The Green Revolution and the Water Crisis in India: An ...home.hiroshima-u.ac.jp/fukito/water crisisB.pdf1. Basic Model Rice is produced by the two identical competitive firm, aiming

The Green Revolution and the Water Crisis in India:

An Economic Analysis

Toshitaka Fukiharu (Faculty of Economics, Hiroshima University)

November, 2006

In[1]:= Off@Solve::ifun, General::spell1, General::spellDIntrocuction

Green Revolution implies an innovation in the agriculture: the invention of new seedling for crops. It started inMexico for the wheat and in Philipine for the rice in the 1950s. Norman Borlorg was awarded the Nobel Prize forPeace in 1970 for the promotion of the revolution. The new seedling, invented through crossbreeding, allows farmersto produce the greater output of rice (wheet) than the old seedling. In Africa, aflicted by the food shortage, the promo-tion of this revolution has been attempted. In India, however, it is reported that there are farmers who commit suicideafter running into debt in the process of the revolution, while the farmers may well expect greater profit from the GreenRevolution. In this paper, it is examined in terms of general equilibrium theory if the innovation always make itpossible for the users of innovation to obtain the greater profit. In Section 1, we start with the basic model withoutseedling. In Section 2, seedling industry is introduced into the basic model. For the purpose of examining the robust-ness of the conclution in Section 2, the assumption on production function is modified in Section 3. In the GreenRevolution, water, as the capital good, is important. In Section 4, the accumulation of the innovation-embodyingcapital good, water, is examined, utilizing game theory. Section 4 concludes this paper.

water crisisB.nb 1

1. Basic Model

Rice is produced by the two identical competitive firm, aiming at the profit maximization. Their identical productionfunction is assumed as in what follows:

q=f(Y1 , L)=kY1a1 La2 a1 +a2 <1 (1)

where q is output, k is the level of technology, Y1 is the fixed amount of water, and L is the labor input. By the profitmaximization, the labor demand function, Ld , the supply function of rice, xs , and the profit, p0 , are computed as inwhat follows.

In[2]:= q = k ∗ y1^Ha1L∗ l^Ha2L;pi = p ∗ q − w ∗ l;sol1 = PowerExpand@Solve@D@pi, lD 0, lD@@1DDD;

In[5]:= "Ld=" H ld = l ê. sol1LOut[5]= Ld= a2− 1

−1+a2 k− 1−1+a2 p− 1

−1+a2 w 1−1+a2 y1− a1

−1+a2

In[6]:= "xs=" H PowerExpand@Simplify@qs = q ê. sol1DDLOut[6]= xs= a2 a2

1−a2 k1+ a21−a2 p a2

1−a2 w a2−1+a2 y1a1− a1 a2

−1+a2

In[7]:= "π=" Simplify@PowerExpand@Simplify@pi0 = pi ê. sol1DDDOut[7]= π= I−a2 1

1−a2 + a2 a21−a2 M k 1

1−a2 p 11−a2 w a2

−1+a2 y1− a1−1+a2

Suppose that there is one (aggregate) household and it maximizes utility under income constraint. Two firms areowned by the (aggregate) household. Specifically, the household is assumed to maximize utility under income con-straint:

max u[x, l ]=xb1 lb2

s.t. px=w(N0 -l )+2p0 (2)

where u[x, l ] is the utility function, x is the quantity of rice, l is the leisure, N0 is the household's initial endowment ofleisure hours, p is the commodity price, w is the wage rate, and p0 is the profit distribution. From the assumption ofutility function as Cobb-Douglas type in (2) the demand function of rice, xd , is computed as in what follows.

In[8]:= u = x^Hb1L∗ le^Hb2L;sol2 = PowerExpand@

Solve@8D@u, xDê D@u, leD p êw, p ∗ x == w ∗Hl0 − leL + 2 ∗ pi0<, 8x, le<D@@1DDD;xd = Simplify@x ê. sol2D; ls = Simplify@l0 − le ê. sol2D; xd

Out[10]=b1 p a2

1−a2 y1− a1−1+a2 I−2 a2 1

1−a2 k 11−a2 w a2

−1+a2 + 2 a2 a21−a2 k 1

1−a2 w a2−1+a2 + l0 p 1

−1+a2 w y1 a1−1+a2 M

b1 + b2

Meanwhile, the labor supply function, Ls , is computed as in what follows.

In[11]:= ls

Out[11]=1

b1 + b2 Iy1− a1−1+a2 I2 a2 1

1−a2 b2 k 11−a2 p 1

1−a2 w 1−1+a2 − 2 a2 a2

1−a2 b2 k 11−a2 p 1

1−a2 w 1−1+a2 + b1 l0 y1 a1

−1+a2 MM

water crisisB.nb 2

General equilibrium (GE) requires

2xs =xd (Commodity Market) (3) Ls =2Ld (Labor Market) (4) From (3) the GE rice price is computed as in what follows, where w=1.

In[12]:= f1 = PowerExpand@xd − 2 qs ê. w → 1D; sol3 = PowerExpand@Simplify@Solve@f1 0, pDD@@1DDDOut[12]= 9p → 2−1+a2 b11−a2 Ia2 1

1−a2 b1 + a2 a21−a2 b2M−1+a2 k −1+a2

1−a2 l01−a2 y1−a1=It is easy to ascertain that the GE rice price satisfies (4).

In[13]:= Simplify@PowerExpand@ls − 2 ld ê. sol3 ê. w → 1DDOut[13]= 0

From the computation of GE rice price, it is clear that as the innovation is promoted the GE price declines due to theincreased output. As the innovation is promoted, the output increases, while the rice price declines. What is the effectof the promotion of innovation on the profit of the farmers. The profit, derived in what follows, show that it is indepen-dent of the promotion of innovation. Note, however, that at GE only the relative price is determined. In this paper wagerate, w, is selected as the numeraire. In the actual world, when the innovation is promoted, the nominal profit may wellincrease. The result in this section implies that wage rate also rises, and the real profit remains the same.

In[14]:= Simplify@PowerExpand@pi0 ê. sol3 ê. w → 1DDOut[14]= −

H−1 + a2L b1 −2+a2−1+a2 Ia2 b1 a2

−1+a2 + b1 1−1+a2 b2M l0

2 Ha2 b1 + b2L2

The promotion of innovation contributes to the promotion of the welfare of the households, as shown by the derivedutility level at GE.

In[15]:= Simplify@PowerExpand@u ê. sol2 ê. sol3DDOut[15]= 2b1−a2 b1 a2 b1+b2

1−a2 b1b1+a2 b1+b2 b2b2 Hb1 + b2L−b1−b2 Ia2 11−a2 b1 + a2 a2

1−a2 b2M−a2 b1−b2

kb1 l0a2 b1+b2 w−b2 Jw +b2 w

a2 b1 + J−1 +1

a2 N w a2−1+a2 Nb1+b2

y1a1 b1

2. Introduction of the Seedling Industry: Cobb-Douglas-Type Production Function Case

In this section, somewhat sophisticated aspect is introduced into the basic model. In the basic model, it is implicitlyassumed that the availability of seedling is fixed. In the actual world, the seedling would be provided by firms, espe-cially when the new seedling is invented by the crossbreeding. While the farmers can change the amount of seedlinginput, the seedling itself must be purchased.

As in the previous section, rice is produced by the two identical competitive firms, aiming at the profit maximization.However, their identical production function is assumed as in what follows:

q=f(Y2 , L)=HkY2La1 La2 a1 +a2 <1 (5)

water crisisB.nb 3

where q is output, Y2 is the seedling input, k is the level of innovation, and L is the labor input. By the profit maximiza-tion, the labor demand function, Ld , the seedling demand function, Y2 d , the supply function of rice, xs , and the profit,p1 , are computed as in what follows, where r is the price of seedling.

In[16]:= q = Hk ∗ yL^Ha1L ∗ l^Ha2L;pi = p ∗ q − r ∗ y − w ∗ l;sol1 = Solve@8D@pi, yD 0, D@pi, lD 0<, 8y, l<D@@1DD

Out[18]= 9l →−a1 Log@a1D−Log@a2D+a1 Log@a2D−a1 Log@kD−Log@pD+a1 Log@rD+Log@wD−a1 Log@wD

−1+a1+a2 ,

y →−Log@a1D+a2 Log@a1D−a2 Log@a2D−Log@kD+a2 Log@kD−Log@pD+Log@rD−a2 Log@rD+a2 Log@wD

−1+a1+a2

k =In[19]:= q = Hk ∗ yL^Ha1L ∗ l^Ha2L;

pi = p ∗ q − r ∗ y − w ∗ l;sol1 = Solve@8D@pi, yD 0, D@pi, lD 0<, 8y, l<D@@1DD;sol1 =8y → a1^HH−1 + a2LêH−1 + a1 + a2LL ∗ a2^H−a2êH−1 + a1 + a2LL∗ p^H−1êH−1 + a1 + a2LL∗

r^HH1 − a2LêH−1 + a1 + a2LL ∗ w^Ha2êH−1 + a1 + a2LL∗ k^HH−1 + a2LêH−1 + a1 + a2LLê k,l −> a1^HH−a1LêH−1 + a1 + a2LL∗ a2^HH−1 + a1LêH−1 + a1 + a2LL ∗ p^H−1êH−1 + a1 + a2LL ∗

r^Ha1êH−1 + a1 + a2LL ∗ w^HH1 − a1LêH−1 + a1 + a2LL∗ k^H−a1êH−1 + a1 + a2LL<;ld = l ê. sol1; "Ld="

ld

Out[23]= Ld= a1− a1−1+a1+a2 a2 −1+a1

−1+a1+a2 k− a1−1+a1+a2 p− 1

−1+a1+a2 r a1−1+a1+a2 w 1−a1

−1+a1+a2

In[24]:= "Y2 d=" Hyd = y ê. sol1LOut[24]= Y2 d= a1 −1+a2

−1+a1+a2 a2− a2−1+a1+a2 k−1+ −1+a2

−1+a1+a2 p− 1−1+a1+a2 r 1−a2

−1+a1+a2 w a2−1+a1+a2

In[25]:= "xs=" PowerExpand@Simplify@qs = q ê. sol1DDOut[25]= xs= a1 a1 H−1+a2L

−1+a1+a2 − a1 a2−1+a1+a2 a2 H−1+a1L a2

−1+a1+a2 − a1 a2−1+a1+a2 k a1 H−1+a2L

−1+a1+a2 − a1 a2−1+a1+a2 p− a1+a2

−1+a1+a2 r a1 H1−a2L−1+a1+a2 + a1 a2

−1+a1+a2 w H1−a1L a2−1+a1+a2 + a1 a2

−1+a1+a2

In[26]:= "π1=" Simplify@PowerExpand@Simplify@pi0 = pi ê. sol1DDDOut[26]= π1= I−a1 −1+a2

−1+a1+a2 a2− a2−1+a1+a2 + a1− a1

−1+a1+a2 I−a2 −1+a1−1+a1+a2 + a2− a2

−1+a1+a2 MM k− a1−1+a1+a2 p− 1

−1+a1+a2 r a1−1+a1+a2 w a2

−1+a1+a2

In this section, the seedling provider, firm 3, is introduced. The firm 3 provides the seedling, aiming at the profitmaximization. Its production function is assumed as in what follows:

Y2 =g(L)=Lc1 0<c1 <1 (6)

where Y2 is the seedling output, L is the labor input, and k2 is the level of innovation. By the profit maximization, thelabor demand function, L2 d , the seedling supply function, Y2 s , and the profit, p3 , are computed as in what follows,where r is the price of seedling.

In[27]:= y2 = l2^Hc1L;pi2 = r ∗ y2 − w ∗ l2;sol1A = Solve@D@pi2, l2D 0, 8l2<D@@1DD;"L2 d=" Hl2d = PowerExpand@l2 ê. sol1ADL

Out[30]= L2 d= c1− 1−1+c1 r− 1

−1+c1 w 1−1+c1

water crisisB.nb 4

In[31]:= "Y2 s=" Hy2s = PowerExpand@y2 ê. sol1ADLOut[31]= Y2 s= c1− c1

−1+c1 r− c1−1+c1 w c1

−1+c1

In[32]:= "π3=" HSimplify@PowerExpand@pi01 = pi2 ê. sol1ADDLOut[32]= π3= I−c1 1

1−c1 + c1 c11−c1 M r 1

1−c1 w c1−1+c1

Only minor modification is required for the specification of the household. There is one (representative) householdand it maximizes utility under income constraint. Three firms are owned by the household. Specifically, the householdis assumed to maximize utility under income constraint:

max u[x, l ]=xb1 lb2

s.t. px=w(N0 -l )+2p0 + p3 (7)

From the assumption of utility function as Cobb-Douglas type in (7) the demand function of rice, xd , and the laborsupply function, Ls , are computed as in what follows.

In[33]:= u = x^Hb1L∗ le^Hb2L;sol2 = Simplify@PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x == w ∗Hl0 − leL + 2 ∗ pi0 + pi01<, 8x, le<D@@1DDDD;Simplify@xd = x ê. sol2; ls = l0 − le ê. sol2D;

General equilibrium (GE) requires

2xs =xd (Commodity Market) (3) Ls =2Ld +L2 d (Labor Market) (4') Y2 s =2Y2 d (Seedling Market) (8)

In order to examine the effect of promoted innovation, we compare the profits when k2 =1 and k2 =10, specifying theparameters as in what follows.

a1 =1/4, a2 =1/4, b1 =b2 =1/2, c1 =1/3, N0 =10 (9)

Under (9), xd -2xs in (3) is derived as in what follows when k2 =1.

In[36]:= f1 = PowerExpand@xd − 2 qs ê. w → 1D;f2 = PowerExpand@Simplify@PowerExpand@ls − H2 ld + l2dL ê. w → 1DDD;f10 = Simplify@PowerExpand@

f1 ê. a1 → 1ê4 ê. a2 → 1ê 4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê. k2 → 1DDOut[37]=

360 − 27 è!!!!k p2è!!!!r + 8 è!!!3 r3ê2

72 p

In the same way, under (9), Ls -(2Ld +L2 d ) in (4') is derived as in what follows when k2 =1.

In[38]:= f20 = Simplify@PowerExpand@ls − H2 ld + l2dL ê. a1 → 1ê 4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê.

l0 → 10 ê. k2 → 1DDOut[38]= 5 −

è!!!k p2

4 è!!!r −2 r3ê2

3 è!!!3

water crisisB.nb 5

Thus, under (9), from (3) and (4'), the GE price price, p, and the GE seedling price, r, are computed as in what followswhen k2 =1.

In[39]:= sol4 = Solve@8f10 0, f20 0<, 8p, r<D@@2DDOut[39]= 9r →

3 52ê3

4 , p →33ê4 52ê3è!!!2 k1ê4 =

It is easy to confirm that these GE prices satsisfy the seedling market equilibrium condition, (8).

In[40]:= Simplify@H2 ∗ yd − y2sL ê. a1 → 1ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê.sol4 ê. l0 → 10 ê. k2 → 1D

Out[40]= 0

Under (9), the profit for the farmer producing rice is 1.875, as shown in what follows when k2 =1.

In[41]:= N@pi0k = pi0 ê. a1 → 1ê 4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê. sol4 ê.l0 → 10D

Out[41]= 1.875

Under (9), the profit for the seedling provider is 1.25, as shown in what follows when k2 =1.

In[42]:= N@pi01 ê. a1 → 1ê 4 ê. a2 → 1 ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê. sol4 ê.l0 → 10 ê. k2 → 1D

Out[42]= 1.25

Under (9), the GE utility level of the household rises as the level of technology rises, say, when k2 rises from 1 to 10,as shown in what follows.

In[43]:= u ê. sol2 ê. sol4 ê. l0 → 10 ê. a1 → 1ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê.w → 1

Out[43]=35ê8 52ê3 k1ê8

23ê4

3. Introduction of the Seedling Industry: CES-Type Production Function Case

In this section, we replace the assumption of Cobb-Douglas type produnction function by the CES type one, andexamine if the conclusion in the previous section remains the same. As in the previous section, rice is produced by the two identical competitive firms, aiming at the profit maximization.However, their identical production function is assumed as in what follows: q=f(kY2 , L)=HHkY2L-t + L-tL-nêt (5')

where n is the degree of homogeneity, and t is the value of constant elasticity of substitution.

water crisisB.nb 6

ü 3.1. The Case of t=-1/2

Suppose that t=-1/2, and n=1/2, in order to guarantee the positive profit. By the profit maximization, the labor demandfunction, Ld , the seedling demand function, Y2 d , the supply function of rice, xs , and the profit, p1 , are computed as inwhat follows, where r is the price of seedling.

In[44]:= Clear@wD;m = 1ê2; t = −1 ê2;q = HHk ∗ yL^H−tL + l^H−tLL^H−mêtL;pi = p ∗ q − r ∗ y − w ∗ l;sol1 = Solve@8D@pi, yD 0, D@pi, lD 0<, 8y, l<D@@1DD;ld = l ê. sol1;PowerExpand@Simplify@qs = q ê. sol1DD;yd = y ê. sol1;pi0 = PowerExpand@Simplify@pi ê. sol1DD;

In this section, the seedling provider, firm 3, is introduced. The firm 3 provides the seedling, aiming at the profitmaximization. Its production function is assumed as in what follows:

Y2 =g(L)=Lc1 c1 <1 (6')

where Y2 is the seedling output, L is the labor input, and k2 is the level of innovation. By the profit maximization, thelabor demand function, L2 d , the seedling supply function, Y2 s , and the profit, p3 , are computed as in what follows,where r is the price of seedling.

In[53]:= y2 = l2^Hc1L;pi2 = r ∗ y2 − w ∗ l2;sol1A = Solve@D@pi2, l2D 0, 8l2<D@@1DD;l2d = PowerExpand@l2 ê. sol1AD;y2s = PowerExpand@y2 ê. sol1AD;Simplify@PowerExpand@pi01 = pi2 ê. sol1ADD;

We assume exactly the same specification of the household as in the previous section. There is one (representative)household and it maximizes utility under income constraint. Three firms are owned by the household. Finally, thehousehold is assumed to maximize utility under income constraint: (7). From the assumption of utility function as Cobb-Douglas type in (7) the demand function of rice, xd , and the laborsupply function, Ls , are computed as in what follows.

In[59]:= u = x^Hb1L∗ le^Hb2L;sol2 = Simplify@PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x == w ∗Hl0 − leL + 2 ∗ pi0 + pi01<, 8x, le<D@@1DDDD;Simplify@xd = x ê. sol2; ls = l0 − le ê. sol2D;

As in the previous section, the GE requires (3), (4'), and (8). In order to examine the effect of promoted innovation, wecompare the profits when k2 =1 and k2 =10, specifying the parameters as in (9). Under (9), from (3) and (4') the GE riceprice, p, and GE seedling peice, r, are derived as in what follows when k2 =1.

water crisisB.nb 7

In[62]:= f1 = Simplify@PowerExpand@xd − 2 qs ê. w → 1DD;f2 = PowerExpand@Simplify@PowerExpand@ls − H2 ld + l2dL ê. w → 1DDD;sol3 = PowerExpand@Simplify@Solve@f1 0, pDDD;check1 = Simplify@

PowerExpand@8f1 0, f2 0< ê. a1 → 1ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê.c1 → 1 ê3 ê. l0 → 10 ê. k2 → 1DD;

sol4 = Solve@check1, 8p, r<D;

In[67]:= N@sol4 ê. k → 1D@@2DDOut[67]= 8r → 1.73927, p → 2.14631<It is easy to ascertain that these GE prices satsisfy the seedling market equilibrium condition, (8).

In[68]:= N@2 yd − y2s ê. a1 → 1ê 4 ê. a2 → 1 ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê. w → 1 ê.

sol4@@2DD ê. k → 1DOut[68]= −4.44089 × 10−16

Under (9), the GE utility level of the household rises as the level of technology rises, say, when k rises from 1 to 10. Inorder to show this, first, the utility level when k=1 is computed as shown in what follows.

In[69]:= u ê. sol2 ê. a1 → 1ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê.N@sol4@@2DDD ê. w → 1 ê. k → 1

Out[69]= 4.95229

Under (9), the profit for the farmer producing rice is 1.81381, as shown in what follows when k =1.

In[70]:= N@pi0k = pi0 ê. a1 → 1ê 4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê.N@sol4@@2DDD ê. l0 → 10 ê. k → 1D

Out[70]= 1.81381

Under (9), the profit for the seedling provider is 0.88287, as shown in what follows when k =1.

In[71]:= N@pi01 ê. a1 → 1ê 4 ê. a2 → 1 ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê.N@sol4@@2DDD ê. l0 → 10 ê. k → 1D

Out[71]= 0.88287

In exactly the same way, under (9), from (3) and (4'), the GE rice price, p, and the GE seedling price, r, are computedas in what follows when k =10.

In[72]:= f10A = Simplify@PowerExpand@f1 ê. a1 → 1ê 4 ê. a2 → 1 ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê. k → 10DD;

f20A = Simplify@PowerExpand@ls − H2 ld + l2dL ê. a1 → 1ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê.b2 → 1ê 2 ê. c1 → 1 ê3 ê. l0 → 10 ê. w → 1 ê. k → 10DD;

sol4A = N@Solve@8f10A 0, f20A 0<, 8p, r<DD@@2DDOut[74]= 8r → 3.05178, p → 1.37065<Under (9), the profit for the farmer producing rice is 2.00867, as shown in what follows when k =10.

water crisisB.nb 8

In[75]:= N@pi0kA =

pi0 ê. a1 → 1ê4 ê. a2 → 1ê 4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê. sol4A ê.l0 → 10 ê. k → 10D

Out[75]= 2.00867


In[76]:= N@pi01 ê. a1 → 1ê 4 ê. a2 → 1 ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê. sol4A ê.l0 → 10 ê. k → 10D

Out[76]= 2.052

Under (9), the GE utility level when k=10 is computed as shown in what follows.

In[77]:= u ê. sol2 ê. a1 → 1ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê. sol4A ê.w → 1 ê. k → 10

Out[77]= 6.86286

Thus, under (9), when k rises, say from 1 to 10, the profits for the firms rise, as well as the household utility, contraryto the previous section.

ü 3.2. The Case of t=1/2

In this subsection, we examine the case in which t=1/2 and n=1/2 are assumed, in order to guarantee the positive profit.By the profit maximization, the labor demand function, Ld , the seedling demand function, Y2 d , the supply function ofrice, xs , and the profit, p1 , are computed as in what follows, where r is the price of seedling.

In[78]:= Clear@wD;m = 1ê 2; t = 1ê 2;q = HHk ∗ yL^H−tL + l^H−tLL^H−mêtL;pi = p ∗ q − r ∗ y − w ∗ l;f1 = Simplify@PowerExpand@8D@pi, yD 0, D@pi, lD 0<DD;sol1 = Solve@8D@pi, yD 0, D@pi, lD 0<, 8y, l<D;ld = l ê. sol1@@1DD ê. w → 1;qs = q ê. sol1@@1DD ê. w → 1;yd = y ê. sol1@@1DD ê. w → 1;pi0 = pi ê. sol1@@1DD ê. w → 1;y2 = l2^Hc1L;pi2 = r ∗ y2 − w ∗ l2;sol1A = Solve@D@pi2, l2D 0, 8l2<D@@1DD;l2d = PowerExpand@l2 ê. sol1AD;y2s = PowerExpand@y2 ê. sol1AD;pi01 = Simplify@PowerExpand@pi2 ê. sol1ADD;

We assume exactly the same specification of the household as in the previous subsection. There is one(representative) household and it maximizes utility under income constraint. Three firms are owned by the household.Finally, the household is assumed to maximize utility under income constraint: (7). From the assumption of utility function as Cobb-Douglas type in (7) the demand function of rice, xd , and the laborsupply function, Ls , are computed as in what follows.

water crisisB.nb 9

In[94]:= u = x^Hb1L∗ le^Hb2L;w = 1;sol2A = Simplify@PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x == w ∗Hl0 − leL + 2 ∗ pi0A + pi01A<, 8x, le<D@@1DDDD;sol2 = sol2A ê. 8pi0A → pi0, pi01A → pi01<;xd = x ê. sol2; ls = l0 − le ê. sol2;

As in the previous section, the GE requires (3), (4'), and (8). In order to examine the effect of promoted innovation, wecompare the profits when k2 =1 and k2 =10, specifying the parameters as in (9). Under (9), from (3) and (4') the GE riceprice, p, and GE seedling peice, r, are derived, in terms of Newton method, as in what follows when k =1.

In[99]:= f1 = PowerExpand@xd − 2 qs ê. w → 1D;f2 = H2 ld + l2dL − ls ê. w → 1;check1 =8f1 0, f2 0< ê. a1 → 1 ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê.

k → 1;sol4 = FindRoot@check1, 8p, 1<, 8r, 1<, WorkingPrecision → 34, MaxIterations → 1000D

Out[102]=8p → 9.925936526067923619692516086124228, r → 2.433063890650822553968008078777221<It is easy to ascertain that these GE prices satsisfy the seedling market equilibrium condition, (8).

In[103]:=2 yd − y2s ê. a1 → 1ê 4 ê. a2 → 1ê 4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê. w → 1 ê.

k → 1 ê. sol4

Out[103]=0. × 10−30

Under (9), the GE utility level of the household when k =1 is derived as in what follows.

In[104]:=u ê. sol2A ê. 8pi01A → pi01, pi0A → pi0< ê. sol2A ê. a1 → 1ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê.

b2 → 1 ê2 ê. c1 → 1 ê3 ê. l0 → 10 ê. w → 1 ê. k → 1 ê. sol4

Out[104]=2.4251374064054069951435103947

For later use, the stability of price dynamics is examined when k =1. The following differential equation is the Walra-sian tatonnement process of the two markets: the rice market and the seedling market, where t stands for time.

d@p@tDDÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅdt =xd -2xs d@r@tDDÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅdt =2Y2 d -Y2 s (10)

The price trajectory of p[t] on (10), starting from p[0]=1 and r[0]=1, is depicted in what follows. It is convergent. Theprice trajectory of r[t] on (10), starting from p[0]=1 and r[0]=1, is depicted in what follows. It is also convergent.Thus, the Walrasian tatonnement process is stable.

water crisisB.nb 10

In[105]:=Clear@d1, d2D;d1 = Hf1 ê. a1 → 1ê4 ê. a2 → 1ê 4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê. k → 1L ê.8p → p@sD, r → r@sD<;d2 = H2 yd − y2s ê. a1 → 1ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê.

w → 1 ê. k → 1L ê. 8p → p@sD, r → r@sD<;solD = NDSolve@8D@p@sD, sD d1, D@r@sD, sD d2, p@0D 1, r@0D 1<,8p@sD, r@sD<, 8s, 0, 1000<D;Plot@Hp@sD ê. solDL, 8s, 0, 100<, AxesLabel → 8"s", "p@sD"<D;

20 40 60 80 100s

2

4

6

8

10p@sD

In[110]:=Plot@Hr@sD ê. solDL, 8s, 0, 100<, AxesLabel → 8"s", "r@sD"<D;

20 40 60 80 100s

0.75

1.25

1.5

1.75

2

2.25

r@sD

Under (9), the profit for the farmer producing rice is 1.91013, as shown in what follows when k =1.

In[111]:=N@pi0k = pi0 ê. a1 → 1ê 4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê. sol4 ê.

l0 → 10 ê. k → 1DOut[111]=

1.91013


In[112]:=N@pi01 ê. a1 → 1 ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê. sol4 ê.

l0 → 10 ê. k → 1DOut[112]=

1.46076

water crisisB.nb 11

In exactly the same way, under (9), from (3) and (4'), the GE rice price, p, and the GE seedling price, r, are computedas in what follows when k =4.

In[113]:=f10A = f1 ê. b1 → 1 ê2 ê. b2 → 1 ê2 ê. c1 → 1ê3 ê. l0 → 10 ê. k → 4;f20A = ls − H2 ld + l2dL ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê. w → 1 ê. k → 4;check2 = 8f10A 0, f20A 0<;sol4A = FindRoot@check2, 8p, 1<, 8r, 1<, WorkingPrecision → 50, MaxIterations → 1000D

Out[116]=8p → 6.5341759658088092794970608957833779867681330906354,r → 2.0026867265590724109586266291779457297324515899518<

It is easy to ascertain that these GE prices satsisfy the seedling market equilibrium condition, (8).

In[117]:=2 yd − y2s ê. a1 → 1ê 4 ê. a2 → 1ê 4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê. w → 1 ê.

k → 4 ê. sol4A

Out[117]=−0. × 10−47

Under (9), the GE utility level of the household when k =4 is derived as in what follows. This utility level is higher thanthe one when k2 =1. Thus, when the green revolution takes place, the household becomes better off.

In[118]:=u ê. sol2A ê. 8pi01A → pi01, pi0A → pi0< ê. sol2A ê. a1 → 1ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê.

b2 → 1 ê2 ê. c1 → 1 ê3 ê. l0 → 10 ê. w → 1 ê. k → 4 ê. sol4A

Out[118]=2.892533623790402869182184367720163481907674791

The stability of price dynamics is examined when k =4. In the same way as above, the Walrasian tatonnement processof the two markets, (10), is assumed. When k =4, the price trajectory of p[t] on (10), starting from p[0]=1 and r[0]=1, isdepicted in what follows. It is convergent. The price trajectory of r[t] on (10), starting from p[0]=2 and r[0]=2, isdepicted in what follows. It is also convergent. Thus, the Walrasian tatonnement process is stable when k =4.

water crisisB.nb 12

In[119]:=Clear@d1, d2, solDD;d1 = Hf1 ê. a1 → 1ê4 ê. a2 → 1ê 4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê. k → 4L ê.8p → p@sD, r → r@sD<;d2 = H2 yd − y2s ê. a1 → 1ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. l0 → 10 ê.

w → 1 ê. k → 4L ê. 8p → p@sD, r → r@sD<;solD = NDSolve@8D@p@sD, sD d1, D@r@sD, sD d2, p@0D 2, r@0D 2<,8p@sD, r@sD<, 8s, 0, 100<D;Plot@Hp@sD ê. solDL, 8s, 0, 1<, AxesLabel → 8"s", "p@sD"<D;

0.2 0.4 0.6 0.8 1s

2.25

2.5

2.75

3

3.25

3.5p@sD

In[124]:=Plot@Hr@sD ê. solDL, 8s, 0, 40<, AxesLabel → 8"s", "r@sD"<, PlotRange → AllD;

10 20 30 40s

1.31.41.51.61.71.81.9

r@sD

Under (9), the profit for the farmer producing rice is 1.84848, as shown in what follows when k =4. This profit is lowerthan the one when k =1.

In[125]:=N@

pi0kA = pi0 ê. a1 → 1 ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê. sol4A ê.l0 → 10 ê. k → 4D

Out[125]=1.84848

Under (9), the profit for the seedling provider is 1.09086, as shown in what follows when k =4. This profit is lower thanthe one when k2 =1.

water crisisB.nb 13

In[126]:=N@pi01 ê. a1 → 1 ê4 ê. a2 → 1ê4 ê. b1 → 1ê2 ê. b2 → 1ê2 ê. c1 → 1ê3 ê. w → 1 ê. sol4A ê.

l0 → 10 ê. k → 4DOut[126]=

1.09086

Thus, under (9), when k rises, say from 1 to 4, the profits for the firms fall, while the household's utility level rises.This result is different from the previous two cases.

4. Accumulation of Innovation-Embodying Capital Good: Water

Another aspect of innovation can be examined by the Nash-type non-cooperative game. Any innovation needs capitalgoods to apply it for production. In the case of "green revolution" the capital good is water. It is one of the shortcom-mings of "green revolution" that it requires a large amount of water. In other words, the water is like machine in theclothe manufacturing. Also, unlike the western countries, the water is not provided collectively in India: Indian farmersdig wells individually for use in raising rice. In this section, it is assumed that the capital good, the level of watersupply, is adjusted sporadically, since it requires huge amount of money. Y1 in section 1 is regaded as the water supply.Suppose that 2 farmers raised rice utilizing water, Y1 , and labor, L, with the same production function, q=f(Y1 , L). Forsimplicity, the seedling firm is omitted from the formulation in this section. Starting from the situation, in which thetwo firms possess the same level of water supply, suppose that one of the firms expands the water supply, utilizing itsown investment fund. We examine what would happen to the both firms'profits. First, the profits for the both firms arecomputed when Y1 =1 for each firm.

ü 4.1. Cobb-Douglas Production Function Case When Y1 =1 for the Both Firms

Suppose that the production function is of the Cobb-Douglas type and Y1 =1. The profits for the both firms arecomputed as in what follows. First of all, the GE price of the commodity market is computed from the commoditymarket equilibrium.

In[127]:=y = 1; l0 = 10;q = y^H1 ê4L∗ l^H1ê4L;pi = p ∗ q − w ∗ l;sol1 = Solve@D@pi, lD 0, lD@@1DD;ld = l ê. sol1;Simplify@qs = q ê. sol1D;Simplify@pi0 = pi ê. sol1D;u = x^H1 ê2L ∗ le^H1ê 2L;sol2 = Solve@8D@u, xDêD@u, leD pêw, p ∗ x == w ∗Hl0 − leL + 2 ∗ pi0<, 8x, le<D@@1DD;xd = x ê. sol2; ls = l0 − le ê. sol2;PowerExpand@8xd 2 qs, ls 2 ld<D;sol3 = Solve@8xd 2 qs ê. w → 1<, 8p<D@@1DD

Out[138]=8p → 4< For the purpose of checking the GE solution, the GE price of the commodity market is computed from the labormarket equilibrium. We may conclude that the GE rice price is 4.

water crisisB.nb 14

In[139]:=Solve@8ls 2 ld ê. w → 1<, 8p<D@@1DD

Out[139]=8p → 4<Under this assumtion, the profits for the both firms are computed as 3.

In[140]:=pi0e1 = pi0 ê. sol3 ê. w → 1

Out[140]=3

ü 4.2. CES Production Function with t=-1/2 Case When Y1 =1 for the Both Firms

Suppose that the production function is of the CES type with t=-1/2 and Y1 =1. The profits for the both firms arecomputed as in what follows. First of all, the GE price of the commodity market is computed from the commoditymarket equilibrium.

In[141]:=y = 1; l0 = 10;m = 1ê2; t = −1ê 2;q = Hy^H−tL + l^H−tLL^H−m ê tL;pi = p ∗ q − w ∗ l;sol1 = Solve@D@pi, lD 0, lD@@1DD;ld = l ê. sol1;PowerExpand@Simplify@qs = q ê. sol1DD;Simplify@PowerExpand@pi0 = pi ê. sol1DD;u = x^H1 ê2L ∗ le^H1ê 2L;sol2 = PowerExpand@

Solve@8D@u, xDêD@u, leD pêw, p ∗ x == w ∗Hl0 − leL + 2 ∗ pi0<, 8x, le<DD@@1DD;xd = x ê. sol2; ls = l0 − le ê. sol2;PowerExpand@8xd 2 qs, ls 2 ld<D ê. w → 1;sol3 = Solve@8xd 2 qs ê. w → 1<, 8p<D@@1DD

Out[153]=8p → 2< For the purpose of checking the GE solution, the GE price of the commodity market is computed from the labormarket equilibrium. We may conclude that the GE rice price is 2.


Out[154]=8p → 2<Under this assumtion, the profits for the both firms are computed as 3.

In[155]:=pi0e1 = pi0 ê. sol3 ê. w → 1

Out[155]=3

water crisisB.nb 15

ü 4.3. CES Production Function with t=1/2 Case When Y1 =1 for the Both Firms

Suppose that the production function is of the CES type with t=1/2 and Y1 =1. The profits for the both firms arecomputed as in what follows. First of all, the GE price of the commodity market is computed from the commoditymarket equilibrium.

In[156]:=Clear@y, m, t, q, piD;y = 1; l0 = 10;m = 1ê2; t = 1 ê2;q = Hy^H−tL + l^H−tLL^H−m ê tL;pi = p ∗ q − w ∗ l;sol1 = Simplify@PowerExpand@Solve@D@pi, lD 0, lD@@1DDDD;ld = l ê. sol1;PowerExpand@Simplify@qs = q ê. sol1DD;Simplify@PowerExpand@pi0 = pi ê. sol1DD;u = x^H1 ê2L ∗ le^H1ê 2L;sol2 = PowerExpand@

Solve@8D@u, xDêD@u, leD pêw, p ∗ x == w ∗Hl0 − leL + 2 ∗ pi0<, 8x, le<DD@@1DD;xd = x ê. sol2; ls = l0 − le ê. sol2;f1 = Simplify@PowerExpand@8xd 2 qs ê. w → 1<DD;sol3 = FindRoot@f1, 8p, 1<D

Out[169]=8p → 8.< For the purpose of checking the GE solution, it is ascertained that the GE price of the commodity market satisfies thelabor market equilibrium. We may conclude that the GE rice price is 8.

In[170]:=N@ls − 2 ld ê. 8w → 1, p → 8<D

Out[170]=0.

Under this assumtion, the profits for the both firms are computed as 3.

In[171]:=pi0e1 = pi0 ê. sol3 ê. w → 1

Out[171]=3.

Next, we examine what would happen if one of the firms expand its capital to 10, while the another firm does notfollows suit. The profits for the both firms are computed when Y1 =1 for one firm and and Y1 =10 for the another firm.

ü 4.4. Cobb-Douglas Production Function Case When Y1 =1 for One Firm and Y1 =10 for the Another Firm

Suppose that the production function is of the Cobb-Douglas type and Y1 =1 for one firm and and Y1 =10 for theanother firm.. The profits for the both firms are computed as in what follows. First of all, the GE price of the commod-ity market is computed from the commodity market equilibrium.

water crisisB.nb 16

In[172]:=y1 = 1; y2 = 10; l0 = 10; w = 1;q1 = y1^H1ê 4L ∗ l^H1 ê4L;pi1 = p ∗ q1 − w ∗ l;sol11 = Solve@D@pi1, lD 0, lD@@1DD;ld1 = l ê. sol11;Simplify@qs1 = q1 ê. sol11D;Simplify@pi01 = pi1 ê. sol11D;q2 = y2^H1ê 4L ∗ l^H1 ê4L;pi2 = p ∗ q2 − w ∗ l;sol12 = Solve@D@pi2, lD 0, lD@@1DD;ld2 = l ê. sol12;Simplify@qs2 = q2 ê. sol12D;Simplify@pi02 = pi2 ê. sol12D;u = x^H1 ê2L ∗ le^H1ê 2L;sol2 = PowerExpand@

Solve@8D@u, xDêD@u, leD p êw, p ∗ x == w ∗Hl0 − leL + pi01 + pi02<, 8x, le<D@@1DDD;xd = x ê. sol2; ls = l0 − le ê. sol2;PowerExpand@8xd qs1 + qs2, ls ld1 + ld2<D;sol3 = FindRoot@8xd qs1 + qs2<, 8p, 1<D

Out[189]=8p → 2.84211< For the purpose of checking the GE solution, the GE price of the commodity market is computed from the labormarket equilibrium. We may conclude that the GE rice price is 2.84211.

In[190]:=FindRoot@8ls ld1 + ld2<, 8p, 1<D

Out[190]=8p → 2.84211<Under this assumtion, the profit for the firm with Y1 =1 is computed as 1.90208.

In[191]:=pi0e1 = pi01 ê. sol3

Out[191]=1.90208

Meanwhile, the profit for the firm with Y1 =10 is computed as 4.09792.

In[192]:=pi0e1 = pi02 ê. sol3

Out[192]=4.09792

ü 4.5. CES Production Function with t=-1/2 Case When Y1 =1 for One Firm and Y1 =10 for the Another Firm

Suppose that the production function is of the CES type with t=-1/2, and Y1 =1 for one firm and and Y1 =10 for theanother firm.. The profits for the both firms are computed as in what follows. First of all, the GE price of the commod-ity market is computed from the commodity market equilibrium.

water crisisB.nb 17

In[193]:=Clear@yD;y1 = 1; y2 = 10; l0 = 10; w = 1;m = 1ê2; t = −1ê 2;q = Hy^H−tL + l^H−tLL^H−m ê tL;pi = p ∗ q − w ∗ l;sol1 = Solve@D@pi, lD 0, lD@@1DD;ld = l ê. sol1;PowerExpand@Simplify@qs = q ê. sol1DD;Simplify@PowerExpand@pi0 = pi ê. sol1DD;ld1 = l ê. sol1 ê. y → y1;qs1 = q ê. sol1 ê. y → y1;Simplify@pi01 = pi ê. sol1 ê. y → y1D;ld2 = l ê. sol1;Simplify@qs2 = q ê. sol1 ê. y → y2D;Simplify@pi02 = pi ê. sol1 ê. y → y2D;u = x^H1 ê2L ∗ le^H1ê 2L;sol2 = PowerExpand@

Solve@8D@u, xDêD@u, leD p êw, p ∗ x == w ∗Hl0 − leL + pi01 + pi02<, 8x, le<D@@1DDD;xd = x ê. sol2; ls = l0 − le ê. sol2;PowerExpand@8xd qs1 + qs2, ls ld1 + ld2<D;sol3 = Solve@8xd qs1 + qs2<, 8p<D@@1DD

Out[212]=9p →13 J−1 − è!!!!!!10 +

"########################71 + 2 è!!!!!!10 N=In[213]:=

N@%DOut[213]=8p → 1.54372< For the purpose of checking the GE solution, the GE price of the commodity market is computed from the labormarket equilibrium. We may conclude that the GE rice price is 1.54372.

In[214]:=Simplify@Hls − ld1 − ld2L ê. sol3D

Out[214]=0

Under this assumtion, the profit for the firm with Y1 =1 is computed as 2.13949.

In[215]:=Simplify@pi0e1 = pi01 ê. sol3D

Out[215]=1

18 J35 − 4 è!!!!!!10 + 5 "########################71 + 2 è!!!!!!10 −"##############################710 + 20 è!!!!!!10 N

In[216]:=N@%D

Out[216]=2.13949

Meanwhile, the profit for the firm with Y1 =10 is computed as 5.47744.

water crisisB.nb 18

In[217]:=N@pi0e1 = pi02 ê. sol3D

Out[217]=5.47744

ü 4.6. CES Production Function with t=1/2 Case When Y1 =1 for One Firm and Y1 =10 for the Another Firm

Suppose that the production function is of the CES type with t=1/2, and Y1 =1 for one firm and and Y1 =10 for theanother firm.. The profits for the both firms are computed as in what follows. First of all, the GE price of the commod-ity market is computed from the commodity market equilibrium.

In[218]:=y1 = 1; y2 = 10; l0 = 10; w = 1;m = 1ê2; t = 1 ê2;q = Hy^H−tL + l^H−tLL^H−m ê tL;pi = p ∗ q − w ∗ l;sol1 = Solve@D@pi, lD 0, lD@@1DD;ld = l ê. sol1;PowerExpand@Simplify@qs = q ê. sol1DD;Simplify@PowerExpand@pi0 = pi ê. sol1DD;ld1 = Simplify@l ê. sol1 ê. y → y1D;qs1 = Simplify@q ê. sol1 ê. y → y1D;pi01 = PowerExpand@Simplify@pi ê. sol1 ê. y → y1DD;ld2 = PowerExpand@Simplify@l ê. sol1 ê. y → y2DD;qs2 = Simplify@q ê. sol1 ê. y → y2D;pi02 = Simplify@pi ê. sol1 ê. y → y2D;u = x^H1 ê2L ∗ le^H1ê 2L;sol2 = Simplify@PowerExpand@

Solve@8D@u, xDê D@u, leD pêw, p ∗ x == w ∗Hl0 − leL + pi01 + pi02<, 8x, le<DD@@1DDD;xd = x ê. sol2; ls = l0 − le ê. sol2;sol3 = FindRoot@8xd qs1 + qs2<, 8p, 1<D


In[236]:=Simplify@Hls − ld1 − ld2L ê. sol3D

Out[236]=0.


In[237]:=Simplify@pi0e1 = pi01 ê. sol3D

Out[237]=1.77001


water crisisB.nb 19

In[238]:=N@pi0e1 = pi02 ê. sol3D

Out[238]=3.30651

As derived so far, if one of the firms expand its capital to 10, while the another firm does not follows suit, then theprofit for firm with Y1 =1 declines, while the profit for the firm with Y1 =10 rises. The latter firm may well expand itslevel of water supply for the rice production. Suppose that it does follow suit with Y1 =10 for this firm. Now, weexamine what would happen to their profits if both firms expand their capital good to 10.

ü 4.7. Cobb-Douglas Production Function Case When Y1 =1 for the Both Firms

Suppose that the production function is of the Cobb-Douglas type and Y1 =10. The profits for the both firms arecomputed as in what follows. First of all, the GE price of the commodity market is computed from the commoditymarket equilibrium.

In[239]:=w = 1;Clear@yD; y = 10; l0 = 10;q = y^H1 ê4L∗ l^H1ê4L;pi = p ∗ q − w ∗ l;sol1 = Solve@D@pi, lD 0, lD@@1DD;ld = l ê. sol1;Simplify@qs = q ê. sol1D;PowerExpand@pi0 = pi ê. sol1D;u = x^H1 ê2L ∗ le^H1ê 2L;sol2 = Solve@8D@u, xDêD@u, leD pêw, p ∗ x == w ∗Hl0 − leL + 2 ∗ pi0<, 8x, le<D@@1DD;PowerExpand@xd = x ê. sol2; ls = l0 − le ê. sol2D;PowerExpand@8xd 2 qs, ls 2 ld<D;sol3 = Solve@8xd 2 qs ê. w → 1<, 8p<D@@1DD

Out[251]=9p →2 23ê451ê4 =

In[252]:=N@%D



Out[253]=9p →2 23ê451ê4 =

Under this assumtion, the profits for the both firms are computed as 3, exactly the same as in the case for Y1 =1.

water crisisB.nb 20

In[254]:=pi0e2 = pi0 ê. sol3 ê. w → 1

Out[254]=3

ü 4.8. CES Production Function with t=-1/2 Case When Y1 =10 for the Both Firms


In[255]:=y = 10; l0 = 10;m = 1ê2; t = −1ê 2;q = Hy^H−tL + l^H−tLL^H−m ê tL;pi = p ∗ q − w ∗ l;sol1 = Solve@D@pi, lD 0, lD@@1DD;ld = l ê. sol1;PowerExpand@Simplify@qs = q ê. sol1DD;Simplify@PowerExpand@pi0 = pi ê. sol1DD;u = x^H1 ê2L ∗ le^H1ê 2L;sol2 = PowerExpand@

Solve@8D@u, xDêD@u, leD pêw, p ∗ x == w ∗Hl0 − leL + 2 ∗ pi0<, 8x, le<DD@@1DD;xd = x ê. sol2; ls = l0 − le ê. sol2;PowerExpand@8xd 2 qs, ls 2 ld<D ê. w → 1;sol3 = Solve@8xd 2 qs ê. w → 1<, 8p<D@@1DD

Out[267]=9p →23 I5 − è!!!!!!10 M=

In[268]:=N@%D



Out[269]=9p →23 I5 − è!!!!!!10 M=

Under this assumtion, the profits for the both firms are computed as 4.24951, greater than the ones in the case forY1 =1.

In[270]:=N@pi0e1 = pi0 ê. sol3 ê. w → 1D

Out[270]=4.24951

water crisisB.nb 21

ü 4.9. CES Production Function: t=1/2 Case When Y1 =10 for the Both Firms


In[271]:=Clear@y, m, t, q, piD;y = 10; l0 = 10;m = 1ê2; t = 1 ê2;q = Hy^H−tL + l^H−tLL^H−m ê tL;pi = p ∗ q − w ∗ l;sol1 = Simplify@PowerExpand@Solve@D@pi, lD 0, lD@@1DDDD;ld = l ê. sol1;PowerExpand@Simplify@qs = q ê. sol1DD;Simplify@PowerExpand@pi0 = pi ê. sol1DD;u = x^H1 ê2L ∗ le^H1ê 2L;sol2 = PowerExpand@

Solve@8D@u, xDêD@u, leD pêw, p ∗ x == w ∗Hl0 − leL + 2 ∗ pi0<, 8x, le<DD@@1DD;xd = x ê. sol2; ls = l0 − le ê. sol2;f1 = Simplify@PowerExpand@8xd 2 qs ê. w → 1<DD;sol3 = FindRoot@f1, 8p, 1<D


In[285]:=ls − 2 ld ê. 8w → 1< ê. sol3

Out[285]=−3.55271 × 10−15

Under this assumtion, the profits for the both firms are computed as 2.32051, smaller than the ones in the case forY1 =1.

In[286]:=pi0e1 = pi0 ê. sol3 ê. w → 1

Out[286]=2.32051

So far, after assuming that the both firms have the same level of capital good: Y1 =1, we examined what would happenif one of the firms expand its capital to 10, while the another firm does not follows suit. The result was that the profitfor the firm with Y1 =1 declines, while the profit for the firm with Y1 =10 rises. Then, the latter firm may well expand itslevel of water supply for the rice production. Assuming that it does follow suit with Y1 =10 for this firm, we examinewhat would happen to their profits if both firms expand their capital good to 10. The result depends on the assumptionon the production function. When the production function is of Cobb-Douglas type, the profits for the case, in whichboth firms expand their capital good to 10, are exactly the same as in the case, in which both firms keep their capitalgood to 1. Or, when the production function is of CES type with t=1/2, the profits for the case, in which both firmsexpand their capital good to 10, are smaller than the ones in the case, in which both firms keep their capital good to 1.

water crisisB.nb 22

Is there any possibility, in which both firms return to the original case, in which both firms keep their capital good to 1?From the viewpoint of Nash-type non-cooperative game, there is no possibility of returning to the original case, as isclear from the following pay-off matrix for the Nash-type non-cooperative game.

ü 4.10. Pay-Off Matrix for the Cobb-Douglas Production Function Case

When the first firm is named the player A and the second one, the player B, their strategy sets are the same, {Y1 i =1,Y1 i =10}, {i=A, B}. From the above computation, the pay-off matrix for the Nash-type non-cooperative game is givenby the following matrix.. ikjjjjjjjj A \ B Y1 B = 1

Y1 A = 1 3Y1 A = 10 4.09792

Y1 B = 101.90208

3

y{zzzzzzzz ikjjjjjjjj A \ B Y1 B = 1Y1 A = 1 3

Y1 A = 10 1.90208

Y1 B = 104.09792

3

y{zzzzzzzzPlayer A' s Pay − Off Matrix Player B' s Pay − Off Matrix

The solution to this game is {Y1 A =10, Y1 B =10}.

ü 4.11. Pay-Off Matrix for the CES with t=-1/2 Production Function Case


Y1 A = 1 3Y1 A = 10 5.47744

Y1 B = 102.139494.24951


Y1 A = 10 2.13949

Y1 B = 105.477444.24951


The solution to this game is {Y1 A =10, Y1 B =10}.

ü 4.12. Pay-Off Matrix for the CES with t=1/2 Production Function Case


Y1 A = 1 3Y1 A = 10 3.30651

Y1 B = 101.770012.32051


Y1 A = 10 1.77001

Y1 B = 103.306512.32051


The solution to this game is {Y1 A =10, Y1 B =10}, which is the "Prisionners' Dilemma".

water crisisB.nb 23

5. Conclusions

In this paper, the examination of innovation was attepted. Traditionally, the effect of innovation has been examinedon the economic growth, assuming the constant returns to scale on the production function. This assumption produceszero profit for the firms. In this paper, the focus was placed on the relation between the innovation and profits for thefirms, by assuming the diminishing returns to scale. Taking "green revolution" as an example of innovation, it wasshown by the simulation approach that whether the profit increases after the innovation depends on the assumption onthe production function. If we assume that the production function is of the CES type with t>0, the final profitdecreases in spite of the innovation. Next, the relation between the accumulation of innovation-embodying capital and the profit was examined. Whenone of the firms expand its capital good, while the another firm does not follows suit, the profit of the former increasesindependently of the assumption on the production function. Since the profit of the latter firm decreases, independentlyof the assumption on the production function, this firm may well follow suit. When the production function is of theCES type with t>0, the final profit decreases for both firms, compared with the original situation. There is no incentiveto return to the original situation, since the solution to the Nash game is that both firms expand their investment, inspite of the reduced profit. Thus, in India, where farmers possess their wells individually for supplying water to theirrice fields, farmers expand their investment on the wells. The resource problem, the shortage of water, emerges inIndia, where some of the farmers committed suicide, disappointed by the dried wells through water shortage (NHKSpecial,『ウォーター・クライシス－水は誰のものか』2,「涸れ果てる大地」, August 27, 2005). The analysis in this paper applies to the"over investment" in semi-conductor industry. After the US-Japan tradefriction on the semi-conductor, the Japanese engineers moved to Korea, and Korea expanded investment on R&D,expanding capacity investment (NHK Special『日本の群像』8, 「トップを奪い返せ」, 2005). The competitionbetween Japan, Korea, and other counties, led to the predicament in the industry. The tendency of over-investment canbe explained by the above game-theoretic analysis.

In[287]:=TimeUsed@D

Out[287]=122.036

References

Time, "Seeds of Hope", September 25, 2006, pp.42-3.Time, "Asia's Great Science Experiment", October 30, 2006, pp.46-54.

water crisisB.nb 24

Documents

The Green Revolution and the Water Crisis in India: An ...home.hiroshima-u.ac.jp/fukito/water crisisB.pdf1. Basic Model Rice is produced by the two identical competitive firm, aiming