For the most part, the corruption literature tells us that poor countries are more likely to suffer from market failures. The literature further describes its qualitative form as risk of expropriation, property rights, contract enforcement, institutions, efficiency of markets, institutional stability, etc ([1] [3] ). These however suggest that market efficiency is a function of institutions ([10] ). This is quite a reasonable stance since much has been written on the importance of institutions to economic development ([11] ). [12] builds a clear connection be- tween institutions, bureaucratic corruption, and development.

Clearly, institutions matter as do market failures at explaining development. However, this paper takes the position that the mechanism that creates market failures is also the mechanism that generates the existing institutional properties and an economies level of development. That is, the mechanism that determines level of development also determines the size of the market failure.

The source of market failures is not central to this paper. However, the fact that it occurs is. Here it is assumed that some degree of market failures exists everywhere. Again, making reference to [4] who show that market failures are tied to the level of corruption, it is said here that corruption manifests itself in both rich and poor countries alike. No matter how rich or poor, corruption will be manifested in some form2. Work by [4] shows that “preventing all corruption is excessively costly, so intervention with some corruption is the best option3”.

This is a little controversial in that it claims that the size of the market failure is neither a factor of poverty nor poverty a factor of market failure. Rather, market failures and poverty are no more interrelated than ice-cream consumption and crime ([14] ). This is controversial for a number of reasons, but most because it says that institutional change cannot be a policy to improve economic performance. Rather, policy that tackles market failures ignores its dynamic force and hence will be ineffective as a policy.

Economists face a number of shortcomings when analyzing market failures or corruption primarily due to how it is measured (degree of institutional transparency, information asymmetries, cronyism, etc.)4. This paper employs a much more general measure of market failures. Market failure is assumed captured as the difference between that which is taken by the state (confiscation or taxation) and what is given back to society as a whole. To this end market failure in this paper is a measure, as suggested by [16] in defining corruption, as the un-internalized negative externality generated by market inefficiencies.

This paper argues that the existing models on corruption, while descriptively correct (see [16] for a summary of the literature) fail to provide a reasonable theoretical framework to explain both its measured quantitative form and its determinants. While this paper blurs the line between market failure and corruption it should be noted that the model presented here is really measuring the size of the market failure and corruption is postulated to imply government intervention to mitigate its effect and with it giving rise to bureaucratic corruption ([4] ). In this respect, policies that are meant to alleviate market failures should not be looked at as a determinant variable of economic development. Rather, the size of market failures should be looked at as a gauge of a much grander dynamic system. This is important because work by [9] has shown that institution building, along the lines of tackling transparency and corruption, has been unsuccessful at combating deep rooted poverty5.

The paper is organized as follows. The next section presents the basic model and economic participants. Section 3 derives the full dynamics of the model. Section 4 derives the various levels of corruption. Finally, Section 5 provides a brief summary and concluding remarks.

Government taxes workers income and use tax-generated income to consume public-goods from a public-goods producer and invest in public education. To simplify the exposition, the tax rate is assumed exogenously determined and fixed as too are the fraction of tax revenues devoted to public education and public-goods. The total population is fixed (zero population growth) and total tax revenues is given by, where and is the tax revenue per worker. For simplicity it is assume that government has no access to capital markets from which to draw fiscal deficits and at the end of each period it does not carry forward any fiscal surpluses. In other words, this government is in fiscal balance (tax revenues = fiscal expen − ditures) each period.

Without any loss to generality it is assumed that total tax revenues is spent on public education and public-goods. The value of all public-goods consumed is given by. Where, is the contract price of public-goods produced by the public-goods producer. To simplify the analysis it is assumed that both prices and the quantity of public-goods are negotiated, agreed upon, and delivered in the same period.

Output is composed of two goods, each produced in one of two sectors. Each sector produces either a consumer good or public good. In the consumer goods market output is assumed to take place in a perfectly competitive one-good sector where producers earn zero economic profits and the price of the consumer good, , is the numéraire. Each firm employs physical and human capital in the production process. For simplicity, let production follow the standard neoclassical constant returns to scale Cobb-Douglas technology.

where and are effective output and effective capital per unit of labor in the consumer goods sector. All inputs earn their respective marginal products:

where, and are labor’s and capital’s wage and rental rates respectively.

The public-goods producer is small relative to the consumer goods market and hence it must pay market prices for both inputs. In addition, public goods operate under an imperfectly competitive framework where only a few (for simplicity, one) firm produces public goods. This profit maximizing firm is an input price taker and pays for both labor and capital at the rates determined by Equations (2) and (3) respectively. For simplicity, it is assumed that the public good producer earns zero economic profits. The imperfectly competitive public-goods producer maximizes the following profit function:

where, is the effective capital stock per worker employed in the public-goods sector and. In addition, Equation (4) drops the sector input wages and rental rate since they are the same

across the two sectors. The public-good producer’s output is. He pays labor and

capital rental rates based on Equations (2) and (3) respectively and the firm owner extracts value from the firm. Here, represents the negative non-internalized externality extracted from society for the benefit of a very few. In this paper how is pocketed is of little interest.

Via the current literature the value of can be found in a system of bureaucrats as in [4] [6] . Where, the firm owners have private information about the true costs of fulfilling the contract and the bureaucrats, either the contract awarding agent or the overseer of the contract, has an incentive to participate in a sharing arrangement with the owners. [4] provide an excellent model showing just this interaction. The purpose of this paper is not to model such a system. Instead, the reader is asked to assume that such a system is in place. That is to say, the purpose of this paper is to present an alternative approach to derive the market failure level, as captured through, without explicitly modeling the interactions of bureaucrats and owners of the public-goods firm. This paper intends to show that under this more general construct the outcomes in the current corruption literature can also be derived without formally modeling the interactions.

The economy is composed of consumers in each period. The consumer lives in a two period overlapping generations (OLG) model framework. In the first period it consumes, invests in one child, , human capital, and saves for retirement,. In the second period he lives off his retirement savings and thereafter expires. Aside from the initial decision to invest in child’s human capital there is no bequest from parent to child or child to parent and there are no government transfers (i.e. social security).

Under Assumption 1 returns of human capital investment maintain the assumption of diminishing returns while still generating a sizable impact on human capital accumulation. Assumption 2 assumes that investment in human capital has a low impact on human capital accumulation. It is also the corner solution and is included here for completeness. However, the central analysis of this paper is based on Assumption 1.

Under assumption, all households within a given economy (defined by its type population) earn a wage and rental rate of capital given by Equations (2) and (3) regardless of the sector of employment. That is, capital stock in period is additive over all its population. Such that, per worker effective physical and human capital is given by Equations (11) and (12).

To simplify notation let the systems map for both physical and human capital be defined by Equation (13).

The dynamic systems at steady state explicit form are given by Equations (14) and (15) and the phase diagram of these are shown in Figure 1.

where,.

The arrows of motion under Assumptions 1 and 2 are attracting toward point A and attracting to point B in the area between the two curves and are respectively shown in Figure 1(a) and Figure 1(b)6.

Under Assumption 1, an interior solution for human and physical capital, point A in Figure 1(a), is given by Equations (16) and (17).

where, , , and in Equation (17) is given by Equation (16).

Based on parameter values outlined in Figure 1 it can be shown that point A in Figure 1(a) under Assumption 1 is a saddle. Computing the Jacobian at produces the following real eigenvalues.

where, represents the characteristic roots of the polynomial under Assumption 1. Furthermore, the elements of the Jacobian are derived from Equation (13) whose map is defined as. Given that the sum characteristic polynomial evaluated at one is and evaluated at negative one; it follows that the steady state point is a saddle.

The steady state point under Assumption 2 is a sink. By assumption human capital accumulation, Equation (8), has at minimum one unit of human capital for all insufficient levels of both human capital, , and physical capital,. Such that steady state physical and human capital poverty trap of Figure 1(b) has. Evaluating the steady state (point B in Figure 1(b)) produces two positive and real characteristics roots that lie within the unit circle.

Before exploring the size of the market failure the establishment of the behavior in the neighborhood of the steady state under Assumption 1 needs to be established7.

In what follows the existence of a periodic orbit around the steady state point and the steady state undergoing a bifurcation maintaining its orbit will be shown. To simplify the analysis it is assumed that there are two types of households, poor and rich households (as defined by).

Let there be two types of households whose existence is based on Assumption 1 and whose level of development is subject to the household type value of parameters. To this it is said that there is a family of parameter values whose type is defined in (i.e. poor household and rich household) such that is the value of a family of parameters by household type8. In an attempt to simplify the dynamics and to maintain the model tractable the following assumption is made:

Assumption 3 At equilibrium, (that is, at point A in Figure 1(a), given its parameter family value.)

human capital is such that.

Assumption 3 states that around or at in some small neighborhood human capital is non- changing. That is, additional knowledge is neither created nor destroyed in the neighborhood. Such that the orbital map is one dimensional in physical capital.

Given Assumption 3 the nonlinear Equation can be rewritten as:

where, is a map and its solution is a saddle point solution at. Next consider an exposition of the saddle-node bifurcation under a change in the parameter family that characterizes the two assumed level of developments. Let, the rich households have a consumption discount preference in the first period of 0.6 and poor households.9 Then a change in as defined by produces a saddle-node bifurcation. This is seen as a result of the hyperbolicity around. That is, Equation (19) is hyperbolic if none of its eigenvalues are on the unit circle. From it is clear that none lie on the unit circle. This is also true after altering. Figure 1 provides diagram of the above.

Figure 2(a) and Figure 2(b) show that under each household its solution to Equation (19), , has a family that is also its bifurcation value. That is, any changes to family experiences a bifurcation. Next the orbit around is evaluated.

Given the use of a discrete dynamic system it is clear that any orbit in some local region in neighborhood is subject to chaotic behavior. However, based on the arrows of motion in Figure 1(a) while point A is not a sink it is nonetheless stable near it. Whether the orbit is homoclinic and includes or a periodic orbit around is not of great concern here. For the purpose of this paper all that is needed is the existence of at least one orbit that shows properties of stability. However, the stability need not be on the orbit but rather on the region in. To this it is known that at least one such region exists based on the arrows of motion. Let this region in be defined as such that that is defined under Assumptions 1 and 3 is locally stable. Furthermore, the region in the local neighborhood is identical in size to the area around and

0.34 0.32

(a) (b)

Figure 2. Bifurcation point for poor and rich households is reached at respectively. (a) and (b) show the bifurcation diagram of Equation (19) where is in the y-axis and a range of around for poor and rich household’s in the -axis.

. This states that in there is a periodic point around that is equal in distance (i.e.

). For simplicity let the positive and negative periodic points be of equal

distance from.

The statement of distance is quite reasonable since under bifurcations undertaken by the eigenvalues, stability, and arrows of motion do not change as a result of the location change in. That is, the changes in are location changes and not changes in the mass of the steady state point. Let the following proposition be a summary of the above:

Proposition 1 The positive and negative periodic point in the region around within the stable subspace is of equal distance in the dimension of,.

Poor and rich households have already been defined over the parameter value set. Consider then, the size of the market failure given these two levels of development.

Proposition 2 (a) The size of the market failure is larger in Poor households than rich ones.

(b) Poor households have higher variance in market failure than do rich ones.

Proof. The proof to Proposition 2 is easier found my numerical computation. For instance, Proposition 2(a) is

found by showing that. For simplicity, let the periodic point

in its outer range from center in, , be one unit away from. Such that, using the parameter values employed in the previous section for poor and rich households generates and respectively.

In addition, the spread between and also decreases as a household moves to a higher level of development (Proposition 2 (b)). That is, for an equal number of randomly allotted periodic points in within each household type generates sets of that produces the set of market failure size points that are decreasing in variance as a household moves from low to higher levels of development. Given the parameterized model this implies for poor and rich households respectively. Or a reduction in variance of about 0.01 as one moves from the poor to the rich economies steady state.

The theoretical model presented above provides a number of specific findings.

Implication 1 Poor households will have higher levels of corruption relative to more developed households.

Implication 2 Poor households will have higher variance in corruption levels than developed households.

Implication 3 Corruption abatement can only take place if the household moves to a higher steady state.

Implication 4 Corruption is a map and not a point.

Implication 5 Higher levels of human capital are present in more developed households and hence higher levels of human capital are associated with less corruption.

Figure 1 in [1] (page 435) shows growth rate in the y-axis and perception of corruption in the x-axis. Their figure is telling of Implications 1 and 2 in that it clearly shows a negative regression line in the level of perceived corruption to economic growth (See also [8] ). Moreover, it also shows a greater dispersion (variance) of growth levels at the higher units of perceived corruption.

In addition, in [4] figure 3 they show a u-shaped relationship between income and benefit of government intervention. The argument is that in poor countries there is a higher opportunity costs associated with diverting scarce resources to bureaucratic monitoring. This suggests that in poor countries one expects to see higher incidences of corruption.

Implication 3 is what this paper terms the [9] paradox. This implication provides a good interpretation as to why institutional reform (corruption abatement) in less developed nations has not led to lasting development. The present theory states that reform aimed at producing sustainable economic development must target individuals’ preferences and not institutions. This is quite a strong statement since it suggests that the chicken and egg question is no question at all rather that preferences are what define existent institutions and not the other way around. From a practical stance it is clear that both are important determinants of a successful development policy.

[4] introduces a system of bureaucrats and shows that there exists a socially optimal level of corruption. This work extends their findings by showing that there exists a map of corruption points that corresponds to a given level of development (as summarized by Implication 4).

Implication 5 is consistent with the findings in [1] [7] that show negative correlation between proxies of corruption and proxies of human capital. This implication is also consistent with the literature that link (negatively) corruption to growth ([1] [3] [8] ).