Dynamics of Income Distribution in a Market Economy: Possibilities for Poverty Allevation

Appendix

Model Description

Wage rate WR is assumed to adjust over period WRAT towards indicated wage rate IWR.

$$\mskip2mu\mathrm{d}/\text{d} t[\textit{WR}] = (\textit{IWR}-\textit{WR})/\textit{WRAT}$$

(1)

IWR depends on the wage‐bargaining position of the workers, which is determined by their opportunity cost of accepting wage‐employment. It is assumed that the opportunity cost of transferring a self‐employed worker to wage-work is zero when wage offered is equal to the current consumption expenditure per worker averaged over the whole workforce.

$$\textit{IWR} = [(R_{\text{s}}*(1-\textit{SP}_{\text{s}})+(\textit{AS}_{\text{s}}/\textit{LAS}))/\textit{TW}]\:,$$

(2)

where ${ R_{\text{s}} }$, ${ \textit{SP}_{\text{s}} }$ and ${ {\text{AS}}_{\text{s}} }$ are, respectively, income share, saving propensity and accumulated unspent savings of the self‐employed sector. LAS and TW are, respectively, life of accumulated unspent savings and total workforce. Subscripts s and f designate, respectively, self‐employed and capitalist sectors.

Ownership of land and capital as well as contribution to labor are the bases for claim to income while absentee ownership is possible through leasing arrangements. Thus, ${ R_{\text{s}} }$ is computed by adding together the value of output produced by the self‐employed sector ${ \textit{VQ}_{\text{s}} }$ and the wage payments received by the wage‐workers ${ W_{\text{f}} }$, and subtracting from the sum the rent payments made to the absentee owners. ${ R_{\text{f}} }$ is given by adding together the value of output produced by the capitalist sector ${ \textit{VQ}_{\text{f}} }$ and the rent payments it receives from the self‐employed sector, and subtracting from the sum the wage‐payments it makes.

$$R_{\text{s}} = \textit{VQ}_{\text{s}}+ \textit{WR} *W_{\text{f}}-\textit{LR}*\textit{RL} - \textit{KR}*\textit{RK}\:,$$

(3)

$$R_{\text{f}} = \textit{VQ}_{\text{f}} -\textit{WR} *W_{\text{f}}+\textit{LR}*\textit{RL}+\textit{KR}*\textit{RK} \:,$$

(4)

where LR, RL, KR, and RK, are, respectively, land rent, rented land, capital rent, and rented capital.

KR and LR depend, respectively, on the long-term averages of the marginal revenue products of capital and land (AMRPK and AMRPL) in the economy, and the demand for renting capital and land (RKD and RLD) as compared with the supply of rentable assets (RK and RL). The demand for renting, in turn, depends on the lack of ownership of adequate resources for productively employing the workers in the self‐employed sector.

$$\textit{KR} = \textit{AMRPK}*f_{1} [\textit{RKD}/\textit{RK}]\:;\quad f^{\prime}_{1} >0$$

(5)

$$\textit{RKD} = \textit{DKE}_{\text{s}} - KO_{\text{s}}\:.$$

(6)

Where ${ \textit{DKE}_{\text{s}} }$ is desired capital to be employed in the self‐employed sector and ${ \textit{KO}_{\text{s}} }$ is capital owned by it. Land rent LR and demand for renting land RLD are determined similarly.

The saving propensity of all households in not uniform. Since capitalist households associated with the capitalist sector receive incomes which are much above subsistence, their saving propensity is stable. On the other hand, the saving propensity of the worker households depends on their need to save for supporting investment for self‐employment and on how their absolute level of income compares with their inflexible consumption. Thus, ${ \textit{SP}_{\text{s}} }$ in the model is determined by the utility of investment in the self‐employed sector arising from a comparison of worker productivity in the sector with the wage rate in the capitalist sector, and the rent burden of this sector compared with the factor contribution to its income from land and capital.

$$\begin{aligned}\textit{SP}_{\text{s}} & = \mu * f_{2} [\textit{MRPW}_{\text{s}}/\textit{WR}]*f_{3} [(\textit{LR}*\textit{RL}+\textit{KR}*\textit{RK})\\&\quad / (\textit{VQ}_{\text{s}} -\textit{MRPW}_{\text{s}} *W _{\text{s}})]\:,\\[-5mm]\end{aligned}$$

(7)

$$\textit{SP}_{\text{f}} = \mu\:,$$

(8)

where ${ f^{\prime}_{2} > 0 }$, ${ f^{\prime}_{3} < 0 }$, μ is a constant, and MRPW is marginal revenue product of workers.

AS represent the balance of unspent savings, which determine the availability of liquid cash resources for purchase of assets. AS are consumed over their life LAS whether or not any investment expenditure occurs.

$$\mskip2mu\mathrm{d}/\text{d} t[\textit{AS}_{i}] \\ = R_{i}*\textit{SP}_{i}-\textit{AS}_{i}/\textit{LAS}-\textit{LA}_{i} *\textit{PL}- \sum _{j} \textit{KA}_{i}^{j}*\textit{GPL}\:;\\[-2mm]i = {\text{s,f}}\:;\quad j={\text{m,t}} \:,$$

(9)

where LA, PL, KA, and GPL are, respectively, land acquisitions, price of land, capital acquisitions, and general price level. Subscript i refers to any of the two sectors, self‐employed (s) and capitalist (f), and superscript j to the type of capital, modern (m) or traditional (t).

${ W_{\text{f}} }$ is assumed to adjust towards indicated workers ${ IW_{\text{f}} }$ given by desired workers ${ DW_{\text{f}} }$ and total workforce TW. TW is assumed to be fixed, although, relaxing this assumption does not alter the conclusions of this paper. All workers who are not wage‐employed must be accommodated in self‐employment. Thus ${ W_{\text{s}} }$ represents the remaining workers in the economy.

$$\mskip2mu\mathrm{d}/\text{d} t[W_{\text{f}}] = (\textit{IW}_{\text{f}} - W_{\text{f}})/\textit{WAT}$$

(10)

$$\textit{IW}_{\text{f}} = \textit{TW}*f _{4} (\textit{DW}_{\text{f}}/\textit{TW})$$

(11)

$$W_{\text{s}} = \textit{TW}-W_{\text{f}}$$

(12)

where ${ 1 \geq f_{4}\geq 0 }$, and ${ f^{\prime}_{4} > 0 }$. WAT is worker adjustment time.

The desired workers in each sector ${ DW_{i} }$ is determined by equating wage rate with the marginal revenue product of workers. A modified Cobb–Douglas type production function is used.

$$\textit{DW}_{i} = E_{i}^{\text{w}}*\textit{VQ}_{i}/\textit{WR}\:,$$

(13)

where ${ E_{i}^{\text{w}} }$ is the elasticity of production of workers in a sector.

Land and capital owned by the capitalist sector (${ \textit{LO}_{\text{f}} }$ and ${ \textit{KO}_{\text{f}} }$) are allocated to commercial production (${ \textit{KE}_{\text{f}} }$ and ${ \textit{LE}_{\text{f}} }$) and renting (RK and RL) activities depending on the desired levels of these factors in each activity. Thus,

$$\textit{RK} = (\textit{DRK}/(\textit{DRK} +\textit{DKE}_{\text{f}}))*\textit{KO}_{\text{f}}$$

(14)

$$\textit{RL} = (\textit{DRL}/(\textit{DRL}+\textit{DLE}_{\text{f}}))*\textit{LO}_{\text{f}}$$

(15)

$$\textit{KE}_{\text{f}} = \textit{KO}_{\text{f}}- \textit{RK}$$

(16)

$$\textit{LE}_{\text{f}} = \textit{LO}_{\text{f}}- \textit{RL}$$

(17)

Capital and land employed by the self‐employed sector consist of these production factors owned by them and those rented from the capitalist sector.

$$\textit{KE}_{\text{s}}= \textit{Ko}_{\text{s}}+ \textit{RK}$$

(18)

$$\textit{LE}_{\text{s}}= \textit{Lo}_{\text{s}}+ \textit{RL}\:.$$

(19)

Desired capital and land to be employed in any sector (${ \textit{DKE}_{i} }$ and ${ \textit{DLE}_{i} }$) are determined on the basis of economic criteria.

$$\mskip2mu\mathrm{d}/\text{d} t(\textit{DKE}_{i}) / \textit{KE}_{i}=f_{6}[\textit{MRPK}_{i}/\textit{MFCK}]$$

(20)

$$\mskip2mu\mathrm{d}/\text{d} t(\textit{DLE}_{i})/\textit{LE}_{i}=f_{5}[\textit{MRPL}_{i}/\textit{MFCL}]\:,$$

(21)

where ${ f^{\prime}_{5} }$ and ${ f^{\prime}_{6} > 0 }$. ${ \textit{MRPL}_{i} }$ and ${ \textit{MRPK}_{i} }$ are respectively marginal revenue products of land and capital in a sector, and MFCL AND MFCK are respectively marginal factor costs of land and capital.

$$\textit{MRPL}_{i}= (E_{i}^\text{l}*\textit{VQ}_{i}/\textit{LE}_{i})$$

(22)

$$\textit{MRPK}_{i}= (E_{i}^{\text{k}}*\textit{VQ}_{i}/\textit{KE}_{i})$$

(23)

$$\textit{MFCL} = \textit{PL}* \textit{IR}$$

(24)

$$\textit{MFCK} = \textit{IR}+(1/\textit{LK})*\textit{GPL} \:,$$

(25)

where ${ E_{i}^\text{l} }$ and ${ E_{i}^{\text{k}} }$ are, respectively, elasticities of production of land and capital in a sector. PL is price of land, IR is exogenously defined interest rate, LK is life of capital and GPL is general price level.

Changes in the quantities of capital and land desired to be rented out (DRK and DRL) depend on their respective rents KR and LR compared with their marginal factor costs MFCK and MFCL.

$$\mskip2mu\mathrm{d}/\text{d} t[\textit{DRK}]/\textit{RK} = f_{7} [\textit{KR}/\textit{MFCK}]\:;\quad f^{\prime}_{7} > 0$$

(26)

$$\mskip2mu\mathrm{d}/\text{d} t[\textit{DRL}]/\textit{RL} = f_{8} [\textit{LR}/\textit{MFCL}]\:;\quad f^{\prime}_{8} > 0\:.$$

(27)

The value of output produced by each sector is given by the product of the quantity it produces Q _i and the general price level GPL.

$$\textit{VQ}_{i}= Q_{i} *GPL$$

(28)

$$Q_{i}= A_{i}*K_{i}^{E^{\text{ki}}}*L_{i}^{E^{\text{li}}}*W_{i}^{E^{\text{wi}}}\:,$$

(29)

where K _i , L _i , and W _i represent capital, land and workers employed by a sector. A _i represent technology constants, which increase with the use of modern capital.

$$A_{i}= \textit{\AA}*f_{9} \big[K_{i}^{m}/(K_{i}^{t}+K_{i}^{m})\big]\:,$$

(30)

where ${ f^{\prime}_{9} > 0 }$ and Å is a scaling factor based on initial conditions of inputs and output of the production process.

Ownership is legally protected and the financial market is fragmented by households. Thus, purchase of any productive assets must be self‐financed by each sector through cash payments. Land ownership ${ \textit{LO}_{i} }$ of each sector changes through acquisitions ${ \textit{LA}_{i} }$ from each other. Each sector bids for the available land on the basis of economic criteria, its current holdings, and the sector's liquidity.

$$LA_{i}= \mskip2mu\mathrm{d}/\text{d} t[LO_{i}]$$

(31)

$$\textit{LO}_{i}= \bigg(\textit{DLO}_{i}/ \sum _{i}\textit{DLO}_{i}\bigg)*TL\:,$$

(32)

where ${ \textit{DLO}_{i} }$ is desired land ownership in a sector and TL is total land which is fixed,

$$\textit{DLO}_{i}=\textit{LO}_{i}*f_{6}[\textit{MRPL}_{i}/\textit{MFCL}]*f_{11}[\textit{CA}_{i}]\:,$$

(33)

where ${ f^{\prime}_{11}[\textit{CA}_{i}] }$ is ${ > 0 }$, and ${ \textit{CA}_{i} }$ is cash adequacy of a sector.

Cash adequacy of a sector ${ \textit{CA}_{i} }$ is given by the ratio of its accumulated unspent savings to the desired savings. The latter is computed by multiplying cash needed to finance investment and the traditional rate of consumption of savings in the sector by cash coverage CC.

$$ CA_{i}= AS_{i}/\Bigg(\bigg(\left(AS_{i}/LAS\right) + \left(LA_{i}*PL\right) \\[-3mm]+\Big(\sum _{j}KA_{ij}*GPL\Big)\bigg)*CC\Bigg)\:. $$

(34)

Capital ownership in a sector ${ \textit{KO}_{i}= \textit{KO}_{i}^{t}+ \textit{KO}_{i}^{m} }$ changes through acquisitions ${ \textit{KA}_{i}^{j} }$ and decay. Although there is a preference for modern capital, its acquisition ${ \textit{KA}_{i}^{m} }$ depends on the ability to accommodate the technology represented by it. Inventory availability of each type of capital ${ \textit{KIA}^{j} }$ also limits its purchases.

$$\mskip2mu\mathrm{d}/\text{d} t[\textit{KO}_{i}] = \sum_{j}\textit{KA}_{i}^{j}-\textit{KO}_{i}/\textit{LK} \:,$$

(35)

$$\textit{KA}_{i}^{j}= \textit{DKA}_{i}^{j}*\textit{KIA}^{j}\:,$$

(36)

$$\textit{DKA}_{i}^{m}= (\textit{KO}_{i}/\textit{LK})*f_{5}[\textit{MRPK}_{i}/\textit{MFCK}]* f_{11}[\textit{CA}_{i}]*\textit{TCF}_{i} \:,$$

(37)

$$\begin{aligned}\textit{DKA}_{i}^{t} & = (\textit{KO}_{i}/\textit{LK})*f_{5}[\textit{MRPK}_{i}/\textit{MFCK}]\\&\quad *f_{11} [\textit{CA}_{i}]*(1-\textit{TCF}_{i})\:,\end{aligned}$$

(38)

where ${ \textit{DKA}_{i} }$ are desired capital acquisitions, ${ f^{\prime}_{11} \geq 0 }$, and LK is life of capital. ${ \textit{TCF}_{i} }$ represent exogenously defined technological capability. ${ 0 < \textit{TCF}_{i} < 1 }$.

$$\textit{KIA}^{j}=f_{12} \left[\textit{KI}^{j}/\left(\sum _{i}\textit{DKA}_{i}^{j}\right)* \textit{KIC}\right] \:,$$

(39)

where ${ 0 \leqslant f_{12}\leqslant 1 }$, ${ f^{\prime}_{12} > 0 }$, and KIC is capital inventory coverage

$$\mskip2mu\mathrm{d}/\text{d} t[ \textit{KI}^{j}] = \textit{KQ}^{j}-\sum _{i}\textit{KA}_{i}^{j}\:,$$

(40)

where ${ \textit{KQ}^{j} }$ represent supply of capital. ${ \textit{KQ}^{m} }$ is imported, while ${ \textit{KQ}^{t} }$ is created within the economy by allocating a part of the capacity to its production.

$$\textit{KQ}^{t}=\sum Q_{i}*\bigg(\sum _{i}\textit{DKA}_{i}^{t}/\textit{TD}\bigg)\:.$$

(41)

The price of land PL is assumed to adjust towards indicated price of land IPL which is given by the economy‐wide average of the marginal revenue product of land AMRPL, interest rate IR and the desired land ownership in each sector ${ \textit{DLO}_{i} }$

$$\mskip2mu\mathrm{d}/\text{d} t[\textit{PL}] = (\textit{IPL}-\textit{PL})/\textit{LPAT}\:,$$

(42)

$$\textit{IPL} = (\textit{AMRPL}/\textit{IR})*f_{13}\left[\sum _{i}\textit{DLO}_{i}/\textit{TL}\right]\:;\\{\text{where}}\quad f^{\prime}_{13} > 0\:.$$

(43)

General price level GPL is determined by supply and demand considerations.

$$\mskip2mu\mathrm{d}/\text{d} t[\textit{GPL}] = \textit{GPLN}*f_{14}[\textit{TD}/\sum_{i}Q_{i}]$$

(44)

where ${ f^{\prime}_{14} > 0 }$. GPLN is normal value of GPL and TD is total demand for goods and services to be produced within the economy. TD is given by adding up non-food consumption C _i , traditional capital acquisition ${ \textit{KA}_{i}^{t} }$ and production of traditional capital for inventory, food demand FD and government spending G which is equal to taxes, if any, collected.

$$\begin{aligned}[b]\textit{TD} & = \sum C_{i}+\sum _{i}\textit{DKA}_{i}^{t}\\&\quad + \left(\left(\textit{KIC}*\sum _{i} \textit{DKA}_{i}^{t} -\textit{KI}^{j}\right)/\textit{IAT}\right)\\& \quad +\textit{FD}+G \:,\end{aligned}$$

(45)

$$\mskip2mu\mathrm{d}/\text{d} t(C_{i}) = \\\frac{[(((R_{i}*(1-SP_{i})+AS_{i}/LAS)/GPL)*FNFC_{i})-C_{i}]}{CAT}\:,\hspace*{-3mm}$$

(46)

where IAT is inventory adjustment time, ${ \textit{FNFC}_{i} }$ fraction non-food consumption, and CAT is consumption adjustment time. Food demand FD is given by multiplying population P with normal per capita food demand NFPCD and a function f ₁₅ representing a weak influence of price.

$$\textit{FD} = P*\textit{NFPCD}*f_{15} [\textit{GPL}/\textit{GPLN}]\:,$$

(47)

where ${ f^{\prime}_{15}<0 }$ and P bears a fixed proportion with total workforce TW.

The elasticity of production of land ${ E_{i}^\text{l} }$ is assumed to be constant as is suggested by empirical evidence concerning agricultural economies [Strout 1978, Heady and Dillon 1961]. Elasticity of production of capital ${ E_{i}^{\text{k}} }$ depends on the technology of production, which is determined by the proportions of traditional and modern capital employed. Since constant returns to scale are assumed, ${ E_{i}^{\text{w}} }$ is given by (47).

$$E_{i}^{\text{k}}=f_{16}[K_{i}^{m}/(K_{i}^{t}+K_{i}^{m})]\:;\quad f^{\prime}_{16} > 0\,,$$

(48)

$$E_{i}^{\text{w}}= 1-E_{i}^{\text{k}}-E_{i}^\text{l} \,.$$

(49)

Behavioral Relationships

Sixteen behavioral relationships [${ f_{1}\cdots f_{16} }$] have been incorporated into the model. The slope characteristics of these relationships have already been described in above equations. The graphical forms of the functions representing these relationships are shown in Figs. 11 and 12 placed below. General considerations for specifying such relationships are discussed in [63].

Figure 11

Behavioral relationships f ₁ through f ₈

Figure 12

Behavioral relationships f ₉ through f ₁₆