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Information and Communication Technologies and the Income

Distribution: A General Equilibrium Simulation

Toshitaka Fukiharu (Faculty of Economics, Hiroshima University)

November, 2007

In[1]:= Off@General::spell, General::spell1, Solve::svars, Solve::ifun, N:: meprec D

ICT.nb 1

Introduction

In the traditional economics, the innovation was characterilized by the modification in the production process: e.g.the shifts in the production functions. This paper focusses on the innovation in information and communication technol-ogies (ICTs). The special feature of the innovation in the ICTs implies the improvement in the quality of consumptionof goods and services including leisure hours, as well as the modification in the production process. As an example ofthe analysis on the traditional innovation, we may refer to Fukiharu [2007], which examined the relationship betweenthe Green Revolution (GR) , one of the innovations, and the profit. It is not certain whether the farmers' profit rises dueto the GR, since it raises supply of grain, while reducing production cost. In Fukiharu [2007] it was shown that whenthe production function is of CES (Constant Elasticity of Substitution) Type, the farmers' profit might fall with theassumption of positive substitution. It has been argued that the innovation of ITCs expands the unfairness of income distribution. In the traditionalexpression, this may be refrased that the wealthy capitalists class becomes relatively better off, while the poor workingclass becomes relatively worse off. In this paper, assuming three social classes; the entrepreneurs, the capitalists, andthe workers, how the relative shares of these classes in the national income changes due to the innovation of ITCs.Following the Classicals' framework, the capitalists have capital goods and do not work, while the workers have nocapital goods and earn income by providing his initial endowment of leisure for the other members. If the productionfunction is assumed to be under constant returns to scale, the payment of rent for the capital goods and the one of thewage for the labor supply occupies the whole revenue of products. In this paper, however, the production function isassumed to be under decreasing returns to scale, following Fukiharu [2007]. This assumption guarantees the positiveprofit: surplus. Thus, in this paper, the entrepreneurs class exists. By computing the General Equilibrium (GE) prices,the comparative statics analysis is conducted. First, examining three cases on the assumption on the production func-tion, how the shares of three social classes change due to the innovation of the ITCs is analyzed. Second, the utilitychange of three social classes due to the innovation of the ITCs is analyzed. We start with the Cobb-Douglas functioncase.

1: Cobb-Douglas Production Function Case I

In this paper, it is assumed that there is only one firm, producing consumption good, q, utilizing the capital goods, K,and the labor input, L.The production function is assumed as in what follows:

q=f(K , L)=Hck KLa1 Hcl LLa2 a1 +a2 <1 (1-1)

where q is output, ck is the level of ICTs with respect to the capital input in the production process, K is the capitalinput, cl is the level of ICTs with respect to the labor input and L is the labor input. It is assumed that the production isconducted under decreasing returns to scale, so that the positive profit is guaranteed. By the profit maximization of thefirm, the capital demand function, Kd , the labor demand function, Ld , the supply function of the consumption good, qs ,and the profit, p0 , are computed.

ü Firm

First, the capital demand function, Kd , is computed as in what follows, where p is the price of the consumption good,w is the wage rate, and r is the rental price of the capital good.

ICT.nb 2

In[2]:= q = Hck ∗ kL^Ha1L∗Hcl ∗ lL^Ha2L;pi = p ∗ q − w ∗ l − r ∗ k;sol1 = Solve@8D@pi, lD 0, D@pi, kD 0<, 8l, k<D@@1DD;Solve@8D@pi, lD 0, D@pi, kD 0<, 8l, k<D;kd = Factor@k ê. sol1D

Out[6]= a1−1

−1+a1+a2 +a2

−1+a1+a2 a2−a2

−1+a1+a2 ck−1− 1−1+a1+a2 +

a2−1+a1+a2 cl−

a2−1+a1+a2 p− 1

−1+a1+a2 r1

−1+a1+a2 −a2

−1+a1+a2 wa2

−1+a1+a2

The labor demand function, Ld , is computed as in what follows.

In[7]:= ld = Simplify@Factor@l ê. sol1DD

Out[7]= a1−a1

−1+a1+a2 a2−1+a1

−1+a1+a2 ck−a1

−1+a1+a2 cl−a2

−1+a1+a2 p−1

−1+a1+a2 ra1

−1+a1+a2 w1−a1

−1+a1+a2

The supply function of consumption good, qs , is computed as in what follows.

In[8]:= qs = Simplify@PowerExpand@Factor@q ê. sol1DDD

Out[8]= a1−a1

−1+a1+a2 a2−a2

−1+a1+a2 ck−a1

−1+a1+a2 cl−a2

−1+a1+a2 p−a1+a2

−1+a1+a2 ra1

−1+a1+a2 wa2

−1+a1+a2

The profit accrues to the firm after paying the rental cost for capital goods and labor cost, and the profit function, p0 , iscomputed as in what follows.

In[9]:= pi0 = Simplify@PowerExpand@Factor@pi ê. sol1DDD

Out[9]= a1−1+a1

−1+a1+a2 a2−1+a2

−1+a1+a2 I−a1a1+a2

−1+a1+a2 a21

−1+a1+a2 + a11

−1+a1+a2 Ia21

−1+a1+a2 − a2a1+a2

−1+a1+a2 MMck−

a1−1+a1+a2 cl−

a2−1+a1+a2 p−

1−1+a1+a2 r

a1−1+a1+a2 w

a2−1+a1+a2

ü Workers

It is assumed that there is the (aggregate) household, who has the initial endowment of leisure hours, L0 =100, with noprofit distribution from the firm. This class aims at utility maximization subject to income constraint:

max u[x, l ]=xc1 b1 l c2 b2

s.t. px=w(L0 -l ) (2)

where u[x, l ] is the utility function, x is the quantity of consumption good, l is the leisure consumption, L0 is thehousehold's initial endowment of leisure hours, p is the commodity price, w is the wage rate. Note that the innovationof the ICTs causes the modification of the utility function. The level of ITCs with respect to the consumption good isexpressed by c1 , while the one with respect to leisure time is expressed by c2 . From the assumption of utility functionas Cobb-Douglas type in (2) the demand function of consumption good, xdW , is computed as in what follows.

In[10]:= l0 = 100;u = x^Hc1 ∗ b1L∗ le^Hc2 ∗ b2L;sol2 =

PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x w ∗Hl0 − leL<, 8x, le<D@@1DDD;xdW = Simplify@x ê. sol2D

Out[13]=100 b1 c1 w

b1 c1 p + b2 c2 p

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

ICT.nb 3

In[14]:= ls = Simplify@l0 − le ê. sol2D

Out[14]=100 b1 c1

b1 c1 + b2 c2

ü Entrepreneurs

It is assumed that the firm is owned by the (agregate) entrepreneur, who receives the whole profit. This class aims atthe utility maximization subject to the income receipt, p0 , consuming the consumption good, x, and hiring a part of theworkers, L, for this class, without working itself outside the production process.

max u[x, L]=xc1 b1 L c2 b2

s.t. px+wL=p0 (3)

Note that the innovation of the ICTs causes the modification of the utility function in exactly the same way as in thecase of household. The level of ITCs with respect to the consumption good is expressed by c1 , while the one withrespect to leisure time is expressed by c2 . From the assumption of utility function as Cobb-Douglas type in (3), theentrepreneur's demand function of consumption good, xdE , is computed as in what follows.

In[15]:= u = x^Hc1 ∗ b1L∗ le^Hc2 ∗ b2L;sol2E =

PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x + w ∗ le pi0<, 8x, le<D@@1DDD;xdE = Simplify@x ê. sol2ED

Out[17]=1

b1 c1 + b2 c2Ia1−

1+a1−1+a1+a2 a2−

1+a2−1+a1+a2 I−a1

a1+a2−1+a1+a2 a2

1−1+a1+a2 + a1

1−1+a1+a2 Ia2

1−1+a1+a2 − a2

a1+a2−1+a1+a2 MM

b1 c1 ck−a1

−1+a1+a2 cl−a2

−1+a1+a2 p− a1+a2−1+a1+a2 r

a1−1+a1+a2 w

a2−1+a1+a2 M

The entrepreneur's demand function for labor, LdE , is computed as in what follows.

In[18]:= ldE = Expand@Factor@Simplify@le ê. sol2EDDD

Out[18]=1

b1 c1 + b2 c2

Ia1−a1

−1+a1+a2 a2−a2

−1+a1+a2 b2 c2 ck−a1

−1+a1+a2 cl−a2

−1+a1+a2 p−1

−1+a1+a2 ra1

−1+a1+a2 w1

−1+a1+a2 −a1

−1+a1+a2 M − 1b1 c1 + b2 c2

Ia1−1

−1+a1+a2 +a2

−1+a1+a2 a2−a2

−1+a1+a2 b2 c2 ck−a1

−1+a1+a2 cl−a2

−1+a1+a2 p−1

−1+a1+a2 ra1

−1+a1+a2 w1

−1+a1+a2 −a1

−1+a1+a2 M −1

b1 c1 + b2 c2

Ia1−a1

−1+a1+a2 a2−1

−1+a1+a2 +a1

−1+a1+a2 b2 c2 ck−a1

−1+a1+a2 cl−a2

−1+a1+a2 p−1

−1+a1+a2 ra1

−1+a1+a2 w1

−1+a1+a2 −a1

−1+a1+a2 M

ICT.nb 4

ü Capitalists

It is assumed that there is the (aggregate) capitalist, who has the initial endowment of capital good, K0 =100, with noprofit distribution from the firm, consuming the consumption good, x, and hiring a part of the workers, L, for this classwithout working itself outside the production process. This class aims at the utility maximization subject to incomeconstraint:

max u[x, L]=xc1 b1 L c2 b2

s.t. px+wL=rK0 (4)

Note that the innovation of the ICTs causes the modification of the utility function in exactly the same way as in thecase of household. The level of ITCs with respect to the consumption good is expressed by c1 , while the one withrespect to leisure time is expressed by c2 . From the assumption of utility function as Cobb-Douglas type in (4) thecapitalist's demand function of consumption good, xdK , is computed as in what follows.

In[19]:= k0 = 100;u = x^Hc1 ∗ b1L∗ le^Hc2 ∗ b2L;sol2K =

PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x + w ∗ le r ∗ k0<, 8x, le<D@@1DDD;xdK = Simplify@x ê. sol2KD

Out[22]=100 b1 c1 r

b1 c1 p + b2 c2 p

The capitalist's demand function for labor, LdK , is computed as in what follows.

In[23]:= ldK = Simplify@le ê. sol2KD

Out[23]=100 b2 c2 r

b1 c1 w + b2 c2 w

ü General Equilibrium

The general equilibrium (GE) prices, p, r, and w are computed from the three equations, which stem from the threemarket equilibria.

Kd =K0 (capital market equilibrium) (5) xd =xdW +xdE +xdK =qs (commodity market equilibrium) (6) Ld +LdE +LdK =Ls (labor market equilibrium) (7)

Assuming that w=1, we compute GE prices, p* and r*. From (5) and (6), p* is computed as in what follows.

ICT.nb 5

In[24]:= f1 := kd k0;f2 := Hxd = Simplify@xdK + xdE + xdW ê. k → kdDL qs;f3 := Simplify@ld + ldK + ldE ê. k → kdD ls;f11 = f1 ê. w → 1;sol3 = Simplify@PowerExpand@Solve@f11, rD@@1DDDD;Simplify@PowerExpand@Factor@Simplify@PowerExpand@f2 ê. w → 1DD ê. sol3DDD;

f12 = 100a1

1−a2 a21+H−2+a1L a2+a22H−1+a2L H−1+a1+a2L b1 c1 +

100a1

1−a2 a2a1

H−1+a2L H−1+a1+a2L b2 c2 − 100 a2H−1+a2L a2+a1 H1+a2LH−1+a2L H−1+a1+a2L b1 c1 ck

a1−1+a2 cl

a2−1+a2 p−1+ a2

−1+a2 ;sol3A = PowerExpand@Simplify@PowerExpand@Solve@f12 0, pD@@1DDDDD

Out[31]= 9p → 1001+ a1 H−1+a2L1−a2 −a2 a2−

H−1+a2L a2+a1 H1+a2L−1+a1+a2 b11−a2

c11−a2 Ja21+H−2+a1L a2+a22H−1+a2L H−1+a1+a2L b1 c1 + a2

a1H−1+a2L H−1+a1+a2L b2 c2N

−1+a2

ck−a1 cl−a2=

When a1 =a2 = b1 =b2 =1/3, for example, p* is expressed as in what follows. The innovation in ICTs reduces commod-ity prices.

In[32]:= sol3p = sol3A ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3<

Out[32]= 9p →102ê3 c12ê3

3 31ê3 I c127 è!!!!3

+ c29 è!!!!3

M2ê3ck1ê3 cl1ê3

=

Substituting p* into (5), r* is computed as in what follows. Note that while the innovation in ICTs reduces rental priceof capital goods, the enhanced efficiency in the production function has no effect in this reduction. Only the enhancedefficiency in the utility function has effect in this reduction.

In[33]:= sol3r = Simplify@PowerExpand@Simplify@Factor@sol3 ê. sol3ADDDD

Out[33]= 9r →a1 a2

a1H−1+a2L H−1+a1+a2L b1 c1

a21+H−2+a1L a2+a22H−1+a2L H−1+a1+a2L b1 c1 + a2

a1H−1+a2L H−1+a1+a2L b2 c2

=

When a1 =a2 = b1 =b2 =1/3, for example, r* is expressed as in what follows.

In[34]:= sol3r ê. 8a1 → 1ê 3, a2 −> 1 ê3, b1 → 1ê3, b2 → 1ê3<

Out[34]= 9r →c1

27 è!!!3 I c127 è!!!!3

+ c29 è!!!!3

M=

It is ascertained that {p*,r*} satisfies (7): the labor market equilibrium.

In[35]:= Simplify@PowerExpand@Factor@Simplify@Hld + ldK + ldE ê. k → kdL − ls ê. sol3A ê. sol3r ê. w → 1 ê.

8a1 → 1ê 3, a2 −> 1ê 3, b1 → 1ê3, b2 → 1ê3<DDDD

Out[35]= 0

ü Income Distribution

Under {p*,r*,w*} the total income for the workers class, w*Ls , the total income for the entrepreneus class, p0 , the onefor the capitalists class, r*K0 , are determined. The shares of them, {w*Ls /p*qs , p0 /p*qs ,r*K0 /p*qs }, are computed asin what follows.

ICT.nb 6

In[36]:= Simplify@Factor@Simplify@Factor@H8sW = w ∗ ld, sE = pi0, sK = r ∗ kd<êHp ∗ qsLL ê. sol3A ê. sol3r ê. w → 1DDDD

Out[36]= 8a2, 1 − a1 − a2, a1<

It was shown that

w*Ls /p*qs = a2 , (8-1) p0 /p*qs = 1-a1 -a2 , (9-1) r*K0 /p*qs =a1 . (10-1) Thus, the innovation in ICT causes no effect on the income distribution.

ü Utility Change

For each social class, the utility change due to the innovation in ITCs is examined. Suppose that

a1 =a2 = b1 =b2 =1/3 (5-1)

When ck =cl =c1 =c2 =1, the utility level of workers is computed as follows.

In[37]:= N@uL0 = Simplify@PowerExpand@u ê. sol2 ê. sol3A ê. w → 1DD ê.8a1 → 1ê3, a2 −> 1ê 3, b1 → 1ê3, b2 → 1ê3< ê. 8ck → 1, cl → 1, c1 → 1, c2 → 1<D

Out[37]= 7.67671

The utility level of workers class for c1 =c2 =1, and arbitrary ck and cl is computed respectively as in what follows. Itshows that the utility level of workers rises due to the innovation in ITCs.

In[38]:= N@uL1A = Simplify@PowerExpand@u ê. sol2 ê. sol3A ê. w → 1DD ê.8a1 → 1 ê3, a2 −> 1 ê3, b1 → 1ê3, b2 → 1ê3, c1 → 1, c2 → 1<D

Out[38]= 7.67671 ck1ê9 cl1ê9

The utility level of workers class for ck =cl = c1 =1, and c2 =5/4 , and the one for ck =cl = c2 =1, and c1 =5/4 are com-puted respectively as in what follows. It shows that the utility level of workers rises due to the innovation in ITCs.

In[39]:= 8N@uL1A = u ê. sol2 ê. sol3A ê. w → 1 ê.8a1 → 1ê3, a2 −> 1ê 3, b1 → 1ê3, b2 → 1ê3, ck → 1, cl → 1, c1 → 1, c2 → 5ê4<D,

N@uL1A = u ê. sol2 ê. sol3A ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3,b1 → 1ê3, b2 → 1 ê3, ck → 1, cl → 1, c1 → 5ê4, c2 → 1<D<

Out[39]= 811.1008, 8.85721<

Under (5), the utility level of the entrepreneurs class for c1 =c2 =1, and arbitrary ck and cl is computed respectively asin what follows. It shows that the utility level of entrepreneurs rises due to the innovation in ITCs.

In[40]:= N@PowerExpand@u ê. sol2E ê. sol3p ê. sol3r ê. w → 1 ê.8a1 → 1 ê3, a2 −> 1 ê3, b1 → 1ê3, b2 → 1ê3, c1 → 1, c2 → 1<DD

Out[40]= 3.0465 ck1ê9 cl1ê9

ICT.nb 7

Under (5), the utility level of the entrepreneurs class for ck =cl = c1 =1, and c2 =5/4 , and the one for ck =cl = c2 =1, andc1 =5/4 are computed respectively as in what follows. It shows that the utility level of entrepreneurs rises due to theinnovation in ITCs.

In[41]:= N@8u ê. sol2E ê. sol3p ê. sol3r ê. w → 1 ê.8a1 → 1 ê3, a2 −> 1 ê3, b1 → 1ê3, b2 → 1ê3, ck → 1, cl → 1, c1 → 1, c2 → 5ê4<,

u ê. sol2E ê. sol3p ê. sol3r ê. w → 1 ê.8a1 → 1 ê3, a2 −> 1 ê3, b1 → 1ê3, b2 → 1ê3, ck → 1, cl → 1, c1 → 5ê4, c2 → 1<<D

Out[41]= 83.45013, 3.53743<

Note, however, the profit declines when c2 rises, while it rises when c1 rises.

In[42]:= pi1 = pi0 ê. sol3p ê. sol3r ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3<

Out[42]=100 c1

27 è!!!3 I c127 è!!!!3

+ c29 è!!!!3

M

In[43]:= Simplify@D@pi1, c1DD

Out[43]=300 c2

Hc1 + 3 c2L2

Under (5), the utility level of the capitalists class for c1 =c2 =1, and arbitrary ck and cl is computed respectively as inwhat follows. It shows that the utility level of capitalists class rises due to the innovation in ITCs. Since p0 =r*K0 under(5), the utility leve for the capitalists is exactly the same as the one for the entrepreneurs.

In[44]:= N@PowerExpand@u ê. sol2K ê. sol3p ê. sol3r ê. w → 1 ê.8a1 → 1 ê3, a2 −> 1 ê3, b1 → 1ê3, b2 → 1ê3, c1 → 1, c2 → 1<DD

Out[44]= 3.0465 ck1ê9 cl1ê9

Under (5), the utility level of the capitalists class for ck =cl = c1 =1, and c2 =5/4 , and the one for ck =cl = c2 =1, andc1 =5/4 are computed respectively as in what follows. It shows that the utility level of capitalists rises due to the innova-tion in ITCs.

In[45]:= N@8u ê. sol2K ê. sol3p ê. sol3r ê. w → 1 ê.8a1 → 1 ê3, a2 −> 1 ê3, b1 → 1ê3, b2 → 1ê3, ck → 1, cl → 1, c1 → 1, c2 → 5ê4<,

u ê. sol2K ê. sol3p ê. sol3r ê. w → 1 ê.8a1 → 1 ê3, a2 −> 1 ê3, b1 → 1ê3, b2 → 1ê3, ck → 1, cl → 1, c1 → 5ê4, c2 → 1<<D

Out[45]= 83.45013, 3.53743<

ICT.nb 8

2: Cobb-Douglas Production Function Case II

In this section, it is assumed that there is only one firm, producing consumption good, q, utilizing the capital goods, K,and the labor input, L.The production function is assumed as in what follows:

q=f(K , L)=Kck a1 Lcl a2 a1 +a2 <1 (1-2)

where q is output, ck is the level of ICTs with respect to the capital input in the production process, K is the capitalinput, cl is the level of ICTs with respect to the labor input and L is the labor input. In (1-2), the innovation may causethe modification of marginal rate of substitution between capital and labor, while it is not the case in (1-1). It isassumed, however, that the production is conducted under decreasing returns to scale, so that the positive profit isguaranteed.

ü Firm

First, the capital demand function, Kd , is computed as in what follows, where p is the price of the consumption good,w is the wage rate, and r is the rental price of the capital good. Then, the labor demand function, Ld , and the supplyfunction of consumption good, qs , are computed. The profit accrues to the firm after paying the rental cost for capitalgoods and labor cost, and finally the profit function, p0 , is computed.

In[46]:= q = Hk^Hck ∗ a1LL∗HHlL^Hcl ∗ a2LL;pi = p ∗ q − w ∗ l − r ∗ k;sol1 = Solve@8D@pi, lD 0, D@pi, kD 0<, 8l, k<D@@1DD;kd = Simplify@Factor@k ê. sol1DD;ld = Simplify@Factor@l ê. sol1DD;qs = Simplify@PowerExpand@Factor@q ê. sol1DDD;pi0 = Simplify@PowerExpand@Factor@pi ê. sol1DDD;

ü Workers

In the same way as in the previous section, it is assumed that there is the (aggregate) household, who has the initialendowment of leisure hours, L0 =100, with no profit distribution from the firm. This class aims at utility maximizationsubject to income constraint, (2). From the assumption of utility function as Cobb-Douglas type in (2) the demandfunction of consumption good, xdW , and the labor supply function, Ls , are computed.

In[53]:= l0 = 100;u = Hx^Hc1 ∗ b1LL∗Hle^Hc2 ∗ b2LL;sol2 =

PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x w ∗Hl0 − leL<, 8x, le<D@@1DDD;xdW = Simplify@x ê. sol2D; ls = Simplify@l0 − le ê. sol2D;

ICT.nb 9

ü Entrepreneurs

In the same way as in the previous section, it is assumed that the firm is owned by the (agregate) entrepreneur, whoreceives the whole profit. This class aims at the utility maximization subject to the income receipt, p0 , consuming theconsumption good, x, and hiring a part of the workers, L, for this class, without working itself outside the productionprocess, (3). From the assumption of utility function as Cobb-Douglas type in (3), the entrepreneur's demand functionof consumption good, xdE , and the demand function for labor, LdE , are computed.

In[57]:= u = Hx^Hc1 ∗ b1LL∗Hle^Hc2 ∗ b2LL;sol2E =

PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x + w ∗ le pi0<, 8x, le<D@@1DDD;xdE = Simplify@x ê. sol2ED; ldE = Simplify@le ê. sol2ED;

ü Capitalists

In the same way as in the previous section, it is assumed that there is the (aggregate) capitalist, who has the initialendowment of capital good, K0 =100, with no profit distribution from the firm, consuming the consumption good, x,and hiring a part of the workers, L, for this class without working itself outside the production process. This class aimsat the utility maximization subject to income constraint, (4). From the assumption of utility function as Cobb-Douglastype in (4) the capitalist's demand function of consumption good, xdK , and the demand function for labor, LdK , arecomputed.

In[60]:= k0 = 100;u = Hx^Hc1 ∗ b1LL∗Hle^Hc2 ∗ b2LL;sol2K =

PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x + w ∗ le r ∗ k0<, 8x, le<D@@1DDD;xdK = Simplify@x ê. sol2KD; ldK = Simplify@le ê. sol2KD;


The general equilibrium (GE) prices, p, r, and w are computed from the three equations, which stem from the threemarket equilibria, (5), (6), and (7). Assuming that w=1, we compute GE prices, p* and r*. From (5) and (6), p* iscomputed as in what follows.

ICT.nb 10

In[64]:= f1 := kd k0f2 := Hxd = Simplify@xdK + xdE + xdW ê. k → kdDL qsf3 := Simplify@ld + ldK + ldE ê. k → kdD lsf11 = f1 ê. w → 1;sol3 = Simplify@PowerExpand@Solve@f11, rD@@1DDDD;f12 = Simplify@PowerExpand@

Simplify@Factor@Hxd = Simplify@xdK + xdE + xdW ê. k → kdDL − qs ê. sol3 ê. w → 1DDDD;

f14 = 100 a2a1 ck+a2 cl H1+a2 clL

H−1+a2 clL H−1+a1 ck+a2 clL b1 c1 cla1 ck+a2 cl H1+a2 clL

H−1+a2 clL H−1+a1 ck+a2 clL −

J100a1 ck

1−a2 cl a21+a22 cl2

H−1+a2 clL H−1+a1 ck+a2 clL b1 c1 cl1+a22 cl2

H−1+a2 clL H−1+a1 ck+a2 clL +

100a1 ck

1−a2 cl a22 a2 cl+a1 Hck−a2 ck clL

H−1+a2 clL H−1+a1 ck+a2 clL b2 c2 cl2 a2 cl+a1 Hck−a2 ck clL

H−1+a2 clL H−1+a1 ck+a2 clL N p1− a1 ck+a2 cl−1+a1 ck+a2 cl + a1 ck

H1−a2 clL H−1+a1 ck+a2 clL ;

sol3p = Expand@Factor@PowerExpand@Simplify@Factor@Solve@f14 0, pDDD@@1DDDDD

Out[71]= 9p → 100H1−a2 clL I1+ a1 ck−1+a2 cl M a2

H1−a2 clL Ha1 ck+a2 cl H1+a2 clLLH−1+a2 clL H−1+a1 ck+a2 clL b11−a2 cl c11−a2 cl

clH1−a2 clL Ha1 ck+a2 cl H1+a2 clLL

H−1+a2 clL H−1+a1 ck+a2 clL Ja21+a22 cl2



a22 a2 cl+a1 Hck−a2 ck clL


H−1+a2 clL H−1+a1 ck+a2 clL N−1+a2 cl

=

Substituting p* into (5), r* is computed as in what follows.

In[72]:= sol3r = PowerExpand@Simplify@Factor@sol3 ê. sol3pDDD

Out[72]= 9r → 1001−a1 ck−a2 cl

1−a2 cl − −1+a1 ck+a2 cl−1+a2 cl a1 a2

a2 cl1−a2 cl −

a1 ck+a2 cl H1+a2 clLH1−a2 clL H−1+a1 ck+a2 clL b1 c1 ck

cla2 cl

1−a2 cl −a1 ck+a2 cl H1+a2 clL

H1−a2 clL H−1+a1 ck+a2 clL Ja21+a22 cl2





H−1+a2 clL H−1+a1 ck+a2 clL N−1+a2 cl1−a2 cl =


In[73]:= N@Factor@Simplify@Hld + ldK + ldE ê. k → kdL − ls ê. sol3p ê. sol3r ê. w → 1D ê.8a1 → 1ê3, a2 −> 1ê 3, b1 → 1ê3, b2 → 1ê3<D ê.

8c1 → 12ê10, c2 → 9ê 10, ck → 3ê4, cl → 11ê10<, 20D

Out[73]= 0.×10−68



In[74]:= checkCD =

Simplify@Factor@H8sW = w ∗ ld, sE = pi0, sK = r ∗ kd<êHp ∗ qsLL ê. sol3p ê. sol3r ê. w → 1DD

Out[74]= 9a2 cl, 1 − a1 ck − a2 cl, Ia1 a22 a2 cl+a1 Hck−a2 ck clL

H−1+a2 clL H−1+a1 ck+a2 clL ck cl2 a2 cl+a1 Hck−a2 ck clL

H−1+a2 clL H−1+a1 ck+a2 clL Hb2 c2 + a2 b1 c1 clLM í

Ja21+a22 cl2





H−1+a2 clL H−1+a1 ck+a2 clL N=

It was shown that

ICT.nb 11

w*Ls /p*qs = a2 cl , (8-2) p0 /p*qs = 1-a1 ck -a2 cl , (9-2) r*K0 /p*qs =a1 ck . (10-2)

In the framework of this section, the altered production process causes the income distribution, whereas the modifica-tion of utility function does not.

ü Utility Change

When the innovation of ICTs improves the efficiency in production, it raises the shares in national income, as is clearfrom (8-2) and (10-2). This, however, reduces the share of the profit in national income. In this situation, what happensto each of the social classes. We examine the utility change, starting from the case of workers class. First, it is assumed

a1 =a2 = b1 =b2 =1/3, c1 =c2 = ck =cl =1. (5-2) The utility level of the workers class at GE is computed as in what follows.

In[75]:= N@uL0 = u ê. sol2 ê. sol3p ê. sol3r ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3< ê.8ck → 1, cl → 1, c1 → 1, c2 → 1<D

Out[75]= 7.67671

When ci increases to 5/4 from (5-2), then the utility level of the workers class increases, as shown in what follows (i=1,2, k, l).

In[76]:= uL1 = u ê. sol2 ê. sol3p ê. sol3r ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3<;N@8uL1 ê. 8ck → 5 ê4, cl → 1, c1 → 1, c2 → 1<, uL1 ê. 8ck → 1, cl → 5ê4, c1 → 1, c2 → 1<,

uL1 ê. 8ck → 1, cl → 1, c1 → 5ê4, c2 → 1<, uL1 ê. 8ck → 1, cl → 1, c1 → 1, c2 → 5ê4<<D

Out[76]= 88.7243, 8.76165, 8.85721, 11.1008<

When (5-2) holds, the utility level of the entrepreneurs class at GE is computed as in what follows.

In[77]:= N@uE0 = u ê. sol2E ê. sol3p ê. sol3r ê. w → 1 ê.8a1 → 1ê3, a2 −> 1ê 3, b1 → 1ê3, b2 → 1ê3< ê. 8ck → 1, cl → 1, c1 → 1, c2 → 1<D

Out[77]= 3.0465

When ck or cl increases to 5/4 from (5-2), then the utility level of the entrepreneurs class falls, as shown in whatfollows, although when c1 or c2 increases to 5/4 from (5-2), then the utility level of the entrepreneurs class falls.

In[78]:= uE1 = u ê. sol2E ê. sol3p ê. sol3r ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3<;N@8uE1 ê. 8ck → 5 ê4, cl → 1, c1 → 1, c2 → 1<, uE1 ê. 8ck → 1, cl → 5ê4, c1 → 1, c2 → 1<,

uE1 ê. 8ck → 1, cl → 1, c1 → 5ê4, c2 → 1<, uE1 ê. 8ck → 1, cl → 1, c1 → 1, c2 → 5ê4<<D

Out[78]= 82.85802, 2.75656, 3.53743, 3.45013<

This phenomenon emerges, since the profit falls, as shown in what follows. When (5-2) holds, the profit at GE iscomputed as 25.

ICT.nb 12

In[79]:= pi00 =

Hpi0 ê. sol2E ê. sol3p ê. sol3r ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3< ê.8ck → 1, cl → 1, c1 → 1, c2 → 1<L

Out[79]= 25

When ck increases to 5/4, then the profit of the entrepreneurs class falls to 75/4, as shown in what follows.

In[80]:= pi01 =

Hpi0 ê. sol2E ê. sol3p ê. sol3r ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3< ê.8ck → 5 ê4, cl → 1, c1 → 1, c2 → 1<L

Out[80]=754

Note that in this model, production does not expand due to the innovation of ICTs. When (5-2) holds, the value ofoutput at GE is computed as 75, while when ck increases to 5/4, it is 75 as shown in what follows.

In[81]:= 8re0 = Hp ∗ qsL ê. sol3p ê. sol3r ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3< ê.8ck → 1, cl → 1, c1 → 1, c2 → 1<,

re1 = Hp ∗ qsL ê. sol3p ê. sol3r ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3< ê.8ck → 5 ê4, cl → 1, c1 → 1, c2 → 1<<

Out[81]= 875, 75<

When the innovation of ICTs improves the quality of life, as exhibited by the rise of c1 , say, from 1 to 5/4, the profitrises and the utility of the entrepreneurs rises, as shown in what follows.

In[82]:= N@uE2 = u ê. sol2E ê. sol3p ê. sol3r ê. w → 1 ê.8a1 → 1ê3, a2 −> 1ê 3, b1 → 1ê3, b2 → 1ê3< ê. 8ck → 1, cl → 1, c1 → 5ê4, c2 → 1<D

Out[82]= 3.53743

When ck increases to 5/4, then the utility level of the capitalists class rises, from 3.0465 to 4.01758, as shown in whatfollows.

In[83]:= N@8HuK0 = u ê. sol2K ê. sol3p ê. sol3r ê. w → 1 ê.8a1 → 1 ê3, a2 −> 1 ê3, b1 → 1ê3, b2 → 1ê3< ê. 8ck → 1, cl → 1, c1 → 1, c2 → 1<L,

HuK1 = u ê. sol2K ê. sol3p ê. sol3r ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3,b1 → 1ê 3, b2 → 1ê3<; uK1 ê. 8ck → 5ê4, cl → 1, c1 → 1, c2 → 1<L<D

Out[83]= 83.0465, 4.01758<

Indeed, when ci increases to 5/4 from (5-2), then the utility level of the capitalists class rises, as shown in what follows(i=l, 1, 2}.

In[84]:= N@8HuK1 ê. 8ck → 1, cl → 5 ê4, c1 → 1, c2 → 1<L,HuK1 ê. 8ck → 1, cl → 1, c1 → 5ê4, c2 → 1<L, HuK1 ê. 8ck → 1, cl → 1, c1 → 1, c2 → 5ê4<L<D

Out[84]= 83.33934, 3.53743, 3.45013<

ICT.nb 13

3: CES Case with t=-1/2

In this section, it is assumed that there is only one firm, producing consumption good, q, utilizing the capital goods, K,and the labor input, L.The production function is assumed as in what follows:

q=f(K , L)=HHck KL-t + Hcl LL-tL-nêt n=1/2, t= –1/2 (1-3)

where q is output, ck is the level of ICTs with respect to the capital input in the production process, K is the capitalinput, cl is the level of ICTs with respect to the labor input and L is the labor input. It is assumed that the production isconducted under decreasing returns to scale, so that the positive profit is guaranteed.

ü Firm

First, the labor demand function, Ld , is computed as in what follows, where p is the price of the consumption good,w is the wage rate, and r is the rental price of the capital good. Then, the capital demand function, Kd , and the supplyfunction of consumption good, qs , are computed. The profit accrues to the firm after paying the rental cost for capitalgoods and labor cost, and finally the profit function, p0 , is computed.

In[85]:= Clear@c, wD;m = 1ê2; t = −1 ê2;q = HHck ∗ kL^H−tL + Hcl ∗ lL^H−tLL^H−m êtL;pi = p ∗ q − r ∗ k − w ∗ l;sol1 = Solve@8D@pi, kD 0, D@pi, lD 0<, 8k, l<D@@1DD;ld = l ê. sol1;PowerExpand@Simplify@qs = q ê. sol1DD;kd = k ê. sol1;pi0 = PowerExpand@Simplify@pi ê. sol1DD;

ü Workers

In the same way as in the previous section, it is assumed that there is the (aggregate) household, who has the initialendowment of leisure hours, L0 =100, with no profit distribution from the firm. This class aims at utility maximizationsubject to income constraint, (2). From the assumption of utility function as Cobb-Douglas type in (2) the demandfunction of consumption good, xdW , and the labor supply function, Ls , are computed. In this section (5-1) is assumed:b1 =b2 =1/3.

In[94]:= l0 = 100; b1 = 1ê3; b2 = 1 ê3;

In[95]:= u = x^Hc1 ∗ b1L∗ le^Hc2 ∗ b2L;sol2 =

PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x w ∗Hl0 − leL<, 8x, le<D@@1DDD;xdW = Simplify@x ê. sol2D; ls = Simplify@l0 − le ê. sol2D;

ICT.nb 14

ü Entrepreneurs


In[98]:= u = x^Hc1 ∗ b1L∗ le^Hc2 ∗ b2L;sol2E =

PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x + w ∗ le pi0<, 8x, le<D@@1DDD;xdE = Simplify@x ê. sol2ED; ldE = Simplify@le ê. sol2ED;

ü Capitalists


In[101]:=

k0 = 100;

In[102]:=

u = x^Hc1 ∗ b1L ∗ le^Hc2 ∗ b2L;sol2K =

PowerExpand@Solve@8D@u, xDêD@u, leD pêw, p ∗ x + w ∗ le r ∗ k0<, 8x, le<D@@1DDD;xdK = Simplify@x ê. sol2KD; ldK = Simplify@le ê. sol2KD;


The general equilibrium (GE) prices, p, r, and w are computed from the three equations, which stem from the threemarket equilibria, (5), (6), and (7). Assuming that w=1, we compute GE prices, p* and r*. From (5) and (6), p* iscomputed as in what follows.

In[105]:=

f1 := kd k0f2 := Hxd = Simplify@xdK + xdE + xdW ê. k → kdDL qsf3 := Simplify@ld + ldK + ldE ê. k → kdD lscheck1 = PowerExpand@8f1, f2< ê. w → 1D;sol3 = Solve@check1, 8p, r<D@@4DD;


ICT.nb 15

In[110]:=

Simplify@Hld + ldK + ldE ê. k → kdL − ls ê. sol3 ê. w → 1DOut[110]=

0

Finally, the stability is examined. When c1 =c2 =ck =cl =1, general equilibrium prices are computed as in what follows.

In[111]:=

sol3 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<Out[111]=

9r →13

, p →203

=

The local stability of this general equilibrium is guaranteed, since all of the eigen values of Jacobian matrix are negaiteat the general equilibrium, as shown in what follows.

In[112]:=

a11 = D@kd − k0 ê. w → 1, rD ê. sol3 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<;a12 = D@kd − k0 ê. w → 1, pD ê. sol3 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<;a21 =

D@HxdK + xdE + xdW ê. k → kdL − qs ê. w → 1, rD ê. sol3 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<;a22 = D@HxdK + xdE + xdW ê. k → kdL − qs ê. w → 1, pD ê. sol3 ê.

8c1 → 1, c2 → 1, ck → 1, cl → 1<;jacobi = 88a11, a12<, 8a21, a22<<;N@Eigenvalues@jacobiDD

Out[117]=

8−601.504, −1.49625<

Price trajectories, {r[s], p[s]}, on the Walrasian tatonnement process shows the stability.

In[118]:=

Clear@tDed1 = kd − k0 ê. w → 1 ê. 8r → r@sD, p → p@sD< ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<;ed2 = HxdK + xdE + xdW ê. k → kdL − qs ê. w → 1 ê. 8r → r@sD, p → p@sD< ê.

8c1 → 1, c2 → 1, ck → 1, cl → 1<;traj1 = NDSolve@8D@r@sD, sD ed1, D@p@sD, sD ed2, r@0D 1, p@0D 1<,

8r@sD, p@sD<, 8s, 0, 10<D@@1DD;

In[122]:=

Plot@r@sD ê. traj1, 8s, 0, 3<, AxesLabel → 8"s", "r@sD"<D;

0.5 1 1.5 2 2.5 3s

0.1

0.2

0.3

0.4

0.5

0.6r@sD

ICT.nb 16

In[123]:=

Plot@p@sD ê. traj1, 8s, 0, 3<, AxesLabel → 8"s", "r@sD"<D;

0.5 1 1.5 2 2.5 3s

1

2

3

4

5

6

r@sD



In[124]:=

data1 =

Simplify@PowerExpand@H8sW = w ∗ ld, sE = pi0, sK = r ∗ kd<êHp ∗ qsLL ê. sol3 ê. w → 1DDOut[124]=

9 −c22 ck +è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

c22 ck Hc22 ck + c12 cl + 2 c1 c2 clL2 Ic1 c2 ck + c22 ck +

è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!c22 ck Hc22 ck + c12 cl + 2 c1 c2 clL M

,

12

,c2 Hc1 + 2 c2L ck

2 Ic1 c2 ck + c22 ck +è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

c22 ck Hc22 ck + c12 cl + 2 c1 c2 clL M=

It was shown that p0 /p*qs =1/2. The expressions for w*Ls /p*qs and r*K0 /p*qs , however, are not straightforward.First, the following holds.

w*Ls /p*qs rises as cl rises, (8-3A) r*K0 /p*qs falls as cl rises. (10-3A) w*Ls /p*qs falls as ck rises, (8-3B) r*K0 /p*qs rises as ck rises. (10-3B)

These results obtain, since we have ∑ (w*Ls /p*qs )/ ∑ cl >0 and ∑ (r*K0 /p*qs )/ ∑ ck <0, as shown by the followingcomputation.

In[125]:=

Factor@Simplify@D@data1@@1DD, clDDDOut[125]=

Hc1 c23 Hc1 + 2 c2L2 ck2L ì ikjjjj4 "###############################################################################c22 ck Hc22 ck + c12 cl + 2 c1 c2 clL

Jc1 c2 ck + c22 ck +"###############################################################################c22 ck Hc22 ck + c12 cl + 2 c1 c2 clL N2y{zzzz

ICT.nb 17

In[126]:=

Factor@Simplify@D@data1@@1DD, ckDDDOut[126]=

−Hc1 c23 Hc1 + 2 c2L2 ck clL ì ikjjjj4 "###############################################################################c22 ck Hc22 ck + c12 cl + 2 c1 c2 clL

Jc1 c2 ck + c22 ck +"###############################################################################c22 ck Hc22 ck + c12 cl + 2 c1 c2 clL N2y{zzzz

Note that (8-3A) ~(10-3B) correspond with (8-2) and (10-2). It was shown that p0 /p*qs =1/2. The difference betweenthis section and the previous section is that in this section w*Ls /p*qs and r*K0 /p*qs depend on c1 and c2 . In order to examine the effect of c1 and c2 on the income distribution, suppose that

ck = cl = 1 (11)

The following holds.

w*Ls /p*qs = c1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ4 Hc1+c2L , (8-4) r*K0 /p*qs = c1+2 c2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ4 Hc1+c2L . (10-4)

In[127]:=

check1 = Simplify@PowerExpand@Simplify@data1 ê. 8ck → 1, cl → 1<DDDOut[127]=

9 c14 Hc1 + c2L , 1

2, c1 + 2 c2

4 Hc1 + c2L =

Thus, the following holds.

w*Ls /p*qs rises as c1 rises when (11) holds, (8-5A) r*K0 /p*qs falls as c1 rises when (11) holds. (10-5A) w*Ls /p*qs falls as c2 rises when (11) holds, (8-5B) r*K0 /p*qs rises as c2 rises when (11) holds. (10-5B)

(8-5A) obtains ∑ (w*Ls /p*qs )/ ∑ c1 >0, as shown by the following computation.

In[128]:=

Simplify@D@check1@@1DD, c1DDOut[128]=

c24 Hc1 + c2L2

ü Utility Change

We have p0 /p*qs =1/2 at general equilibrium, independently of the innovation of ICTs. This, however, does notnecessarily imply that p0 is constant at general equilibrium, independently of the innovation of ICTs. At generalequilibrium, p0 is computed as in what follows.

ICT.nb 18

In[129]:=

pi1 = Simplify@pi0 ê. sol3 ê. w → 1DOut[129]=

−J100 J2 c22 ck + c12 cl + 2 c1 c2 cl − 2 "###############################################################################c22 ck Hc22 ck + c12 cl + 2 c1 c2 clL N

Jc1 c2 ck + c22 ck +"###############################################################################c22 ck Hc22 ck + c12 cl + 2 c1 c2 clL NN í

JHc1 + 2 c2L2 cl Jc22 ck −"###############################################################################c22 ck Hc22 ck + c12 cl + 2 c1 c2 clL NN

The following results obtain from the computation of the difference of profit, in which (a) only c1 rises, (b) only c2

rises, (c) only ck rises, (d) only cl rises starting from c1 =c2 =ck =cl =1. Note that when c2 or cl rises, p0 decreases.This result strengthens the assertion in Fukiharu [2007], which constructs an example in which the innovation does notnecessarily raise profit.

In[130]:=

N@8Hpi1 ê. 8c1 → 5ê 4, c2 → 1, ck → 1, cl → 1<L − Hpi1 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,Hpi1 ê. 8c1 → 1, c2 → 5 ê4, ck → 1, cl → 1<L − Hpi1 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,Hpi1 ê. 8c1 → 1, c2 → 1, ck → 5 ê4, cl → 1<L − Hpi1 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,Hpi1 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 5ê4<L − Hpi1 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L<D

Out[130]=

88.80999, −7.70975, 0.609846, −0.627116<

The following results obtain from the computation of the difference of the workers' utility, in which (a) only c1 rises,(b) only c2 rises, (c) only ck rises, (d) only cl rises starting from c1 =c2 =ck =cl =1. Note that the workers' utility risesby the innovation of ICT.

In[131]:=

uL0 = Simplify@PowerExpand@u ê. sol2 ê. sol3 ê. w → 1DD;

In[132]:=

N@8HuL0 ê. 8c1 → 5ê 4, c2 → 1, ck → 1, cl → 1<L − HuL0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,HuL0 ê. 8c1 → 1, c2 → 5 ê4, ck → 1, cl → 1<L − HuL0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,HuL0 ê. 8c1 → 1, c2 → 1, ck → 5ê4, cl → 1<L − HuL0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,HuL0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 5ê4<L − HuL0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L<D

Out[132]=

80.862069, 3.35502, 0.141114, 0.141009<

The following results obtain from the computation of the difference of the entrepreneurs' utility, in which (a) onlyc1 rises, (b) only c2 rises, (c) only ck rises, (d) only cl rises starting from c1 =c2 =ck =cl =1. Note that the entrepreneurs'utility rises by the innovation of ICT, in spite of the fact that when c2 or cl rises, p0 decreases.

In[133]:=

uE0 = Simplify@PowerExpand@u ê. sol2E ê. sol3 ê. w → 1DD;

In[134]:=

N@8HuE0 ê. 8c1 → 5ê 4, c2 → 1, ck → 1, cl → 1<L − HuE0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,HuE0 ê. 8c1 → 1, c2 → 5 ê4, ck → 1, cl → 1<L − HuE0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,HuE0 ê. 8c1 → 1, c2 → 1, ck → 5ê4, cl → 1<L − HuE0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,HuE0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 5ê4<L − HuE0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L<D

Out[134]=

80.83317, 0.785995, 0.121264, 0.0417479<

ICT.nb 19

The following results obtain from the computation of the difference of the capitalists' utility, in which (a) only c1 rises,(b) only c2 rises, (c) only ck rises, (d) only cl rises starting from c1 =c2 =ck =cl =1. Note that the capitalists' utility fallswhen cl rises.

In[135]:=

uK0 = Simplify@PowerExpand@u ê. sol2K ê. sol3 ê. w → 1DD;

In[136]:=

N@8HuK0 ê. 8c1 → 5ê 4, c2 → 1, ck → 1, cl → 1<L − HuK0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,HuK0 ê. 8c1 → 1, c2 → 5 ê4, ck → 1, cl → 1<L − HuK0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,HuK0 ê. 8c1 → 1, c2 → 1, ck → 5ê4, cl → 1<L − HuK0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L,HuK0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 5ê4<L − HuK0 ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1<L<D

Out[136]=

80.47615, 0.662438, 0.196021, −0.0664897<

4: CES Case with t=1/2

In this section, it is assumed that there is only one firm, producing consumption good, q, utilizing the capital goods,K, and the labor input, L.The production function is assumed as in what follows:

q=f(K , L)=HHck KL-t + Hcl LL-tL-nêt n=1/2, t= 1/2 (1-4)

where q is output, ck is the level of ICTs with respect to the capital input in the production process, K is the capitalinput, cl is the level of ICTs with respect to the labor input and L is the labor input. It is assumed that the production isconducted under decreasing returns to scale, so that the positive profit is guaranteed. The aim of this section is to examine whether (8-3A), (10-3A), (8-3B), and (10-3B) hold when the parameter changesfrom t= –1/2 to t= 1/2. When t= –1/2, however, the computation of GE is not easy. Thus, the Newton method is utilizedin the computation of general equilibrium, and the comparison is made between the case for c1 =c2 =ck =cl =1 and thecases in which (a) only c1 rises, (b) only c2 rises, (c) only ck rises, (d) only cl rises from c1 =c2 =ck =cl =1. We startfrom the computation of general equilibrium.

ü 4.1 The case for c1 =c2 =ck =cl =1

Suppose that

c1 =c2 =ck =cl =1. (12) In this subsection, assuming (12) we compute the income distribution at general equilibrium for the later use in thecomparative statics.

In[137]:=

ck = 1; cl = 1; c1 = 1; c2 = 1;

ICT.nb 20

ü Firm

First, the labor demand function, Ld , is computed as in what follows, where p is the price of the consumption good,w is the wage rate, and r is the rental price of the capital good. Then, the capital demand function, Kd , and the supplyfunction of consumption good, qs , are computed. The profit accrues to the firm after paying the rental cost for capitalgoods and labor cost, and finally the profit function, p0 , is computed. In this section (5-1) is assumed: b1 =b2 =1/3.

In[138]:=

Clear@w, qsD; m = 1ê2; t = 1ê 2; q = HHck ∗ kL^H−tL + Hcl ∗ lL^H−tLL^H−m êtL;pi = p ∗ q − r ∗ k − w ∗ l; f1 = Simplify@PowerExpand@8D@pi, kD 0, D@pi, lD 0<DD;sol1A = Solve@8D@pi, kD 0, D@pi, lD 0<, 8k, l<D; sol1 = sol1A@@1DD;ld = l ê. sol1; kd = k ê. sol1; qs = q ê. sol1; pi0 = pi ê. sol1; b1 = 1ê3; b2 = 1ê3;

ü Workers

In the same way as in the previous section, it is assumed that there is the (aggregate) household, who has the initialendowment of leisure hours, L0 =100, with no profit distribution from the firm. This class aims at utility maximizationsubject to income constraint, (2). From the assumption of utility function as Cobb-Douglas type in (2) the demandfunction of consumption good, xdW , and the labor supply function, Ls , are computed.

In[142]:=

l0 = 100; u = x^Hc1 ∗ b1L∗ le^Hc2 ∗ b2L;sol2 = PowerExpand@Solve@8D@u, xDêD@u, leD pêw, p ∗ x w ∗Hl0 − leL<, 8x, le<D@@1DDD;xdW = Simplify@x ê. sol2D; ls = Simplify@l0 − le ê. sol2D;

ü Entrepreneurs


In[145]:=

u = x^Hc1 ∗ b1L ∗ le^Hc2 ∗ b2L;sol2E = PowerExpand@Solve@8D@u, xDê D@u, leD pêw, p ∗ x + w ∗ le pi0<, 8x, le<D@@1DDD;xdE = x ê. sol2E; ldE = le ê. sol2E;

ü Capitalists


ICT.nb 21

In[147]:=

k0 = 100; u = x^Hc1 ∗ b1L∗ le^Hc2 ∗ b2L;sol2K =

PowerExpand@Solve@8D@u, xDêD@u, leD pêw, p ∗ x + w ∗ le r ∗ k0<, 8x, le<D@@1DDD;xdK = Simplify@x ê. sol2KD; ldK = Simplify@le ê. sol2KD;


The general equilibrium (GE) prices, p, r, and w are computed from the three equations, which stem from the threemarket equilibria, (5), (6), and (7). Assuming that w=1, we compute GE prices, p* and r*. From (5) and (6), {p*,r*} iscomputed as in what follows.

In[150]:=

check1 = H8kd k0, Hxd = xdK + xdE + xdW ê. k → kdL qs< ê. w → 1L;

In[151]:=

sol3 = FindRoot@check1, 8p, 10<,8r, 1ê100<, WorkingPrecision → 50, MaxIterations → 1000D

Out[151]=

8p → 22.500000000000000000000000000000000000000000000000,r → 0.12500000000000000000000000000000000000000000000000<


In[152]:=

Hld + ldK + ldE ê. k → kdL − ls ê. w → 1 ê. sol3

Out[152]=

0.×10−46

In[153]:=

Simplify@Hld + ldK + ldE ê. k → kdL − ls ê. w → 1 ê. 8p → 225ê10, r → 125ê1000<DOut[153]=

0

The local stability of this general equilibrium is guaranteed, since all of the eigen values of Jacobian matrix are nega-tive at the general equilibrium, as shown in what follows.

In[154]:=

a11 =

D@kd − k0 ê. w → 1, rD ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1< ê. 8p → 225ê10, r → 125ê1000<;a12 = D@kd − k0 ê. w → 1, pD ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1< ê.

8p → 225ê 10, r → 125ê 1000<;a21 = D@HxdK + xdE + xdW ê. k → kdL − qs ê. w → 1, rD ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1< ê.

8p → 225ê 10, r → 125ê 1000<;a22 = D@HxdK + xdE + xdW ê. k → kdL − qs ê. w → 1, pD ê. 8c1 → 1, c2 → 1, ck → 1, cl → 1< ê.

8p → 225ê 10, r → 125ê 1000<;jacobi = 88a11, a12<, 8a21, a22<<;N@Eigenvalues@jacobiDD

Out[159]=

8−888.978, −0.13332<

We cannot conduct a graphical simulation of this stability due to the shortage of memory.

ICT.nb 22

ed1=(kd-k0)/.wØ1/.{rØr[s],pØp[s]};ed2=((xdK+xdE+xdW/.kØkd)-qs)/.wØ1/.{rØr[s],pØp[s]};traj1=NDSolve[{D[r[s],s]ãed1,D[p[s],s]ãed2,r[0]ã1/100,p[0]ã10},{r[s],p[s]},{s,0,10},MaxStepsØ10000000][[1]]

No more memory available.Mathematica kernel has shut down.Try quitting other applications and then retry.



In[160]:=

data0 =

N@Simplify@PowerExpand@H8sW = w ∗ ld, sE = pi0, sK = r ∗ kd<êHp ∗ qsLL ê. sol3 ê. w → 1DDDOut[160]=

80.333333, 0.5, 0.166667<

The revenue of the firm at general equilibrium, or GDP: p*qs , is computed as in what follows.

ü Revenue at GE

In[161]:=

revenue0 = N@p ∗ qs ê. sol2 ê. sol3 ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3< ê.8c1 → 1, c2 → 1, ck → 1, cl → 1<D

Out[161]=

75.

ü Profit at GE

The profit of the firm at general equilibrium, p0 , is computed as in what follows.

In[162]:=

profit0 = N@pi0 ê. sol2 ê. sol3 ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3< ê.8c1 → 1, c2 → 1, ck → 1, cl → 1<D

Out[162]=

37.5

ü Utility at GE

The workers' utility at general equilibrium when c1 =c2 =ck =cl =1 is computed as in what follows.

In[163]:=

uL0 = N@u ê. sol2 ê. sol3 ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3< ê.8c1 → 1, c2 → 1, ck → 1, cl → 1<D

Out[163]=

4.8075

ICT.nb 23

The entrepreneurs' utility at general equilibrium when c1 =c2 =ck =cl =1 is computed as in what follows.

In[164]:=

uE0 = N@u ê. sol2E ê. sol3 ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3< ê.8c1 → 1, c2 → 1, ck → 1, cl → 1<D

Out[164]=

2.5

The capitalists' utility at general equilibrium when c1 =c2 =ck =cl =1 is computed as in what follows.

In[165]:=

uK0 = N@u ê. sol2K ê. sol3 ê. k → k0 ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3< ê.8c1 → 1, c2 → 1, ck → 1, cl → 1<D

Out[165]=

1.20187

ü 4.1 The case for ci >1, cj =1 (jπi) (i=l,k,1,2)

In this section we examine how the income distribution changes for each increase of ci (i=l,k,1,2), while the otherparameters remaining at 1. In order to do this, the following function, g[cl , ck , c1 , c2 ] is constructed. This functioncomputes {{w*Ls /p*qs , p0 /p*qs ,r*K0 /p*qs }, p0 , {uL *, uE *, uK *}} when cl , ck , c1 , and c2 are given arbitrarily.

ICT.nb 24

In[166]:=

g@cl_, ck_, c1_, c2_D :=

Module@8q, pi, f1, sol1A, sol1, ld, kd, qs, pi0, u, sol2, xdW, sol2E,xdE, sol2K, xdK, check1, sol3, data0, profit0, uL0, uE0, uK0<,

m = 1 ê2; t = 1ê2;q = HHck ∗ kL^H−tL + Hcl ∗ lL^H−tLL^H−m êtL;pi = p ∗ q − r ∗ k − w ∗ l;f1 = Simplify@PowerExpand@8D@pi, kD 0, D@pi, lD 0<DD;sol1A = Solve@8D@pi, kD 0, D@pi, lD 0<, 8k, l<D;sol1 = sol1A@@1DD;ld = l ê. sol1;kd = k ê. sol1;qs = q ê. sol1;pi0 = pi ê. sol1;b1 = 1ê3; b2 = 1ê 3;l0 = 100;u = x^Hc1 ∗ b1L ∗ le^Hc2 ∗ b2L;sol2 =

PowerExpand@Solve@8D@u, xDêD@u, leD pêw, p ∗ x w ∗Hl0 − leL<, 8x, le<D@@1DDD;xdW = Simplify@x ê. sol2D; ls = Simplify@l0 − le ê. sol2D;u = x^Hc1 ∗ b1L ∗ le^Hc2 ∗ b2L;sol2E =

PowerExpand@Solve@8D@u, xDêD@u, leD pêw, p ∗ x + w ∗ le pi0<, 8x, le<D@@1DDD;xdE = x ê. sol2E; ldE = le ê. sol2E;k0 = 100;u = x^Hc1 ∗ b1L ∗ le^Hc2 ∗ b2L;sol2K =

PowerExpand@Solve@8D@u, xDêD@u, leD pêw, p ∗ x + w ∗ le r ∗ k0<, 8x, le<D@@1DDD;xdK = Simplify@x ê. sol2KD; ldK = Simplify@le ê. sol2KD;check1 = H8kd k0, Hxd = xdK + xdE + xdW ê. k → kdL qs< ê. w → 1L;sol3 = N@FindRoot@check1, 8p, 10<,

8r, 1ê100<, WorkingPrecision → 50, MaxIterations → 1000DD;data0 = N@Simplify@PowerExpand@H8sW = w ∗ ld, sE = pi0, sK = r ∗ kd<êHp ∗ qsLL ê. sol3 ê.

w → 1DDD;profit0 = N@pi0 ê. sol2 ê. sol3 ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3<D;uL0 = N@u ê. sol2 ê. sol3 ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3<D;uE0 = N@u ê. sol2E ê. sol3 ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3<D;uK0 = N@u ê. sol2K ê. sol3 ê. k → k0 ê. w → 1 ê. 8a1 → 1ê3, a2 −> 1ê3, b1 → 1ê3, b2 → 1ê3<D;8data0, profit0, 8uL0, uE0, uK0<<D

First, it is ascertained that when (11) holds, the same result as in the previous subsection is produced by this function.

In[167]:=

g@1, 1, 1, 1DOut[167]=

880.333333, 0.5, 0.166667<, 37.5, 84.8075, 2.5 + 0. , 1.20187<<

First, suppose that cl rises from 1 to 5/4 with other parameters remaining at 1. The share of workers' income declines,while the one of capitalists rises. This result contradicts (8-3A) and (10-3A).

In[168]:=

g@5ê4, 1, 1, 1DOut[168]=

880.322185, 0.5, 0.177815<, 37.8162, 84.89865, 2.5617 + 0. , 1.28586<<

ICT.nb 25

Secondly, suppose that ck rises from 1 to 5/4 with other parameters remaining at 1. The share of workers' incomerises, while the one of capitalists declines. This result contradicts (8-3B) and (10-3B). Note that in this case, the profitand the utility of capitalists decline, while the utility of the entrepreneurs rises.

In[169]:=

g@1, 5 ê4, 1, 1DOut[169]=

880.344208, 0.5, 0.155792<, 37.1966, 84.89191, 2.53016 + 0. , 1.16287<<

Thirdly, suppose that c1 rises from 1 to 5/4 with other parameters remaining at 1. The share of workers' incomedeclines, while the one of capitalists rises, although all the utility levels in the society rise. This result contradicts(8-5A) and (10-5A).

In[170]:=

g@1, 1, 5ê 4, 1DOut[170]=

880.325184, 0.5, 0.174816<, 44.4372, 84.90633, 2.67034 + 0. , 1.21416<<

Fourthly, suppose that c2 rises from 1 to 5/4 with other parameters remaining at 1. The share of workers' incomerises, while the one of capitalists declines. This result contradicts (8-5B) and (10-5B). Note that the profit declines inthis case.

In[171]:=

g@1, 1, 1, 5ê4DOut[171]=

880.341688, 0.5, 0.158312<, 31.4132, 86.97766, 2.92782 + 0. , 1.23581<<

ICT.nb 26

Thus, the results in this section contradict those derived in the previous section.

5: Conclusions

This paper examined the effects of four elements in innovation on the income distribution. Traditional elements in theinnovation argument were restricted to the production process. In this paper, the production function has two factors ofproduction: labor input and capital input. Thus, in this paper, the 1st element is the efficiency improvement of the laborinput through ICTs, and the 2nd element the one of capital input. The special feature of ICTs consists in the fact thatquality of life improves through ICTs. Thus, in this paper, since the utility function has two variables: consumptiongood and leisure hours, the 3rd element is the improvement of the quality of consumption good through ICTs, and the4th is the one of leisure hours through ICTs. In Section 1, the production function is specified by Cobb-Douglas type where the efficiency improvement impliesthe additive type as in the traditional argument: i.e. one unit of input becomes c unit (c>1) after the innovation. Underthis assumption, the innovation of ICTs cannot change the income distribution at all. When measured in terms of utilitychange, however, every element contributes to the enhancement of utility for all the social classes. It was shown thatthe profit declines due to the 4th element. In Section 2, the production function is specified by Cobb-Douglas type where the efficiency improvement impliesthe structural type in the sense that the improvement implies the increased parameter of the Cobb-Douglas productionfunction. It was shown that the 3rd and 4th elements have no effect on the income distribution structure. The 1st and2nd elements can influence the income distribution structure. When measured in terms of utility change, the workersclass and the capitalists class become better of due to any element. It was shown that the utility of entrepreneurs classdeclines due to the first or second element. In this case, the profit was also shown to decline. In Section 3, the production function is specified by CES type with negative substitution parameter. It was shownthat all the four elements influence the income distribution structure, while the income share of the entrepreneurs isconstant. For example, first, the 1st element contributes to the rise of workers' income share, while the 2nd elementcontributes to the rise of capitalists' income share. Second, the 3rd element contributes to the rise of workers' incomeshare, while the 4th element contributes to the rise of capitalists' income share. As for the utility change, except for the3rd element, all the social classes become better off. The 3rd element reduces the utility of capitalists class. It wasshown that the profit falls due to the 1st or 3rd element. In Section 4, the production function is specified by CES type with positive substitution parameter. It was shownthat all the four elements influence the income distribution structure, while the income share of the entrepreneurs isconstant. The results, however, completely contradict those in the previous section. It was shown that the profit fallsdue to the 2nd or 4th element. The local stability is guaranteed for CES type production functions. Thus, it was shown by these simulations, thattheoretically, no definite relation holds between the innovation of ICTs and income distribution (or utility level). Inorder to derive the definite relation, one must examine what type of production function the economy has.

References

Fukiharu, T. [2007], "The Green Revolution and the Water Crisis in India: an Economic Analysis", Farming Systems Design 2007, Int. Symposium on Integrated Analysis on Farm Production Systems, M. Donatelli, J. Hatfield, A. Rizzoli, Eds., Catania (Italy), 10-12, September 2007, Book1, pp.153-4

ICT.nb 27

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