1

Advances of the first chapter of the Phd thesis

(Labor markets of illegal drugs)

Preliminary and incomplete

Labor mobility among the formal, informal

and illegal sectors of an illegal-drug-producer

economy

Student: Omar Fernando Arias Reinoso

Advisor: Hector Sala Lorda

Doctorado en Economía Aplicada

Facultat d’Economia i Empresa-UAB

Bellaterra (Barcelona), Spain

09/06/17

2

Labor mobility among the formal,

informal and illegal sectors of an illegal-

drug-producer economy

Omar F. Arias R.1

Department of applied economics-UAB

2017

Abstract

The purpose of this paper is to study labor mobility among the formal,

informal and illegal sectors of an illegal-drug-producer economy. Formal

and informal sectors trade legal goods; informal and illegal sectors are

unregulated markets; formal and illegal sectors are unrelated. Our

contribution is twofold. On one hand, we merge the literatures of formal,

informal and illegal markets; on the other hand, we study intersectoral

labor mobility in a cocaine producer economy.

Key words: intersectoral labour mobility, formal, informal, illegal, cocaine.

JEL classification: E24, E26, H26, J31, J24

1 I would like to thank Hector Sala, Alfonso Aza, Javier Verdugo and the participants of the EICEA-

seminar of US and UA for their contributions to shape this idea. The usual disclaimer applies.

3

1 Introduction

Government regulation typically looks for social welfare. However, certain activities

avoid that to increase economic profits. Some of them trade legal goods in small-

scale business (e.g. street vendor and home-based workers), others trade illegal

goods (e.g. narcotics and counterfeit) in small and large-scale businesses. The former

are informal; the latter, illegal. It allows us to divide the economy in three sectors:

formal, informal and illegal. Figure 1 represents the relationships among them.

Figure 1: Formal, informal and illegal sectors of an economy

Formal activities are taxed. Employers pay income taxes; employees, payroll ones.

They also have pro-worker policies such as subsidies to unemployment or minimum

wages. Government regulation might reduce coordination of formal wages with the

economic cycle. Wage rigidities allow economic recessions to destroy formal jobs.

4

There are incentives to avoid regulation in small-scale business. Employers would

like to reduce wages; employees, avoid payroll taxes or easily become firm owners.

Low-productive agents look for informality to increase their probability of getting a

job. Informality reduces social welfare. It exposes workers to low wages without social

security. Government uses law enforcement to prevent labor movement toward

informality. Unregulated activities might be a source of illegality. Some agents would

like to increase their profits by trading illegal drugs or counterfeiting. Most of the

labor movement toward illegality would come from informality.

Illegal activities, particularly drug trafficking, are very profitable (see UNODC (2010)).

Governments apply demand and supply-side policies to eliminate them. On one

hand, fostering educational programs to reduce consumption; on the other hand,

curbing production by catching producers and traffickers. Given the inelastic

demand and an imperfect competitive structure, supply-side policies could increase

instead of reduce the profits of illegal drugs2.

The market is also well structured so it facilitates movements towards illegality. For

instance, consider the Colombian cocaine market. According to UNODC (2016) it is

divided in three parts: coca-leaf, cocaine paste and cocaine. Each stage is controlled

by different agents with a specific market structure: coca-leaf, peasants, the base of

the pyramid; cocaine paste, insurgent and contra-insurgent groups, a monopolistic

monopsony; and cocaine, traffickers and/or dealers, oligopoly in the wholesale

market. The market power relies on violence which bounds economic development.

2 Becker, Murphy and Grossman (2006) explains it by using the price elasticity of demand.

Given an inelastic demand, reducing the supply increase the price more than the decrease

of the demanded quantity. At the end, the increase in the price increase the profit of the

supplier.

5

Figure 2: Pyramid of production of cocaine in Colombia

Intersectoral labor mobility is very complex. We expect economic agents to move in

couples formality-informality and informality-illegality rather than formality-

illegality. Governments guide the labor movement towards formality by reducing the

size of informality and illegality. There are basically two ways to do it: education and

punishment policies. It is the purpose of this paper to study their labor market impact

in a search and matching model for an illegal-drug-producer economy.

Education distribute labor force. The marginal productivity of a worker depends

strongly on his educational level. We understand education as the set of techniques

and values that allows to live in society. Workers are distributed among the formal,

informal and illegal sectors according to their educational level. Individuals with

higher education levels will be in the legal sector; those with less education will be

in the illegal sector; and, in the informal sector, will be individuals with an

intermediate level of education.

6

Law enforcement on informality and illegality also distribute labor force. However,

its effect is ambiguous. On one hand, it discourages labor participation; on the other

hand, it stimulates profits since it increases the prices of an inelastic growing

demand. Those profits create balloon effects on the illegal activity. Mejia and

Restrepo (2016) and Mejia and Posada (2007) suggest prohibition policy might be

cost-ineffective because the government is unable to control all the illegal

infrastructure ( see also “Plan Colombia” (DNP (2016) and Dion and Russler (2008)).

The government can also reduce taxes or subsidize formalization. The first policy

affects the flow of labor between the formal and informal sectors. The second policy

is more general and has three problems. First, it is very difficult to calculate the

optimal subsidy that compensates the profits derived from illegality. Second, the

subsidy generates DWL because some “informals” become "illegal" to receive it.

Finally, the legal ones would be "rewarding" the formalization of illegals which is

distributive unfair.

This paper is organized as follows. After this introduction, we explain the general

structure of our model. Then we study the comparative statics and we make some

remarks. Finally, the references.

7

2 The model

We follow closely Pissarides (1990) in the general structure; and Kolm and Larsen

(2001) and Engelhardt (2008) in the 2-dimensional generalization. There are many

firms and workers operating as atomistic competitors. Only unemployed workers

search for jobs. The flow of workers into unemployment is equal to the flow of

workers out of unemployment. Only unemployed workers search for jobs. On-the-

job search does not influence the unemployment rate. Vacant jobs and unemployed

workers become matched to each other.

2.1 Matching

Let us denote formal (white), informal (green) and illegal (black). Let 𝑗 ∈ {𝑤, 𝑔, 𝑏} be

an index for each sector. Labor force in 𝑗 is 𝐿𝑙 . Unemployment and vacancy rates are

𝑢𝑗 and 𝑣𝑗 . The number of job matchings per unit of time is 𝑥(𝑢𝑗 , 𝑣𝑗). The tightness of

the labor market in 𝑗 is 𝜃𝑗 = 𝑣𝑗/𝑢𝑗 , the probability that a firm in 𝑗 fills a vacant is

𝑞(𝜃𝑗) = 𝑥(𝑢𝑗 , 𝑣𝑗)/𝑣𝑗 and the mean duration of a vacant is 𝑞−1(𝜃𝑗). The probability

that an unemployed in 𝑗 find a job is 𝑞(𝜃𝑗)𝜃𝑗 = 𝑥(𝑢𝑗 , 𝑣𝑗)/𝑢𝑗 and the mean duration

of the unemployment is (𝜃𝑗𝑞(𝜃𝑗))−1

.

8

2.2 Unemployment rate

Each sector has an adverse risk 𝑠𝑗 of being separated from the job. The number of

workers who enter unemployment in 𝑗 is 𝑠𝑗(1 − 𝑢𝑗)𝐿𝑗 and the ones who leave

unemployment is 𝜃𝑗𝑞(𝜃𝑗)𝑢𝑗𝐿𝑗 . In the steady state they are equal so the

unemployment rate in 𝑗 is:

𝑢𝑗 =𝑠𝑗

𝑠𝑗+𝜃𝑗𝑞(𝜃𝑗) (1)

2.3 Formal sector

Firms

The firm offers a job contract to an unemployed. If the job is occupied, then the firm

rents 𝑘𝑤 at 𝑟 to produce 𝑓(𝑘𝑤). The price of the commodity is 1. The income tax is a

portion 𝜏𝑖 ∈ (0,1) of the profits. If the job is not occupied, then the firm incurs in a

cost 𝛾 of hiring. The expected cost of a vacant job is 𝛾/𝑞(𝜃𝑙) . The capital depreciation

is 𝛿𝑘𝑤. The cost of labor is 𝑤𝑤.

The present-discounted value of expected profit from an occupied job and vacant

are 𝐽𝑤 and 𝑉𝑤. A job is an asset for the firm. In perfect competition in w:

𝑟𝑉𝑤 + 𝛾⏟ 𝐶𝑜𝑠𝑡 𝑜𝑓 𝑎 𝑣𝑎𝑐𝑎𝑛𝑡

= 𝑞(𝜃𝑤)(𝐽𝑤 − 𝑉𝑤)⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑎 𝑣𝑎𝑐𝑎𝑛𝑡

(2)

9

The profit function is

𝜋𝑤 = (𝑓(𝑘𝑤) − 𝛿𝑘𝑤 − 𝑤𝑤)(1 − 𝜏𝑖)

The equilibrium condition is:

𝐽𝑤(𝑟 + 𝑠)⏟ 𝐶𝑜𝑠𝑡 𝑜𝑓 𝑎 𝑗𝑜𝑏

= 𝜋𝑤 + 𝑠𝑤𝑉𝑤⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑎 𝑗𝑜𝑏

(3)

Since the profits of new jobs are exploited, 𝑉𝑤 = 0. From (2), 𝐽𝑤 =𝛾

𝑞(𝜃𝑤) and we can

write (3) as

𝜋𝑤 =𝛾(𝑟+𝑠𝑤)

𝑞(𝜃𝑤) (4)

Workers

The unemployment insurance is 𝑧. The present-discounted value of the expected

income stream of an unemployed worker is 𝑈𝑤.

𝑟𝑈𝑤 = 𝑧 + 𝜃𝑤𝑞(𝜃𝑤)(𝐸𝑤 − 𝑈𝑤)⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏𝑒𝑐𝑜𝑚𝑖𝑛𝑔 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑

(5)

The payroll tax is a portion 𝜏𝑝 ∈ (0,1) of the wage. The present-discounted value of

the expected income stream of an employed worker is 𝐸𝑤.

10

𝑟𝐸𝑤 = 𝑤𝑤(1 − 𝜏𝑝) + 𝑠𝑤(𝑈𝑤 − 𝐸𝑤)⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏𝑒𝑐𝑜𝑚𝑖𝑛𝑔 𝑢𝑛𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑

(6)

Solving (5) and (6)

𝑟𝑈𝑤 =𝑤𝑤(1−𝜏𝑝)𝜃𝑤𝑞(𝜃𝑤)+𝑧(𝑟+𝑠𝑤)

𝑟+𝑠𝑤+𝜃𝑤𝑞(𝜃𝑤) (7)

𝑟𝐸𝑤 =𝑤𝑤(1−𝜏𝑝)(𝑟+𝜃𝑤𝑞(𝜃𝑤))+𝑧𝑠𝑤

𝑟+𝑠𝑤+𝜃𝑤𝑞(𝜃𝑤) (8)

Wages

Wages in 𝑤 follows a Nash negotiation. Let 𝛽0

𝑤 and 𝛽1𝑤 be two positive

parameters such that 𝛽0𝑤(1 − 𝜏𝑝) + 𝛽1

𝑤(1 − 𝜏𝑖) = 1. The solution comes from

maximizing the following function

(𝐸𝑤 − 𝑈𝑤)𝛽0𝑤(1−𝜏𝑝)(𝐽𝑤 − 𝑉𝑤)

𝛽1𝑤(1−𝜏𝑖)

Its optimality condition is

𝛽0𝑤(1−𝜏𝑝)

𝛽1𝑤(1−𝜏𝑖)

=𝐸𝑤−𝑈𝑤

𝐽𝑤−𝑉𝑤 (9)

From (7) and (8)

𝐸𝑤 − 𝑈𝑤 =𝑤𝑤(1−𝜏𝑝)−𝑧

𝑟+𝑠𝑤+𝜃𝑤𝑞(𝜃𝑤) (10)

11

Replacing (10) in (9), using 𝐽𝑤 = 𝛾/𝑞(𝜃𝑤) and 𝑉𝑤 = 0 we have

𝑤𝑤 =𝛾

𝑞(𝜃𝑤)

𝛽0𝑤

𝛽1𝑤 (𝑟+𝑠𝑤+𝜃𝑤𝑞(𝜃𝑤))

(1−𝜏𝑖)+

𝑧

1−𝜏𝑝 (11)

Equilibrium

We solve the equilibrium in 𝑤 in three steps. First,

max{𝑘𝑤}

𝜋𝑤 = (𝑓(𝑘𝑤) − 𝛿𝑘𝑤 − 𝑤𝑤)(1 − 𝜏𝑖)

With 𝑓′(𝑘𝑤) = 𝑟 + 𝛿 we get 𝑘𝑤∗ . Second, we get 𝑤𝑤

∗ and 𝜃𝑤∗ from (4) and (11). Finally,

we get 𝑢𝑤∗ and 𝑣𝑤

∗ from (1).

2.4 The informal sector

Firms

The firm offers a job contract to an unemployed. The price of the commodity is 1.

The production is a given quantity 𝑦. The probability of catching informal activities

is 𝑝𝑔 ∈ (0,1). The hiring cost is 𝛾 and the expected cost of a vacant job is 𝛾/𝑞(𝜃𝑔) .

The cost of labor is 𝑤𝑔.

Again, the present-discounted value of expected profit from an occupied job and

vacant are 𝐽𝑔 and 𝑉𝑔. In perfect competition in g:

12

𝑟𝑉𝑔 + 𝛾⏟ 𝐶𝑜𝑠𝑡 𝑜𝑓 𝑎 𝑣𝑎𝑐𝑎𝑛𝑡

= 𝑞(𝜃𝑔)(𝐽𝑔 − 𝑉𝑔)⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑎 𝑣𝑎𝑐𝑎𝑛𝑡

(12)

The profit function is

𝜋𝑔 = (𝑦 − 𝑤𝑔)(1 − 𝑝𝑔)

The equilibrium condition is:

𝐽𝑔(𝑟 + 𝑠𝑔)⏟ 𝐶𝑜𝑠𝑡 𝑜𝑓 𝑎 𝑗𝑜𝑏

= 𝜋𝑔 + 𝑠𝑔𝑉𝑔⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑎 𝑗𝑜𝑏

(13)

Since the profits of new jobs are exploited, 𝑉𝑔 = 0. From (12), 𝐽𝑔 =𝛾

𝑞(𝜃𝑔) and we can

write (13) as

𝜋𝑔 =𝛾(𝑟+𝑠𝑔)

𝑞(𝜃𝑔) (14)

Workers

Being informal generates a cost 𝑝𝑔𝐿𝑔. The present-discounted value of the expected

income stream of an unemployed worker is 𝑈𝑔 with

𝑟𝑈𝑔 = 𝜃𝑔𝑞(𝜃𝑔)(𝐸𝑔 − 𝑈𝑔) − 𝑝𝑔𝐿𝑔⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏𝑒𝑐𝑜𝑚𝑖𝑛𝑔 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑

(15)

13

The present-discounted value of the expected income stream of an employed worker

is 𝐸𝑤 with

𝑟𝐸𝑔 = 𝑤𝑔(1 − 𝑝𝑔) + 𝑠𝑔(𝑈𝑔 − 𝐸𝑔)− 𝑝𝑔𝐿𝑔⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏𝑒𝑐𝑜𝑚𝑖𝑛𝑔 𝑢𝑛𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑

(16)

Solving (15) and (16)

𝑟𝑈𝑔 =𝑤𝑔(1−𝑝𝑔)𝜃𝑔𝑞(𝜃𝑔)

𝑟+𝑠𝑔+𝜃𝑔𝑞(𝜃𝑔)− 𝑝𝑔𝐿𝑔 (17)

𝑟𝐸𝑔 =𝑤𝑔(1−𝑝𝑔)(𝑟+𝜃𝑔𝑞(𝜃𝑔))

𝑟+𝑠𝑔+𝜃𝑔𝑞(𝜃𝑔)− 𝑝𝑔𝐿𝑔 (18)

Wages

Wages in 𝑔 also follows Nash negotiation. Let 𝛽0

𝑔 and 𝛽1

𝑔 be two positive

parameters such that 𝛽0𝑔(1 − 𝑝𝑔) + 𝛽1

𝑔(1 − 𝑝𝑔) = 1. The solution comes from

maximizing the following function

(𝐸𝑔 − 𝑈𝑔)𝛽0𝑔(1−𝑝𝑖)(𝐽𝑔 − 𝑉𝑔)

𝛽1𝑔(1−𝑝𝑖)

Its optimality condition is

𝛽0𝑔

𝛽1𝑔 =

𝐸𝑔−𝑈𝑔

𝐽𝑔−𝑉𝑔 (19)

From (17) and (18)

14

𝐸𝑔 − 𝑈𝑔 =𝑤𝑔(1−𝑝𝑔)

𝑟+𝑠𝑔+𝜃𝑔𝑞(𝜃𝑔) (20)

Replacing (20) in (19), using 𝐽𝑔 = 𝛾/𝑞(𝜃𝑔) and 𝑉𝑔 = 0 we have

𝑤𝑔 =𝛾

𝑞(𝜃𝑔)

𝛽0𝑔

𝛽1𝑔 (𝑟+𝑠𝑔+𝜃𝑔𝑞(𝜃𝑔))

(1−𝑝𝑔) (21)

Equilibrium

We solve the equilibrium in 𝑔 in two steps. First, we get 𝑤𝑔

∗ and 𝜃𝑔∗ from (14) and

(21). Finally, we get 𝑢𝑤∗ and 𝑣𝑤

∗ from (1).

2.5 Illegal sector

Firms

The exogenous inverse demand function is 𝑃𝑏(𝑏) = 𝑏

−𝜂 with 𝜂 > 1. There are 𝑛

traffickers with 𝑏 = ∑ 𝑏𝑖𝑛𝑖=1 . If a job is occupied, then firm 𝑖 produces 𝑏𝑖 with a fixed

marginal cost 𝑐. The portion 𝜙 ∈ (0,1) of 𝑑𝑖 survives eradication. The probability of

interdicting illegal trade is 𝑝𝑏 ∈ (0,1). The hiring cost is 𝛾 and the expected cost of a

vacant job is 𝛾/𝑞(𝜃𝑏) . The cost of labor is 𝑤𝑏.

Once more, the present-discounted value of expected profit from an occupied job

and vacant are 𝐽𝑏 and 𝑉𝑏. In perfect competition in b:

15

𝑟𝑉𝑏 + 𝛾⏟ 𝐶𝑜𝑠𝑡 𝑜𝑓 𝑎 𝑣𝑎𝑐𝑎𝑛𝑡

= 𝑞(𝜃𝑏)(𝐽𝑏 − 𝑉𝑏)⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑎 𝑣𝑎𝑐𝑎𝑛𝑡

(22)

The profit function is

𝜋𝑖𝑏 = (𝑑−𝜂𝑏𝑖𝜙 − 𝑐𝑏𝑖 − 𝑤𝑏)(1 − 𝑝𝑏)

The equilibrium condition is:

𝐽𝑏(𝑟 + 𝑠𝑏)⏟ 𝐶𝑜𝑠𝑡 𝑜𝑓 𝑎 𝑗𝑜𝑏

= 𝜋𝑖𝑏 + 𝑠𝑏𝑉𝑏⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑟𝑒𝑡𝑢𝑟𝑛 𝑜𝑓 𝑎 𝑗𝑜𝑏

(23)

Since the profits of new jobs are exploited, 𝑉𝑏 = 0. From (21), 𝐽𝑏 =𝛾

𝑞(𝜃𝑏) and we can

write (22) as

𝜋𝑖𝑏 =𝛾(𝑟+𝑠𝑏)

𝑞(𝜃𝑏) (24)

Workers

Being illegal generates a cost 𝑝𝑏𝐿𝑏. The present-discounted value of the expected

income stream of an unemployed worker is 𝑈𝑏 with

𝑟𝑈𝑏 = 𝜃𝑏𝑞(𝜃𝑏)(𝐸𝑏 − 𝑈𝑏) − 𝑝𝑏𝐿𝑏⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏𝑒𝑐𝑜𝑚𝑖𝑛𝑔 𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑

(25)

16

The present-discounted value of the expected income stream of an employed worker

is 𝐸𝑏 with

𝑟𝐸𝑏 = 𝑤𝑏(1 − 𝑝𝑏) + 𝑠𝑏(𝑈𝑏 − 𝐸𝑏)− 𝑝𝑏𝐿𝑏⏟ 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑏𝑒𝑐𝑜𝑚𝑖𝑛𝑔 𝑢𝑛𝑒𝑚𝑝𝑙𝑜𝑦𝑒𝑑

(26)

Solving (25) and (26)

𝑟𝑈𝑏 =𝑤𝑏(1−𝑝𝑏)𝜃𝑏𝑞(𝜃𝑏)

𝑟+𝑠𝑏+𝜃𝑏𝑞(𝜃𝑏)− 𝑝𝑏𝐿𝑏 (27)

𝑟𝐸𝑏 =𝑤𝑏(1−𝑝𝑏)(𝑟+𝜃𝑏𝑞(𝜃𝑏))

𝑟+𝑠𝑏+𝜃𝑏𝑞(𝜃𝑏)− 𝑝𝑏𝐿𝑏 (28)

Wages

Wages in 𝑏 again follows Nash negotiation. Let 𝛽0

𝑏 and 𝛽1𝑏 be two positive

parameters such that 𝛽0𝑏(1 − 𝑝𝑏) + 𝛽1

𝑏(1 − 𝑝𝑏)(1 − 𝜙) = 1. The solution comes

from maximizing the following function

(𝐸𝑏 − 𝑈𝑏)𝛽0𝑏(1−𝑝𝑏)(𝐽𝑏 − 𝑉𝑏)

𝛽1𝑏(1−𝑝𝑏)(1−𝜙)

Its optimality condition is

𝛽0𝑏

𝛽1𝑏(1−𝜙)

=𝐸𝑏−𝑈𝑏

𝐽𝑔−𝑉𝑔 (29)

From (26) and (27)

17

𝐸𝑏 − 𝑈𝑏 =𝑤𝑏(1−𝑝𝑏)

𝑟+𝑠𝑏+𝜃𝑏𝑞(𝜃𝑏) (30)

Replacing (30) in (29), using 𝐽𝑏 = 𝛾/𝑞(𝜃𝑏) and 𝑉𝑏 = 0 we have

𝑤𝑏 =𝛾

𝑞(𝜃𝑏)

𝛽0𝑏

𝛽1𝑏

1

(1−𝜙)

(𝑟+𝑠𝑏+𝜃𝑏𝑞(𝜃𝑏))

(1−𝑝𝑏) (31)

Equilibrium

We solve the equilibrium in b in three steps. First,

Max{𝑏𝑖}

𝜋𝑖𝑏 = 𝑏−𝜂𝑏𝑖𝜙 − 𝑐𝑏𝑖

The FOC’s are

[𝑏−𝜂 − 𝜂𝑏−𝜂𝑏1]𝜙 = 𝑐⋮

[𝑏−𝜂 − 𝜂𝑏−𝜂𝑏𝑛]𝜙 = 𝑐

Since 𝑏1 = 𝑏2 = ⋯ = 𝑏𝑛 then

[(𝑛𝑏𝑖)−𝜂 − 𝜂(𝑛𝑏𝑖)

−𝜂𝑏1]𝜙 = 𝑐

We then have

18

𝑏𝑖 =1

𝑛[𝜙

𝑐(𝑛 − 𝜂

𝑛)]

1𝜂

The solution of the market is given by:

𝑏 =∑𝑏𝑖

𝑛

𝑖=1

= [𝜙

𝑐(𝑛 − 𝜂

𝑛)]

1𝜂

𝑃𝑑 =𝑐

𝜙(𝑛

𝑛 − 𝜂)

Second, we get 𝜃𝑏∗ from (24) and 𝑤𝑏

∗ from (31). Finally, we get 𝑢𝑏∗ and 𝑣𝑏

∗ from (1).

3 Sector division

Unemployed workers decide the sector in which they would like to work. For this,

they compare the value of working in formal, informal or illegal sectors. It implies

finding a threshold for participating in the sector. We analyze the three possible

combinations: formal-informal, informal-illegal and formal-illegal. We aim to solve

seize of each sector and the economics behind the intersectoral labor mobility.

3.1 Formal-Informal

Evaluating 𝑟𝑈𝑤 and 𝑟𝑈𝑔 from (5) and (15)

𝑧 + 𝜃𝑤𝑞(𝜃𝑤)(𝐸𝑤 − 𝑈𝑤) = 𝜃𝑔𝑞(𝜃𝑔)(𝐸𝑔 −𝑈𝑔) − 𝑝𝑔𝐿𝑔

19

Using (10) and (20) we have

𝑝𝑔𝐿𝑔 =(1−𝑝𝑔)𝜃𝑔𝑞(𝜃𝑔)

𝑟+𝑠𝑔+𝜃𝑔𝑞(𝜃𝑔)𝑤𝑔 −

(1−𝜏𝑝)𝜃𝑤𝑞(𝜃𝑤)

𝑟+𝑠𝑤+𝜃𝑤𝑞(𝜃𝑤)𝑤𝑤 −

(𝑟+𝑠𝑤)

𝑟+𝑠𝑤+𝜃𝑤𝑞(𝜃𝑤)𝑧 (32)

3.2 Informal-Illegal

Evaluating 𝑟𝑈𝑔 and 𝑟𝑈𝑏 from (15) and (25)

𝜃𝑔𝑞(𝜃𝑔)(𝐸𝑔 − 𝑈𝑔) − 𝑝𝑔𝐿𝑔 = 𝜃𝑏𝑞(𝜃𝑏)(𝐸𝑏 − 𝑈𝑏) − 𝑝𝑏𝐿𝑏

Using (20) and (30) we have

𝑝𝑏𝐿𝑏 − 𝑝𝑔𝐿𝑔 =(1−𝑝𝑏)𝜃𝑏𝑞(𝜃𝑏)

𝑟+𝑠𝑏+𝜃𝑏𝑞(𝜃𝑏)𝑤𝑏 −

(1−𝑝𝑔)𝜃𝑔𝑞(𝜃𝑔)

𝑟+𝑠𝑔+𝜃𝑔𝑞(𝜃𝑔)𝑤𝑔 (33)

3.3 Formal-Illegal

Evaluating 𝑟𝑈𝑤 and 𝑟𝑈𝑏 from (5) and (25)

𝑧 + 𝜃𝑤𝑞(𝜃𝑤)(𝐸𝑤 − 𝑈𝑤) = 𝜃𝑏𝑞(𝜃𝑏)(𝐸𝑏 − 𝑈𝑏) − 𝑝𝑏𝐿𝑏

Using (10) and (30) we have

20

𝑝𝑏𝐿𝑏 =(1−𝑝𝑏)𝜃𝑏𝑞(𝜃𝑏)

𝑟+𝑠𝑏+𝜃𝑏𝑞(𝜃𝑏)𝑤𝑏 −

(1−𝜏𝑝)𝜃𝑤𝑞(𝜃𝑤)

𝑟+𝑠𝑤+𝜃𝑤𝑞(𝜃𝑤)𝑤𝑤 −

(𝑟+𝑠𝑤)

𝑟+𝑠𝑤+𝜃𝑤𝑞(𝜃𝑤)𝑧 (34)

3 Comparative statics

Consider the following matching function

𝑥(𝑢𝑤, 𝑣𝑤) = (𝑢𝑤𝑣𝑤)0.5

Then, the corresponding probabilities are

𝑞(𝜃𝑤) =(𝑢𝑤𝑣𝑤)

0.5

𝑣𝑤= (

𝑢𝑤𝑣𝑤)0.5

= (1

𝜃𝑤)0.5

𝜃𝑤𝑞(𝜃𝑤) =(𝑢𝑤𝑣𝑤)

0.5

𝑢𝑤= 𝜃𝑤

0.5

We use the information in the general solution of each sector.

3.1 Formal sector

Let us use 𝑓(𝑘𝑤) = 𝑘𝑤𝛼 . Then 𝑘𝑤

∗ = (𝛼

𝑟+𝛿)

1

1−𝛼, 𝑓(𝑘𝑤) = (

𝛼

𝑟+𝛿)

𝛼

1−𝛼 and

𝜋𝑤 = ((𝛼

𝑟 + 𝛿)

𝛼1−𝛼

− 𝛿 (𝛼

𝑟 + 𝛿)

11−𝛼

− 𝑤𝑤) (1 − 𝜏𝑖)

From (4) and (11) we have

𝑤𝑤 = (𝛼

𝑟 + 𝛿)

𝛼1−𝛼

− 𝛿 (𝛼

𝑟 + 𝛿)

11−𝛼

−𝛾(𝑟 + 𝑠𝑤)

(1 − 𝜏𝑖)𝜃𝑤0.5

21

𝑤𝑤 =𝛾(𝑟 + 𝑠𝑤)

(1 − 𝜏𝑖)

𝛽0𝑤

𝛽1𝑤 𝜃𝑤

0.5 +γ

(1 − 𝜏𝑖)

𝛽0𝑤

𝛽1𝑤 𝜃𝑤 +

𝑧

1 − 𝜏𝑝

The solution of 𝜃𝑤 is

𝜃𝑤∗ =

2𝑎𝑤𝑐𝑤 + 𝑏𝑤2 ±√4𝑎𝑤𝑐𝑤 + 𝑏𝑤2

2𝑐𝑤2

With

𝑎𝑤 = (𝛼

𝑟 + 𝛿)

𝛼1−𝛼

− 𝛿 (𝛼

𝑟 + 𝛿)

11−𝛼

−𝑧

1 − 𝜏𝑝

𝑏𝑤 =𝛾(𝑟 + 𝑠𝑤)

(1 − 𝜏𝑖)[1 +

𝛽0𝑤

𝛽1𝑤]

𝑐𝑤 =𝛽0𝑤

𝛽1𝑤

𝛾

(1 − 𝜏𝑖)

Table 1. Comparative-static properties of equilibrium in the formal sector

Change Unemployment Vacancies Tightness Wages

Income tax + - - -

Payroll tax + + + -

Taxes reduce wages and increase unemployment in the formal sector.

3.2 Informal sector

From (14) and (21) we have

𝑤𝑔 = 𝑦 −𝛾(𝑟 + 𝑠𝑔)𝜃𝑔

0.5

(1 − 𝑝𝑔)

𝑤𝑔 =𝛾(𝑟 + 𝑠𝑔)

(1 − 𝑝𝑔)

𝛽0𝑔

𝛽1𝑔 𝜃𝑔

0.5 +γ

(1 − 𝑝𝑔)

𝛽0𝑔

𝛽1𝑔 𝜃𝑔

22

The solution of 𝜃𝑤 is

𝜃𝑔∗ =

2𝑎𝑔𝑐𝑔 + 𝑏𝑔2 ±√4𝑎𝑔𝑐𝑔 + 𝑏𝑔2

2𝑐𝑔2

With

𝑎𝑔 =𝛾(𝑟 + 𝑠𝑔)

1 − 𝑝𝑔[1 +

𝛽0𝑔

𝛽1𝑔]

𝑏𝑔 =𝛽0𝑔

𝛽1𝑔

𝛾

(1 − 𝑝𝑔)

𝑐𝑔 = 𝑦

Table 2. Comparative-static properties of equilibrium in the informal sector

Change Unemployment Vacancies Tightness Wages

Law enforcement + - - -

Low enforcement reduce wages in the informal sector.

3.3 Illegal sector

The profit function is

𝜋𝑖𝑏 = [1

𝑛(𝜙

𝑐(𝑛 − 𝜂

𝑛))

1−𝜂𝜂

−𝑐

𝑛(𝜙

𝑐(𝑛 − 𝜂

𝑛))

1𝜂

] (1 − 𝑝𝑏)

Using (24)

𝜃𝑏∗ = ([

1

𝑛(𝜙

𝑐(𝑛 − 𝜂

𝑛))

1−𝜂𝜂

−𝑐

𝑛(𝜙

𝑐(𝑛 − 𝜂

𝑛))

1𝜂

](1 − 𝑝𝑏)

𝛾(𝑟 + 𝑠𝑏))

2

23

Table 3. Comparative-static properties of equilibrium in the illegal sector

Change Unemployment Vacancies Tightness Wages

Eradication + - - -

Interdiction + - - +

Interdiction increases illegal wages whereas eradication reduces them.

3.4 Intersectoral labor movement

From (32), (33) and (34) we estimate the labor mobility after considering different

sectoral policies.

Table 4. Comparative-static properties of intersectoral labor movement

Change Formal Informal Illegal

Income tax - + +

Payroll tax - + +

Law enforcement + - +

Interdiction - + +

Eradication + - -

Most of the governmental regulation encourages labor movement towards illegality.

Eradication seems to be a good mechanism to control that movement; however, in

the practice, strong balloons effects may distort the final impact on the general

structure of the labor market.

24

References

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the case of drugs” The Journal of Political Economy 114(1), 38–60.

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Poverty: Explaining Coca Cultivation in Colombia during Plan Colombia”

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25

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