9. The Political Economy of Domestic Child Labor Policies
10.2. The Model
Section 10 unfolds as follows. In section 10.2. we outline the basic model of endoge-nous contingent punitive tariffs. Section 10.3. extends the basic model by introducing a negative consumption externality of child labor. In section 10.4. we introduce income transfers as a second instrument to combat child labor. Section 10.5. concludes.
10.2. The Model
10.2.1. Contingent Punitive Tariffs as Child Labor Policy
In this section, we develop the basic model. We assume a two-country setting with a developed country called "North" where child labor does not exist and a developing country called "South" where child labor exists. Furthermore we assume two-sector economies: sector 1 produces a traded good and a sector 2 produces a non-traded good.
In the North a contingent punitive tariff is determined endogenously as the sole interna-tional policy instrument to combat child labor in the South. The tariff is imposed on imports if it is detected that child labor is used in the South in producing the export good. The traded good 1 is produced in the North by many firms using exclusively adult
labor and by a single firm using both adult and child labor in the South.15 Good 1 is im-ported by the North and the Southern firm produces the good also for its home market.
In the first stage, a contingent punitive tariff rate is determined in the North. This tariff will be imposed on imports from the Southern firm if it is detected of employing child labor in production. In the second stage, the Southern firm decides about the quantity of child and adult labor, taking the contingent punitive tariff determined in the first stage as given. In the third stage, child labor in the South can be detected with a given prob-ability. If child labor is detected, the contingent punitive tariff is imposed, otherwise free trade between North and South prevails. In order to determine a sub-game perfect equilibrium, we solve this game via backward induction.
10.2.1.1. Determination of Northern Prices
We assume that both in the Northern and in the Southern market the demand for the traded good 1 is linear as follows:
(18) d1N = a1N - pN (19) d1S = a1S - pS
d1N and d2N denote the demand in the North and South for good 1, a1N and a1S measure market size in the North and South, and pN and pS are the prices of the good in the North and South, respectively.
If there is no contingent punitive tariff on the imports of the good in the North, we have free trade and in absence of trading costs, good 1 has the same price in both countries.
Thus the following arbitrage condition holds:
(20) pN = pS = p0
15We make the assumption of a single firm in order to avoid coordination problems that arise when indi-vidual firms decide how to respond to a contingent punitive tariff. For the indidual firms avoiding the collective punishment represents a public “bad”, whereas the means of avoiding the implementation of the contingent punitive tariff come, of course, at a private cost.
The free trade price is denoted by p0 . Following Grossman and Horn (1988), we make the assumption that the Southern and Northern firms are output constrained in the short run. Each firm produces only one unit of output. Let nN denote the exogenous number of firms in the North. As assumed above, there is only one large firm in the South con-sisting of nS subsidiaries each producing one unit of output. Hence, the total supply of the good on both markets is nN + nS.
Since we have assumed that the South exports the good 1, the quantity of exports under free trade, x10 , is equal to the fixed Southern supply nS minus Southern demand for good 1:
(21) x10 =nS −d1S =nS + p0 −a1s
From (21) we obtain the producer price of the Southern firm in the Northern market:
(22) p0 =x10 −nS +a1S
If the North now imposes a contingent punitive tariff on its imports from the South, the market price of the traded good in the North amounts to
(23) pNt = pN(1+t)=(x1t −nS +a1S)(1+t),
The producer price of the Southern firm will then be reduced to pNt
/(1+t) where t≥0.
In equilibrium the total supply of good 1 has to be equal to total demand of good 1 and the producer price for the Southern firm is the same in the North and the South:
(24) nS +nN =(a1N − pNt)+(a1S − pS),
where S
t
N p
t
p =
+ ) 1
( .
Solving for pNt, the producer price for Northern firms, we obtain:
Figure 2 portrays the equilibrium of the tradable goods market. The supply curve with the lower slope traces the total supply – domestic production plus imports - of good 1 if there is no contingent punitive tariff. The supply curve with the higher slope traces the total supply of good 1 if a positive contingent punitive tariff is imposed. In this the South will export less case at each price and thus the total supply curve moves inwards.
From Figure 2 and from (25) it can be easily seen that a contingent punitive tariff in-creases the producer price for Northern firms. The producer price of the Southern firm falls; instead of p0, it obtains only pNt
/(1+t) for the Southern firm if in the North a contingent punitive tariff is imposed on imports where pNt
Figure 2: Equilibrium in the Southern and Northern Market for good 1
10.2.1.2. Determination of Child Labor Input
In the second stage, the firm in the Southern sector 1 decides about the quantities of child and adult labor it wants to employ for producing its output. It takes as given the prices it obtains for its products if there is free trade or if the North imposes a contingent punitive tariff on its exports due to Southern firm employing child labor.
We assume that the Southern firm is detected in the act of employing child labor with the probability prob = βL1c < 1. The monitoring agency might be, for example, an inter-national organization or a labeling initiative. The probability of being detected depends positively on the parameter β>0 that represents the efficiency of the monitoring systems of the respective agencies. The probability also depends positively on the number of children employed by the firm, L1c. With a probability of βL1c the Southern firm thus faces the contingent punitive tariff imposed in the North and thus the corresponding producer price, pNt/(1+t), whereas with a probability of (1-βL1c) it faces the free trade price, p0.
The production functions in the Southern sectors 1 and 2 respectively are the following:
(26) q1S =(g+L1c)αL1a1−α, 0<α <1 (27) q2S =υ(L2c)ξ +waL2a , 0<ξ <1
where L1c represents child labor input in sector 1 , L1a adult labor input in sector 1 and g is a positive constant productivity parameter; g > 0 implies that the Southern firm is perfectly able to produce good 1 without the use of child labor.
Sector 2 in the South is assumed to be large and competitive. Assume, for example, sec-tor 2 to represent the agricultural secsec-tor where child labor may also be used. Children are thus assumed to find employment in both sectors. L2c denotes child labor input and L2a denotes adult labor input in sector 2 and υ and wa are constant production parame-ters. Normalizing the price of good 2 to unity, we then obtain as the marginal value product of adult labor, wa. Thus the wage rate rate of adult labor in the Southern sector
2 is wa . Assuming that perfect factor mobility prevails in the South, the marginal value product of adult labor will be the same in sector 1.
The firm in the Southern sector 1 then decides about the optimal quantity of child and adult labor by maximizing its expected profit, πs, under the constraint that each subsidi-ary produces exactly one unit of output and takes p0 , pNt
/(1+t) , the costs of one unit of adult labor, wa , and the costs of child labor ,wc, as given. We justify this by the addi-tional assumption that sector 1 is very small compared to sector 2 and thus cannot influ-ence the child wage rate.
The Southern firms thus faces the following optimization problem for each subsidiary i:
(28) c c a a The first order conditions are
(29) ) ( ) 0
This solves for the following optimal labor inputs Lc* and La*.16
(32) g
Taking the first derivative of L1c* in expression (32) with respect to t yields:
Thus, a higher contingent punitive tariff reduces the amount of child labor used in the production of the export good 1. This is because a higher contingent punitive tariff im-posed by the North on the imports from the South causes the Southern exporting firm to reduce the employment of child labor and substitute adult for child labor. Therefore a higher contingent punitive tariff will reduce child labor in the Southern export sector.
Substituting L1c* and L1a* into (28), we obtain the expected profit of a Southern
Using the envelope theorem, we obtain the following expression for t profit of the Southern exporting firm.
The production functions in the two sectors of the North are modeled as follows:
(37) q1N = L1aN (38) q2N = waNL2aN
16 The solution of the optimisation problem proves to be a maximum
The technology of the Northern firms in sector 1 transforms one unit of adult labor de-noted by L1aN into one unit of output. Since each firm in North is able to produce only one unit of output, each firm only employs one unit of adult labor.
In sector 2 in the North good 2 is produced also by using adult labor only. L2aN
denotes adult labor input in the Northern sector 2 and waN is a constant technology parameter.
Normalizing the price of the good produced in the Northern sector 2 to one, we obtain as the marginal value product of adult labor in the Northern sector waN . Assuming per-fect factor mobility in the North, the marginal value product of adult labor will be the same as in the import competing sector 1.
With waN being the costs of one unit of adult labor in the North and one unit of labor is used to produce the one unit of output, the expected profit of a firm i in the North amounts to:
(39) πNi*=(1−βL1c*)(p 0 −waN)+βLc *(pNt −waN)
If the Southern firm is detected in the act of employing child labor, the Northern firms obtain the higher price pNt for their products in the Northern market, whereas they obtain the lower free trade price pN0
with probability (1 - βL1c*).
Lemma 1: For g sufficiently small, a higher contingent punitive tariff t increases the Northern firms' expected profits ,i.e. * 0
∂ >
Taking the first derivate of πNi * in (39) with respect to the contingent punitive tariff t yields
The first term on the RHS above is greater than the second term if
For g = 0 the inequality above boils down to )
Since all parameters are assumed to be positive, this inequality holds for g = 0 and thus for a sufficiently small g.
A higher contingent punitive tariff on imports from the South implies a higher Northern price if the contingent punitive tariff is imposed and thus a higher expected profit for the Northern firms. On the other hand, a higher contingent punitive tariff on imports from the South results in the Southern firm employing less child labor. This in turn increases the probability for the Northern firms of obtaining the lower free trade price, since a higher contingent punitive tariff reduces the incentive for the Southern exporting firm to employ child labor, which has a negative effect on the probability of being detected.
Thus, the expected profit of the Northern firms is reduced. This negative second effect is dominated by the first positive effect for a sufficiently low g . In the following I as-sume g to be sufficiently small for Lemma 1 to hold.
10.2.1.3. Political-Economic Equilibrium
In the first stage of the game, the equilibrium tariff is determined endogenously. We consider an incumbent government in the North which maximizes its political-support by choosing an appropriate contingent punitive tariff (cf. Hillman, 1982). The Northern government takes into account the interests of the political agents in the North and con-tributions from the Northern firms. Concon-tributions can, for example, be used to finance the government’s re-election campaign in the future.
The political support function of the Northern government is assumed to have the fol-lowing appearance:
(40) M = −ωt2 +∆πN *+µCN(t),
where ∆πN*=πN *(t)−πN *(t=0)−CN(t).
The first negative term on the RHS of (40) represents the weighted disutility of North-ern consumers arising from a contingent punitive tariff. This disutility reflects the frus-tration of consumers bearing the tariff. The disutility for t=0 is zero and increases with a higher t at an increasing rate. The second term is the expected increase in profits of the Northern firms if a contingent punitive tariff t is imposed. The third term denotes the political support that can be bought with the political contributions which the Northern firms give to the Northern government in return for a contingent punitive tariff t. We neither consider the profits nor contributions of firms in the Northern sector 2 since this sector is not directly affected. The weight µ put on CN (t) expresses exogenous prefer-ences of the government for political contributions.
Following Wilson (1990), we assume that politicians specify a demand for contributions from interest groups, which means that the government makes the first step and offers the lobby a certain contingent punitive tariff; the government thus acts as a Stackelberg- leader. In our model, we have one interest group: a Northern lobby consisting of all Northern firms. We thus do not analyze why and how the interests of the Northern firms are coordinated. For each contingent punitive tariff the political support maximizing government will demand the highest possible contributions from the lobby, i.e. the con-tributions at which the lobby is indifferent with regard to whether the contingent puni-tive tariff is imposed or not. The lobby thus does not have any bargaining power vis-À-vis the Northern government and pays the contribution demanded by the Northern gov-ernment in exchange for a contingent punitive tariff t. This weak position of the North-ern firms vis-À-vis the NorthNorth-ern govNorth-ernment can be explained either by their large num-ber or by a geographical dispersion of the firms. The lobby is indifferent if the net gain of lobbying equals zero. We thus have:
(41) CN(t)=nN
(
πNi*(t)−πN *(0))
The first term in the bracket on the RHS of (41) denotes the expected profit of a North-ern firms at t >0. The second term in the bracket on the RHS denotes the profit of a Northern at free trade (t = 0). The difference of these two terms is the rent of a Northern firm arising from lobbying for the contingent punitive tariff t. The Northern firms’ net gain of lobbying is obtained if we subtract the costs of obtaining the contingent punitive tariff t, i.e. the contributions of the Northern lobby CN demanded by the government in return for the contingent punitive tariff t, where we assume that each firm pays the same contribution CN/nN.
This implies that the government uses the Northern firms as a vehicle to extract a rent of size CN(t) from the consumers who suffer a disutility that causes a loss of political sup-port amounting to −ωt2. The political support function (40) thus reduces to
(40’) M =−ωt2 +µCN(t) Northern government for contributions is increasing in t.
Differentiating M in (40’) with respect to the contingent punitive tariff, we obtain the following first order condition for a government maximizing its political support.
(42) t
The negative first term denotes the increase in consumers’ disutility attributable to a higher tariff. The second term represents the increase in political contributions, i.e. the change in the Northern firms' expected total profit caused by an increase in the contin-gent punitive tariff weighted withµ. The political support maximizing tariff rate t* thus crucially depends on how easily the government can convert political contributions into political support, i.e. on the parameter µ as compared to ω.
If µ /ω is very small, there will be a corner solution with the optimal contingent puni-tive tariff t*=0. If µ /ω is sufficiently large, there will be a corner solution with the optimal contingent punitive tariff
_
* t
t = , where
_
t is the prohibitive tariff. For interme-diate values of µ /ω the optimal contingent punitive tariff t* will be between 0 and
_
t . In this case we have an interior equilibrium with respect to t . Figure 3 shows with the help of a numerical example how the optimal contingent punitive tariff t* depends on the political weight O.
0 0,5 1 1,5 2 2,5
0 0,5 1 1,5 2 2,5
̅ t*
Figure 3: Numerical example for the correlation between t* and f with wc =3, wa =4, waN = 5, ̒ =1, a1N = 25, a1S =13, nN = 10, nS = 8, ˺ = 0.5, g = 0.01.
10.2.1.4. Comparative Statics
Proposition 1 discusses how an interior equilibrium * (0, )
_
> α and g sufficiently small, then aN
The proof is in the Appendix 10A1.
We now discuss the comparative-static results derived in Proposition 1 in turn.
(1) The political weights:
a higher weight put on Northern contributions as compared to the weight put on North-ern consumers’ disutility by the NorthNorth-ern govNorth-ernment will result in a higher equilibrium contingent punitive tariff.
(2) The wages:
a) Children’s wages
The positive impact of a higher contingent punitive tariff on Northern firms’ expected profits decreases by a higher child wage rate since the Southern firm employs less child labor if the child wage rate goes up. This lowers the probability of the Northern firms of obtaining the higher price pNt
for their products. However, the negative impact of a higher contingent punitive tariff on Northern firms’ expected profits also decreases by a higher child wage rate. This is because a higher child wage rate causes the Southern
firm to employ less child labor, which increases the marginal product of child labor.
Therefore an increase in the contingent punitive tariff will result in less substitution of child with adult labor and hence the expected loss, which the Northern firms incur by an increase in t, is reduced.
At a very low child wage rate there is a high incidence of child labor with a very low marginal product of child labor. Due to the characteristics of the assumed Cobb-Douglas function a small increase in the child wage rate will then have hardly an effect on the substitution of child with adult labor and on the marginal product of child labor.
Thus an increase in the contingent punitive tariff will cause a massive substitution of child with adult labor. In this case the expected loss, which the Northern firms incur by a higher t, is reduced massively and outweighs the decrease in the positive impact of a higher tariff on Northern firms’ expected profits. Thus for a sufficiently small wc ,
wc
t
∂
∂ * will become positive. Thus it is possible that in countries with a not too large incidence of child labor a higher child wage rate will result in a higher equilibrium tariff in the North. In our model we assume that child labor in the South is not too high with wc be-ing sufficiently large. Thus the first negative effect outweighs the second positive effect.
Summed up, a higher child wage rate results in a substitution of child by adult labor, which results in a reduced probability of Southern firms being detected and in the con-tingent punitive tariff-induced substitution of adult by child labor then being less due to a now larger marginal product of child labor. However, due to our assumptions the ef-fect of a lower probability of the Northern firms of obtaining the higher price pNt for their products outweighs the indirect positive effect. Thus the stakes of the Northern firms decrease, which results in a lower equilibrium contingent punitive tariff set by the Northern government.
b) Adults’ wages
The positive impact of a higher contingent punitive tariff on Northern firms’ expected profits increases by a higher adult wage rate since the Southern firm employs more child labor if the adult wage rate goes up. This increases the probability of the Northern