloth $ 30 cloth $ 150 cloth $ 250 cloth Households: $ 80 labor $ 180 labor $ 40 labor $ 300 labor Output: $ 200 wheat $ 250 cloth $ 300 labor $ 750 total This is the input-output table of the economy as it is usually published together with the National Income data. It allows the comparison of different items in the same column, something that was not possible with the physical input-output table, where

27 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 27 each row had a different denomination. Answer the following questions: Does manufacturing spend more on manufacturing inputs or on agricultural inputs? Which percentage of total household income is spent on manufacturing goods? Both the row sums and the column sums of this economy give the gross output vector. Answer. Manufacturing spends more on agricultural inputs. Households spend 50 percent of their income on manufacturing goods. (vii) Show that the coefficient matrix in the above table (leaving out the row sums and column sums) can be computed as the matrix product diag(p)a diag(x). Answer. (89) = [ [ ] [ 1/4 2/5 11/60 7/50 3/25 1/10 4/5 18/5 2/15 ] [ ] = ] [ [ ] ] =

28 28 HANS G. EHRBAR 7. The Wage-Profit Frontier Assuming a uniform rate of profit, prices and wages must satisfy (90) (91) (1 + r)(p 1 a 11 x 1 + p 2 a 21 x 1 ) + wl 1 x 1 = p 1 x 1 (1 + r)(p 1 a 12 x 2 + p 2 a 22 x 2 ) + wl 2 x 2 = p 2 x 2 After dividing (90) by x 1 and (91) by x 2, the following matrix equation remains: (92) (1 + r)p A + wl = p Problem 9. Show: If A O is indecomposable, l o, w 0, r 1, and p o, then (92) can only hold if r r. Hint: use (53). Answer. Since l o and w 0, (92) implies that (93) (1 + r)p A p and since 1 + r > 0, division by 1 + r maintains the inequality: (94) p A r p

29 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 29 Now use (53) with µ = 1 1, noting that λ = 1+r 1+r : (95) (96) (97) r r 1 + r 1 + r r r If the uniform rate of profit r and the nominal wage w are known, this can be solved (98) p = wl ( I (1 + r)a ) 1 or, writing out the matrix inverse, see (43): (99) [ ] [ ] [ ] 1 1 (1 + r)a22 (1 + r)a p1 p 2 = w l1 l 12 2 f(r) (1 + r)a 21 1 (1 + r)a 11 where f(r) is the determinant (100) f(r) = ( 1 (1 + r)a 11 )( 1 (1 + r)a22 ) (1 + r) 2 a 12 a 21.

30 30 HANS G. EHRBAR Written element by element one gets: (101) p 1 = w ( ) l1 (1 (1 + r)a 22 ) + l 2 (1 + r)a 21 f(r) (102) p 2 = w ( l1 (1 + r)a 12 + l 2 (1 (1 + r)a 11 ) ) f(r) In Woods s notation this is (103) (104) where (105) (106) p 1 = w g(r) f(r) p 2 = w h(r) f(r) g(r) = l 1 (1 (1 + r)a 22 ) + l 2 (1 + r)a 21 h(r) = l 2 (1 (1 + r)a 11 ) + l 1 (1 + r)a 12. Woods chooses the normalization p 1 = 1 and writes p instead of p 2, so that he gets on [22, p. 45] (107) p = h(r) g(r) w = f(r) g(r)

31 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 31 Prices are positive since I (1 + r)a has a positive inverse. Woods proves this on [22, p ] for 2 2 matrices, but we will simply use here this general matrixalgebraic fact without proof. The explicit solution formulas (107) also show that wages and rate of profit are inversely related, and relative prices are either constant with changes in the wage, or they rise monotonically, or they fall monotonically, see the graphs on [22, p. 50]. Woods shows on [22, pp. 46/7] that it depends on the capital-labor ratios whether relative prices fall or rise. The relative price in the industry which has higher capital/labor ratio rises as the wages fall and the profit rate rises. Here capital has to be measured by l A, i.e., these are the prices for a rate of profit of 100 percent, while labor is represented by the vector l. All possible cases in an economy with three products are discussed in [3]. Problem 10. [7] works with the following simple model of a farm and a bakery: (108) (109) a corn 0 bread 1 labor 1 corn b corn 0 bread c labor 1 bread where a > 0 and b > 0. a. 2 points Write down the input coefficient matrix A and the labor coefficient vector l.

32 32 HANS G. EHRBAR Answer. They are the same as in Problem 7: [ ] a b (110) A = 0 0 [ l = 1 ] c b. 3 points What must the vector of gross outputs x be so that the net product is one loaf of bread? Answer. x must satisfy (111) (112) (113) [ ] 0 x Ax = 1 [ ] [ ] [ ] [ ] x 1 a b x 1 0 = x x 2 1 [ ] [ ] [ ] x 1 x 2 ax 1 + bx 2 0 = 0 1 or, written as two scalar equations, (114) (115) x 1 ax 1 bx 2 = 0 x 2 = 1.

33 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 33 Plug x 2 = 1 into (114) to get (116) x 1 = b 1 a i.e., (117) [ ] b/(1 a) x = 1 c. 3 points Show that this economy is productive, i.e., it is able to produce a strictly positive net product y o if and only if a < 1. Don t use the conditions for productivity given later in this Problem, [ but compute the x for a particular y o, e for instance you may choose is y = where e > 0, and see what is necessary 1] to make x o. (Here the symbol indicates that both components of the vector y must be positive.) By the way, the condition of a strictly positive net product does not check whether the net product is big enough for the workers to live on.

34 34 HANS G. EHRBAR [ ] Answer. Without loss of generality we can say that the net product is y = e 1 where e > 0. x must satisfy [ ] e (118) x Ax = 1 [ ] [ ] [ ] [ ] x 1 a b x 1 e (119) = x x 2 1 [ ] [ ] [ ] x 1 ax 1 + bx 2 e (120) = x (121) (122) therefore x 1 ax 1 bx 2 = e x 2 = 1 (123) i.e., (124) x 1 = b + e 1 a [ ] (b + e)/(1 a) x = 1

35 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 35 This is positive if and only if a < 1. Since b > 0, the condition a < 1 is also necessary and sufficient for the case e = 0, i.e., for the production of a net product which has positive amounts of the consumer good. d. 1 point Now introduce the wage bundle [ 0 (125) b =, w] denoting how much bread the laborer has to consume for each hour of labor performed. In other words, w is the real minimum wage. If one considers this bread on the same footing as the corn consumed in production itself, the A-matrix becomes A c = A + bl. Write down A c. Answer. [ ] [ ] [ ] [ ] [ a b 0 [1 ] a b 0 0 a (126) A c = + c = + = 0 0 w 0 0 w cw w ] b cw e. 5 points It was derived in class, equation (39), that a 2 2 matrix A is productive iff a 11 < 1, a 22 < 1, and (127) (1 a 11 )(1 a 22 ) a 12 a 21 > 0.

36 36 HANS G. EHRBAR Verify that for A c to be productive (which means: the technology is efficient enough that the workers can survive), besides a < 1 the following other two conditions are needed: c < 1 w and b < (1 a)( 1 w c). Show that these two conditions are satisfied if and only if 1 (128) w < w max = b 1 a + c This is a mathematical proof involving inequalities. If you are not comfortable with this, don t worry about it, just skip this and go on. Answer. Necessity: start with (129) b < (1 a)( 1 w c) Since a < 1, this implies b 1 a < 1 w c (130) b (131) (132) 1 a + c < 1 w 1 b 1 a + c > w

37 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 37 Now sufficiency: from (133) 1 w < w max = b 1 a + c follows (134) 1 w > b 1 a + c > c i.e., the first condition holds, and also (135) (136) 1 w c > b 1 a (1 a)( 1 w c) > b which is the second condition. f. 6 points Now let s go over to prices and wages. Make bread the numeraire, i.e., its price is 1. If the wage is w (this is now the nominal wage, but since bread is the numeraire, w is also equal to the real wage) show that the price of corn is (137) p = a + (b ac)w b

38 38 HANS G. EHRBAR Answer. Here are the equations for price, wage, and profit rate: (138) (139) (140) (141) p = (1 + r)p A + wl [ ] [ ] [ ] a b [ ] p 1 = (1 + r) p 1 + w 1 c 0 0 ap(1 + r) + w = p bp(1 + r) + cw = 1 Multiply first equation by b and second equation by a: (142) (143) abp(1 + r) + bw = bp abp(1 + r) + acw = a Now subtract (144) (b ac)w = bp a or a + (b ac)w (145) p = b This is a linear relation between price and wage. g. 4 points Show that the uniform rate of profit r satisfies (146) r = 1 a ( b + c(1 a) ) w a + (b ac)w

39 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 39 Answer. Plug (137) into (141): ( a + (b ac)w ) (1 + r) = 1 cw (147) (148) (149) 1 cw 1 + r = a + (b ac)w 1 a (b + c ac)w r = a + (b ac)w h. 2 points What is the maximal rate of profit in the case w = 0, and what is the wage if r = 0? Show that the maximal wage is the same as given in (128). Answer. For w = 0 we get r max = 1 a, and for r = 0 we get a (150) w max = 1 a b + c(1 a) = 1 b 1 a + c As long as 0 < a < 1 and b > 0, both r max and w max are well-defined and positive. i. 3 points Show that the wage-profit frontier given by (146) is downward-sloping. Answer. Start with (151) 1 + r = 1 cw a + (b ac)w

40 40 HANS G. EHRBAR Therefore (152) (153) (154) dr d(1 + r) c( ) a + (b ac)w (1 cw)(b ac) = = ( ) dw dw 2 a + (b ac)w = ac bcw + ac2 w b + ac + bcw ac 2 w ( a + (b ac)w ) 2 = b ( a + (b ac)w ) 2 Since we assumed b > 0, this is downward sloping for all values of w. 8. Morishima s Three Definitions of Value If the input coefficient matrix is productive, one can uniquely define the total labor content (value) of any product. One can think of at least three different mathematical definitions what value should be, which all lead to the same result. According to the first definition, the total labor content of a certain product bundle y consists of: the labor going directly into that bundle, which is l y, plus the labor going into the inputs into that bundle, which is l Ay, plus the labor going into the inputs going into the inputs into that bundle, which is l A 2 y, and so forth. In other words, it is l (I + A + A 2 + )y. It can be shown that this infinite

41 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 41 series converges if A is productive; [1] lists dozens of properties of the so-called M- matrices. Therefore this is a well defined expression, and we will call the vector whose ith position indicates the total labor content of one unit, of the ith product, by λ; it is the value vector. The scalar product λ y gives the total labor content of the bundle y. The vector λ is measured in units of labor time per unit of output, in the same units in which the vector l is measured. If one labor time unit of value is represented in k monetary units, then the monetary expression of the value of a bundle y would be kλ y. The second definition of the value vector is a recursive one. Since the total labor content of a bundle y consists of the direct labor used in the bundle, l y, plus the total labor contained in the products needed to produce y, λ Ay, the value vector λ must satisfy λ y = l y +λ Ay, whatever the bundle y. Hence λ (I A) = l. This equation can indeed be solved and gives a nonnegative solution vector, since I A has a positive inverse: (155) λ = l (I A) 1. One can also define the total labor content of a bundle y using the concept of vertical integration: take the (nonnegative) gross output bundle x whose production creates y as net product, and then calculate the direct labor l x necessary for this production. In other words, one directs the economy to produce output in such proportions that at the end of the production period everything that was used in

42 42 HANS G. EHRBAR production can be replaced from the output, and that the difference of inputs and outputs is exactly the bundle y. The total output necessary to obtain this effect is x = (I A) 1 y. The labor input required for this production is our third definition of the total labor content of the bundle, and indeed, (156) l x = l (I A) 1 y = λ y, where λ is as in (155). Problem 11. [4, p. 54] Consider the following economy: Oil: Bread: 500 barrels of oil 1, 000 workers 3, 000 barrels of oil 2, 000 barrels of oil 2,000 workers 40, 000 loaves of bread Written vertically instead of horizontally, and making explicit the fact that bread does not enter either production process as input, one gets (157) From: \ Into: Oil Bread Oil: 500 barrels of oil 2,000 barrels of oil Bread: 0 loaves of bread 0 loaves of bread Labor: 1,000 workers 2,000 workers

43 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 43 This table describes what was produced in the economy, and what inputs were used in these production processes. The ith column shows the inputs into the ith industry, and the ith row indicates the uses for the outputs of the ith industry. (i) Compute the A-matrix, the l-vector, and the gross product vector x for this economy. The matrix diag(x) is the diagonal matrix which has x as its diagonal vector. Show that the material inputs in the table (157) can be obtained as the matrix product A diag(x), and the labor inputs as the vector l diag(x). Remember, a ij is the amount of commodity i needed to produce one unit of commodity j, and l i is the amount of labor needed to produce one unit of commodity j. Answer. (158) [ ] 1/6 1/20 A = 0 0 [ ] [ ] l 3, 000 = 1/3 1/20 x = 40, 000 (159) (160) [ ] [ ] [ ] 1/6 1/20 3, A diag(x) = = , [ ] [ ] l 3, [ ] diag(x) = 1/3 1/20 = 1, 000 2, , 000

44 44 HANS G. EHRBAR (ii) Compute the net output of this economy. Which gross output would be necessary to make this net output the vector y = [ 50, 000 1, 000 ]? Answer. Given the net output vector y, the gross output vector x must satisfy x = [ y + Ax, from ] which follows x Ax = (I A)x = y, therefore x = (I A) 1 5/6 1/20 y. Now I A =. 0 1 [ ] [ ] Therefore (I A) 1 = 6 1 1/20 6/5 6/100 = and the required gross product is 5 0 5/6 0 1 [ ] [ ] (161) (I A) 1 y = 6/5 6/ , 000 1, 000 (iii) Compute the value vector of this economy. = [ 60, 060 1, 000 ]. Answer. The total labor content of a bundle y consists of the direct labor used in the bundle, l y, plus the total labor contained in the products needed to produce y, λ Ay. The value vector λ must therefore satisfy λ y = l y + λ Ay, whatever the bundle y. Hence λ (I A) = l or (162) λ = l (I A) 1 = [ 1/3 1/20 ] [ 6/5 6/ ] = [ 2/5 7/100 ].

45 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 45 Problem 12. [12, p. 71] Show: if a price vector p together with the wage w gives nonnegative profits in every industry, then p must be greater than or equal to the wage times the value vector, i.e., p wλ. Answer. Nonnegative profits mean (163) p p A + wl. Hence (164) p (I A) wl. Since (I A) 1 O, postmultiplication preserves the sign: (165) p wl (I A) 1 = wλ. 9. Changes in the A-matrix The ith column in the A-matrix, a i, represents the input requirements of the ith industry, and the ith component of the l-vector, l i, the direct labor requirement of the ith industry. Assume there is a change in technology which causes this vector to change. Instead of a i, the new input requirement vector is a i + d. Let e be the vector which has a 1 at the ith place and zeros everywhere else. Then the new A-matrix is A + de. The matrix (I A) 1 is needed to compute the value vector,

46 46 HANS G. EHRBAR and to compute the direct and indirect labor requirements for any given net output. Therefore let us look how this matrix changes due to a change in the ith industry. I.e., we have to compute (I A de ) 1. For this, the so-called Sherman- Morrison-Woodbury theorem comes in handy. It says that (166) is the inverse of (167) (I A) e (I A) 1 d (I A) 1 de (I A) 1 I A de Assume for simplicity that the ith product is produced only in one factory, call it plant i. Then the updating of the matrix (I A) 1 can be done by a distributed computing process in part at this plant i s computer, and in part at the central computer. The central computer maintains a copy of A (which is a sparse matrix with millions of rows and columns), and through an iterative process, also generates a copy of (I A) 1. Plant i does not have nor need the whole matrices A or (I A) 1. It has of course a, the ith column of A, since these data are generated from its own bookkeeping. Let s assume that plant i gets inputs from 100 other industries. If it maintains on its computers a copy of the corresponding columns of (I A) 1, i.e., 100 columns from a square matrix which has millions of rows and columns, then it can compute and transmit to the central computer, along with d,

47 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 47 also (I A) 1 d. This latter vector is no longer sparse but has millions of nonzero entries. But equation (166) makes it easy for the central computer to update its copy of (I A) Choice of Technique The simplest case is: for commodity 1 only one technique is available, but for commodity 2 there are two alternative techniques, α and β. (107) gives us p, the price of the second good, and w, the wage, as functions of the uniform profit rate r in an economy which employs only technique α, and the corresponding formulas for one which only employs technique β. In both cases, prices and wage are normalized such that the first commodity is the numeraire: (168) (169) w (α) = f (α) (r) g (α) (r) p (α) = h(α) (r) g (α) (r) w (β) = f (β) (r) g (β) (r) p (β) = h(β) (r) g (β) (r)

48 48 HANS G. EHRBAR where (170) (171) (172) f (ι) (r) = c 11 c (ι) 22 c(ι) 12 c 21 = 1 (1 + r)(a 11 + a (ι) 22 ) + (1 + r)2 (a 11 a (ι) 22 a(ι) 12 a 21) g (ι) (r) = l 1 c (ι) 22 + l(ι) 2 c 21 h (ι) (r) = l (ι) 2 c 11 + l 1 c (ι) 21 (ι stands for α or β), and (173) (174) c 11 = 1 (1 + r)a 11 c 21 = (1 + r)a 21 c (ι) 12 c (ι) 22 = (1 + r)a(ι) 12 = 1 (1 + r)a(ι) 22 The c ij are not used by Woods, but they simplify the proofs below. If commodity 1 is the wage good, then w can be considered the real wage (since commodity 1 is the numeraire). The equations (168) can therefore be considered the wage-profit frontiers if only technique α or only technique β is used for the second commodity.

49 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 49 Let s first see how the wage-profit curves for the two technologies differ. (175) (176) w (α) w (β) = f (α) (r) g (α) (r) f (β) (r) g (β) (r) g (α) (r)g (β) (r)(w (α) w (β) ) = f (α) (r)g (β) (r) f (β) (r)g (α) (r) = (c 11 c (α) 22 c(α) 12 c 21)(l 1 c (β) 22 + l(β) 2 c 21) (c 11 c (β) 22 c(β) 12 c 21)(l 1 c (α) 22 + l(α) 2 c 21) These are eight terms. Sort them by the l factors they contain, and note that the two terms with l 1 c 11 c (α) 22 c(β) 22 cancel out: (177) = l 1 ( c (α) 12 c 21c (β) 22 + c(β) 12 c 21c (α) 22 ) + + l (α) 2 ( c 11c 21 c (β) 22 + c(β) 12 c2 21) + + l (β) 2 (c 11c 21 c (α) 22 c(α) 12 c2 21).

50 50 HANS G. EHRBAR Each of these terms contains c 21, therefore divide by it: (178) g (α) (r)g (β) (r) (w (α) w (β) ) = l 1 (c (β) 12 c c(α) 22 c(α) 12 c(β) 22 ) 21 l (α) 2 (c 11c (β) 22 c(β) 12 c 21) + + l (β) 2 (c 11c (α) 22 c(α) 12 c 21). Now use (170) to get the formula Woods has in [22, (6.3) on p. 76], but instead of Woods s k(r) we use q(r) = (1 + r)k(r). (179) w (α) w (β) = (1 + r)a 21 ( (β) l g (α) (r)g (β) 2 (r) f (α) (r) l (α) 2 f (β) (r) l 1 q(r) ) where (180) q(r) = c (α) 12 c(β) = (1 + r)a (α) c(β) 12 c(α) 22 ( 1 (1 + r)a (β) 22 ) (1 + r)a (β) 12 ( (α)) 1 (1 + r)a 22 = (1 + r)(a (α) 12 a(β) 12 ) (1 + r)2 (a (α) 12 a(β) 22 a(β) 12 a(α) 22 )

51 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 51 Problem 13. Show that the difference in relative prices has a formula amazingly similar to (179): (181) p (β) p (α) l 1 ( (β) = l g (α) (r)g (β) 2 (r) f (α) (r) l (α) 2 f (β) (r) l 1 q(r) ). Show that p (β) p (α) and w (α) w (β) always have the same sign. Answer. (182) p (β) p (α) = h(β) (r) g (β) (r) h(α) (r) g (α) (r) = h(β) (r)g (α) (r) h (α) (r)g (β) (r) g (α) (r)g (β) (r) etc. until you get (181). Equality of the signs follows from (183) p (β) p (α) = (w (α) w (β) l 1 ). (1 + r)a 21 Now let s look at the decision of the capitalist to change technologies. Assume that process (IIα) is initally employed, at wage rate w and equalized rate of profit r. Now (IIβ) becomes available. Would someone using the new technique with the same prices, wages, and profit rate be able to supply the second good at a lower price than someone using the old technique? This will be our criterion for adoption of the

52 52 HANS G. EHRBAR technology. Woods calls it cost minimization although a profit at the prevailing rate is included here in the cost minimized. Kurz and Salvadori [9] follow Marx in calling it no extra profits. In formulas, technology (β) will be adopted if (184) s 2 (β : α) = p (α) (1 + r)(a (β) 12 + a(β) 22 p(α) ) w (α) l (β) 2 > 0 Now we need some manipulations. s 2 (β : α) = p (α) c (β) 22 c(β) 12 w(α) l (β) 2 = 1 ( (β) c g (α) 22 h(α) c (β) 12 g(α) f (α) l (β) ) 2 = 1 ( (β) c g (α) 22 (l(α) 2 c 11 + l 1 c (α) 12 ) c(β) 12 (l 1c (α) 22 + l(α) 2 c 21 ) f (α) l (β) 2 Now the coefficients of l (α) 2 are exactly f(r), and those of l 1 are q(r), so that one gets ) (185) = 1 g (α) ( l (α) 2 f (β) (r) l (β) 2 f (α) (r) + l 1 q(r) )

53 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 53 Comparing this with (179) one sees that (186) s 2 (β : α) = (w (β) w (α) g (β) (r) ) (1 + r)a 21 which has the same sign as w (β) w (α). This is an important result: if a cost saving technology is introduced, and then profit rates are equalized at the same average profit rate as before, this will lead to a rise in real wages. Since there is a negative relationship between real wages and profit rates, the following holds true too: if profit rates are equalized with the real wages held constant, then the profit rate increases. This is the famous Okishio theorem which is often taken as mathematical proof that the rate of profit cannot fall. Problem 14. Is it possible that someone switches technologies, and after the profit rates have equalized at the new technology (with real wages kept constant) he discovers that at these new prices of production the old technology is cost saving, i.e., he or she would switch back after the benefits of being a newcomer have been exhausted? Answer. If prices of production are formed which keep the real wage the same as before, then the new overall rate of profit is higher than the old. From this we know that switching back cannot be cost saving, because for the new prices of production the same condition holds: for switching back to be cost saving, the old prices of production must have a higher equalized rate of profit than the new.

54 54 HANS G. EHRBAR Problem 15. What does the Okishio theorem say? Answer. The Okishio theorem says: if, starting from a price system with equalized rates of profit a producer switches to a new technology because it is cost saving, and if after this switch profit rates are equalized with the real wage kept constant, then the profit rates after this equalization is higher than the profit rate before. 11. Four Examples All four examples are discussed in detail in [22, chapter 6.7]. Problem 16. Look at the following technique (this is one of the two techniques compared in example 1 in [22, chapter 6.7]): [ ] A (α) 1/2 1/10 (187) = (l (α) ) = [ 1 1 ] 2/5 1/2 (i) Using (42), compute (I A (α) ) 1 and verify that it is a positive matrix. Answer. The inverse of I A (α) is [ ] 1 [ ] (I A (α) ) 1 1/2 1/10 1 1/2 1/10 = = 2/5 1/ /5 1/2 25 [ ] [ ] = 100 1/2 1/10 = (188) /5 1/

55 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 55 (ii) What must the gross product be for a net product of 1239 units of the first product and 0 units of the second product? Answer. The inverse which we just computed helps us here: [ ] 1239 (189) x Ax = (I A)x = 0 [ ] [ ] [ x = (I A) = [ ] [ ] [ ] (190) = = (iii) Compute the value vector. ] = Answer. The same matrix inverse allows us to compute the value vector, using (155): λ = 1 [ ] [ ] = 1 [ ] [ ] (191) (192)

56 56 HANS G. EHRBAR (iv) Compute the maximal wage if r = 0 (with the first good the numeraire). Answer. This maximal w satisfies p A + wl = p. Comparing this with (155) you see that p = wλ. Since we have the convention to normalize p in such a way that its first component is 1, if follows (193) (194) w = 1 λ 1 = (v) Compute the maximal rate of profit if w = 0. Answer. The maximal profit rate, along with its associated price vector, satisfies (195) (1 + r)p A = p ; 1 therefore is an eigenvalue of A. It must be the Frobenius eigenvalue, because no other eigenvalue 1+r has a nonnegative eigenvector associated with it. This Frobenius eigenvalue is the largest eigenvalue. Using (67) one gets frob(a) = 0.7; therefore (196) (197) r =

57 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 57 (vi) Divide the labor input by the value of the means of production in (ii) as an approximate estimate of the maximal rate of profits. Answer. First compute the value of the constant capital needed to produce the above x: λ Ax = 1 [ ] [ ] [ ] 1/2 1/ (198) /5 1/ = 1 [ ] [ ] 2950 (199) = Now let s see how much new value is created to get this x: [ ] [ ] l 2950 (200) x = 1 1 = If wages were zero, then all this new value were profit, therefore the maximal profit rate would be (201) r (vii) Compute the value composition of the two industries. Would you expect the relative price of the second industry (again with the first industry as numeraire) to rise or fall as the profit rate rises?

58 58 HANS G. EHRBAR Answer. Using the value vector we can determine the value composition needed to produce any given gross output x (the value composition is the value of the constant capital divided by the living labor used: (202) c = λ Ax l x A comparison of λ A with l gives us therefore the value compositions of the two industries: λ A = 1 [ ] [ ] 1/2 1/ = 1 [ ] [ ] (203) = [ 21 ] 2/5 1/2 21 (204) l = 1 1 We see that process (IIα) has a much lower value composition (capital intensity) than process I, i.e., it uses a much lower value of constant capital per labor input. From this we should expect that, as the wage falls, the relative price of the output of the second process with relation to the first falls, since the surplus-value created by the workers in the second process is transferred to the capitalists in the first process. The graphs below will show exactly that. The mathematical condition for relative prices to fall is that the value composition computed using l A instead of λ A is lower in process II. This is related to the value composition, and I think it may even be mathematically equivalent. Problem 17. Look at the following technology, which has alternative processes for the second good. This is example 1 in [22, chapter 6.7]. Note that there is a typo in

59 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 59 the book: the labor input for technology (β) is given as 1, whereas it should be 3/2. (205) A = [ a 11 a 21 ] a (α) 12 a (β) 12 a (α) 22 a (β) 22 = [ 1/2 1/10 ] 1/5 2/5 1/2 1/4 l = [ 1 1 3/2 ] B = [ 1 0 ] B is the matrix which shows the output vector for every process. Answer. Process (α) was discussed in question 16. Now look at process (β). Here is the A-matrix which only uses β: [ ] A (β) 1/2 1/5 (206) = 2/5 1/4 [ ] 1 [ ] (I A (β) ) 1 1/2 1/5 1 3/4 1/5 = = 2/5 3/ = 2/5 1/2 25 (207) (208) = λ = 1 59 [ 3/4 1/5 2/5 1/2 ] [ = 1 59 ] [ ] [ ] = 1 [ ] [ ]

60 60 HANS G. EHRBAR I.e., with technology (β) both products contain more labor time than with (α). In terms of labor inputs, (β) is therefore an unambiguously worse technology; it can become more profitable than (α), if at all, only if wages are low. Let s look again at the value composition of the two industries: (209) (210) λ A = 1 [ ] [ ] 1/2 1/ = 1 [ ] [ 59 ] 2/5 1/4 118 l = 1 3/2 Again the first process has higher value composition, versus Here are the wage-profit curves for the two technologies superimposed: w.. r At high wages, (α) is used, which is the more labor-saving technology. At lower wages, (β) is used. As wages fall, the relative price p of the second good with relation to the first good falls, as was explained above. This is true for both technologies. Here are the relative prices of the second good in their dependence on the profit rate:

61 THE TWO-SECTOR FIXED COEFFICIENT PRODUCTION MODEL 61 p.. At some point, as wages are lowered, there is a switch from technology (α) to (β), which uses less of the second good (which is getting cheaper) and more of the first good (which is getting more expensive). This is not what one would expect in a neoclassical framework, and is therefore considered a perverse outcome. But I think it can be explained: the switch was not made because r