Corruption and Growth in a Dynamic General Equilibrium Model

This study introduces some new perspectives of bureaucratic corruption into the dynamic general equilibrium theory recently proposed by Zhang. The main concern of our study is the role of corruption on economic growth and income and wealth distribution between the officials and workers. The economy is composed of the industrial sector and public sector. The population is classified into officials and workers. Corruption takes places through many channels. Officials take brides from producers and households. The model describes dynamic interactions of growth, corruption with fixed tax rate and corruption rates. We simulate the model to demonstrate existence of equilibrium and motion of the dynamic system. We also examine effects of changes in different parameters on the motion of the economic system.


Introduction
Corruption may occur at any part of society.As Lui (1996) pointed, corruption exists in all countries.Economic causes and consequences of corruption are important and difficult issues.There are many theoretical studies on corruption and economic growth (Becker, 1968; Rose-Ackerman, 1999; Shi and Temzelides, 2004; Acemoglu and Verdier, 1998; Dzhumashev, 2014; D'Agostino, etal.2016).There are also many empirical studies on corruption and economic growth (e.g., Mauro, 1995Mauro, , 1998;;Ehrlich and Lui, 1999; Del Monte and Papagni, 2001; Rivera-Batiz, 2001; Glaeser and Saks, 2006;Swaleheen and Stansel, 2007;Teles, 2007;Gyimah-Brempong, 2002;Chea, 2015).The literature is too vast to be properly reviewed here.Different researchers emphasize different aspects of corruption.There is no general convergent conclusion on the role of corruption on economic growth.Chea (2015: 187) reveals "From the theoretical background and empirical evidence from various studies spread in various countries in the world using different methods over the years, there are mostly negative findings of economic growth.Nevertheless, there are also positive findings found on the effects of corruption on economic growth."For instance, Leff (1964) pointed out possible positive effects of corruption on economic growth.Huntington (1968) holds that the "efficient corruption", for instance, in the form of bribery, may enable firms to get works completed in a political economic environment plagued by bureaucratic complexity.Corruption may smooth operations so that the efficiency of the whole system is enhanced (Mo, 2001).Myrdal (1968) pointed out that corruption causes economic inefficiency as officials may delay permissions of projects in order to get bribes.According to the recent review by Dzhumashev (2014: 203), "the existing literature lacks a more general approach for interpreting the role of governance, the size government, and the level of development in the relationship between corruption and growth."As corruption may exist everywhere in society, it is necessary to study corrupt behavior within a dynamic equilibrium framework.
We analyze issues related to corruption and growth in a two-sector two-group economy.The dynamic general equilibrium framework enables us to analyze different aspects of corruption.We introduce a public sector to the neoclassical growth model.Public goods is important for understanding modern economies.(Barro, 1990;andTurnovsky, 2000, 2004).We study the dynamic relationship between government's spending, officials' wage and wealth, and corruption.Different from the traditional

The basic model
We consider that the economy is composed of one industrial sector and one public sector.There is a single good, called industrial good, in the economy and the price of the industrial good is unity.Capital depreciates at a fixed rate, .k  Most aspects of the industrial sector are similar to the standard one-sector neoclassical growth model (Burmeister and Dobell, 1970;Barro and Salai-Martin, 1995).The population consists of officials and workers.Each group is homogenous and the population of each group is fixed.Workers are fully employed by the industrial sector and officials are fully employed by the public sector.Households own assets of the economy and distribute their incomes to consume and save.The industrial sector uses labor and capital inputs to produce goods.Exchanges take place in perfectly competitive markets.Factor markets work well and factors are fully utilized at every moment.Saving is undertaken only by households.Let subscripts 1  j and 2  j stand for workers and officials respectively.We use j N to stand for the fixed population of group .j Let   t T j stand for the work time of a representative household of group j and   t N j for the flow of services used at time t for production.We have   where j h is the fixed level of human capital of group j .Here, we assume that officials' work time is also endogenous.It is considered that a rise in income due to wage increase or corruption affects officials' work time.

The industrial sector
We specify the industrial sector's production function as follows we take account of possible effects of public goods   t G on productivity.We use  to stand for the fixed corruption rate on output level.The tax rate on output  is determined by the government.We introduce . 1 Here, we neglect any effect of corruption on firms' capital and labor force .The marginal conditions are where   t r is the rate of interest and   t w 1 is the worker's wage rate per unit of time.

Workers' disposable income budget constraint
Let   t k 1 stand for the wealth owned by a representative work.If there is no corruption on the worker, the current income is The total value of wealth that the representative household can sell to purchase goods and to save is  .

t k
We assume that selling and buying wealth can be conducted instantaneously without any transaction cost.The disposable income is We assume that the corruption rates on wealth and wage rate are respectively k  and .
w  After paying the corruption fee to the official, the representative household's disposable income where The worker spends the disposable income on saving   t s 1 and consumption  .

t c
The budget constraint implies      .
where    . where is determined by the government budget.The official wage rate is proportional to the worker's market wage rate.The representative official receives the total income due to corruption as follows The representative official's disposable income The official spends the disposable income on saving   t s 2 and consumption  .

The utility functions and optimal decisions
The representative household in each group chooses three variables,  , where ., , ,

The wealth accumulation
According to the definitions of   t s j and  , t k j the change in the household's wealth is given by This equation simply means that change in the wealth is equal to saving minus dissaving.

The public sector
The public sector is financially supported by the government's tax income.The public sector's income is  .

t F 
We assume that the service that the public provides is dependent only on the officials' labor input  , For simplicity of analysis we don't introduce possible corruption of the officials on governance and government expenditures.The interaction between corruption and governance may affect the efficiency of public spending, which will influence growth (Dzhumashev, 2014).

Wealth being owned by the households
The sum of the wealth owned by the two groups is equal to the national capital

The total demand equaling the total supply
The total demand for the product is equal to the supply We built the model.The dynamic general equilibrium model describes the interdependence between wealth accumulation of the workers and officials, public good supply, income and wealth distribution, endogenous labor supply, and economic structure with different ways of corruption.The rest of the paper deals with properties of the model.

The behavior of the model
This section examines properties of the nonlinear dynamic model.We introduce a variable The following lemma shows that the motion of the entire economic system can be described by two differential equations.
The following lemma is checked in the Appendix.

Lemma
The dynamics of the economic system is governed by the following two differential equations with the tax rate as the variable where The ratio between the number of the workers and the number of the officials is 1 : 10 .The representative worker's relative propensity to save and the official's relative propensity to save are respectively The official's propensity to save is higher than the worker's propensity to save.Intuitively this implies that in this neoclassical growth model corruption may benefit economic growth as the wealth is shifted from the group with the lower propensity to save to the group with higher propensity to save.The two groups have the same level of human capital.The tax rate is fixed at one percent.We specify the initial conditions as follows The changes of the variables over time are plotted in Figure 1.The workers supply more labor and the officials work less over time.The national product and wealth rise.The rate of interest falls.The wage rates and wage incomes are enhanced.The equilibrium values of the variables are given as follows , where .The equilibrium point is stable.This result is important as it allows us to effectively conduct comparative dynamic analysis.

Comparative Dynamic Analysis
The previous section plotted the motion of the economic system with the initial condition.This section shows how a change in any parameter alters the motion of the system.As the system is stable and we can plot the motion of the system, we can effectively conduct comparative dynamic analysis.We use the variable,  ,

A rise of the corruption rate on the output level
We first study how the economic system is affected when the corruption rate on the output level is increased as follows:  W are reduced.The official works less and the public supply falls.The official consumes less and has less wealth initially and consumes more and has more wealth in the long term.The representative worker consumes less.The work has more wealth initially and less wealth in the long term.In the long term the worker works longer hours and the nation has more capital.As the wealth accumulated to the corrupt official is partly saved, the national capital is increased.The rate of interest is reduced.The long-term impact is the national economic growth.

A rise of the corruption rate on the capital income
We now allow the corruption rate on the capital income to be increased as follows: . 07 .

A rise of the corruption rate on the wage income
The corruption rate on the wage income is increased as follows: . 07 .0 05 .0 :  w  The long-term effects of the change in the corruption rate on the wage income are similar to the effects of the change in the corruption rate on the output level.The country's output is increased.The officials have more wealth and consume more.The workers have less wealth and consume more.It should be noted that when the corruption rate on the capital income is increased, the worker's wage rate and wage income are increased; when the corruption rate on the wage income is increased, the worker's wage rate and wage income are reduced.

A rise of the officials' human capital
We now increase the officials' human capital as: .6 . 2 5 .

 h
The national output is increased over time.The national output increase is mainly due to the rise in the public service.The rise in the public service is due to the human capital improvement.The ratio between the official's actual wage and the market wage rises initially and falls in the long term.The official's wage rate per unit time

A rise of the officials' propensity to save
Officials might not show their consumption of luxuries and try to save their income, for instance, due to political environment changes.Economically this implies that the officials' propensity to save is enhanced.We now consider the following rise in the officials' propensity to save: . 52 .0 5 .0 : The national output is increased over time.The national output increase is mainly due to the rise in the national capital service.The rise in the national wealth comes from the enhanced propensity to save.The officials work less hours and the total official labor supply is reduced.The public service supply falls.The official's wage rate per unit time and wage income are increased.The corrupt income c w falls initially and rises in the long term.The worker's consumption level and wealth fall initially and rise in the long term.The official's consumption level and wealth rise in the long term.

A rise of the number of the officials
We fix the total population.We now consider that the share of officials in the population is increased.The expansion of the public sector may be due to different reasons.We now consider the following shift of the labor force structure

  N N
The ratio between the official's actual wage and the market wage rises initially and falls in the long term.The corrupt income per official c w falls.The official's wage rate per unit time and wage income are reduced.The worker's wage rate per unit time and wage income are increased.The worker's working time rises initially and changes slightly in the long term.The official's working time falls.The worker's consumption level and wealth fall initially and rise slightly in the long term.The official's consumption level and wealth rise initially and fall in the long term.The workers' total labor supply rise initially and fall in the long term.The officials' total labor supply rise and public service is increased.It should be noted that in the literature of corruption and growth there are opposite conclusions.One argument is that a rise in the government size may produce more opportunities for corruption (e.g., Goel and Nelson, 1998; Rose-Ackerman, 1999; Alesina and Angeletos, 2005).The other argument is that an enlarged government sector may strengthen control of corruption and thus encourage economic growth (Billger and Goel, 2009).Our study shows that the two effects exist -in the short term economic growth is encouraged but in the long-term economic growth is discouraged.

A rise of the workers' human capital
The government tens to encourage workers' human capital, for instance, through subsiding education and providing differential education programs.We now allow the workers' human capital to be changed as follows: .6 . 2 5 .

 h
The ratio between the official's actual wage and the market wage is enlarged.The corrupt income per official rises.The official's wage rate per unit time and wage income are increased.The official works less hours.The officials' total labor supply falls and public service is reduced.The official's consumption level and wealth are increased.The official class benefits.The worker's wage rate per unit time and wage income are increased.The worker's working time is increased.The worker's consumption level and wealth rise initially and fall in the long term.The workers' total labor supply rise.The national wealth and output are increased.The rate of interest rises.We see that the worker class does not benefit in the long term.

Concluding Remarks
This study developed a dynamic general equilibrium model with endogenous wealth and corruption.The economy is composed of one (private) production sector and one public sector.The public service sector is carried out by officials.Its cost is financially supported by taxation on the private sector.Its output affects the private sector's productivity and the people's welfare.The production sector employs the workers' labor inputs and capital.The population is classified into workers and officials.We assume that officials are corrupt in the sense that they take bribes from the private sector and households.The corruption is measured by the corruption rates on the output level of the private sector, the wealth interest returns of the workers, and the wage income of the workers.We simulated the model and examined the transitory and long-run effects of changes in some parameters.The comparative dynamic analysis provided insights into the complicated impact of corruption on economic growth process.For instance, if the workers' human capital is enhanced, then (i) the ratio between the official's actual wage and the market wage is enlarged; (ii) the corrupt income per official rises; (iii) the official's wage rate per unit time and wage income are increased; (iv) the official works less hours and public service is reduced; (v) the official's consumption level and wealth are increased; (vi) the worker's consumption level and wealth rise initially and fall in the long term; (vii) the workers' total labor supply rise, the national wealth and output are increased; and (viii) the official class benefits and the worker class loses in the long term.The enhanced human capital by the workers will not benefit themselves in the long term, even though they benefit in the short term.As mentioned before, although there is a vast literature on corruption, there are relatively few macroeconomic growth models with endogenous wealth in a dynamic general equilibrium framework.Hence, our model can be generalized and extended in different ways.For instance, there are many other possible channels of corruption.We may also introduce institutional structure to allow corruption networks.The workers are not homogenous and the officials are heterogeneous in regard to morality, position and human capital.In an economic system with endogenous knowledge, corruption may not encourage innovative activities as innovators need government's support (e.g., Mo, 2001).
From ( 15) and (A3) we have where we also (A11) and ., where we use (7).Insert (A19) in (A14) and (A15) We show now that all the variables can be expressed as functions of z and G as follows: 0 u by (A19) → We thus proved the lemma.

ˆ1
the three variables as follows


are called respectively the worker's/official's propensity to stay at home, the propensities to consume good and to hold wealth.Maximizing j U subject to budget constraint (6)/(11) yields t z and   t G defined in the appendix.The values of all the other variables are uniquely determined as functions of   t z and   t G at any point in time by the following procedure:   Figure 1.The Motion of the Dynamic System to represent the change rate of the variable,  , t x in percentage due to changes in the parameter value.
where " " stands for "being changed to".The output falls initially and rises in the long term.As the corruption rate is increased, the ratio between the actual wage and the market wage rises.The official's wage rate per unit time


The long-term effects of the change in the corruption rate on the capital income are similar to the effects of the change in the corruption rate on the output level.The country's output is increased.The officials have more wealth and consume more.The workers have less wealth and consume more.

Figure 3 .
Figure 3.A Rise of the Corruption Rate on the Capital Income

2 w and wage income 2 W 1 w and wage income 1 W
are increased.The corrupt income c w falls initially and rises in the long term.The worker's wage rate per unit time are also increased.The official works less hours initially and almost the same hours in the long term.The worker works more hours initially and almost the same hours in the long term.The official supplies more labor and produces more public service.The worker's consumption level and wealth fall initially and rise in the long term.The official's consumption level and wealth fall initially and rise in the long term.

Figure 8 .
Figure 8.A Rise of the Workers' Human Capital From (A1), (1) and (2), we have 0 where we also use the definitions of n ˆ and

 Officials' disposable income and budget constraint
stand for the capital wealth owned by the representative official.We assume that the representative official's wage income per unit of qualified work time is paid in proportion to the worker's wage rate as follows w 