ITQs, Firm Dynamics and Wealth Distribution: Does Full Tradability Increase Inequality?

Abstract

Concerns over the re-distributive effects of individual transferable quotas (ITQ’s) have led to restrictions on their tradability. We consider a general equilibrium model with firm dynamics to evaluate the redistributive impact of changing the tradability of ITQs. A change in tradability would happen, for example, if permits are allowed to be traded as a separate asset from ownership of an active firm. If the property right is associated with ownership of an active firm, the permit can be leased in each period but it is not possible to exit the industry and keep the right. However, allowing the permits to be traded as a separate asset has two effects. First, it leads to a greater concentration of production in the industry. Second, it directly converts a non-tradable asset into a tradable one, and this is equivalent to giving a lump sum transfer to all firms. The first effect implies a concentration in revenues, while the second implies a redistribution of wealth. We calibrate our model to match the observed increase in revenue inequality in the Northeast Multispecies (Groundfish) U.S. Fishery. We show that although observed revenue inequality—measured by the Gini coefficient—increases by 12 %, wealth inequality is reduced by 40 %.

Appendices

Appendix 1: Proof of Proposition 1

Let J(x, t) be the Value function associated with the following problem

$$\begin{aligned} J(c,t)=\displaystyle \max _{d \in \left\{ stay,exit\right\} }\left\{ \max _u \int _{t}^{t+dt}\pi (c,u)e^{-\rho s}ds + E_{dc} J(c+dc,t+dt), p_q q\right\} , \end{aligned}$$

where c follows a Geometric Brownian Motion, $dc=\mu c dt+\sigma c dw$, u is a vector of control variables and $p_q q$ is the termination payoff. Using a Taylor expansion, the following can be written

$$\begin{aligned} J(c+dc,t+dt)= & {} J(c,t)+J_t(c,t) dt+J_c(c,t) dc \\&+\,\frac{1}{2}\left\{ J_{tt}(c,t)dt^2+J_{tc}(c,t)dtdc+J_{cc}(c,t)dc^2\right\} . \end{aligned}$$

Using Ito’s calculus and taking limits, $dt\rightarrow 0$, the Hamilton–Jacobi–Bellman equation is obtained

$$\begin{aligned} -J_t(c,t) = \displaystyle \max _{d \in \left\{ stay,exit\right\} }\left\{ \max _u \pi (c,u)e^{-\rho t} + \mu J_c(c,t) +\frac{\sigma ^2}{2}J_{cc}(c,t), p_q q\right\} . \end{aligned}$$

Given that $\pi (c,u)$ is autonomous, there is a stationary solution $J(c,t)=e^{-\rho t}W(c)$. Then the (stationary) Hamilton–Jacobi–Bellman equation is

$$\begin{aligned} \rho W(c) = \displaystyle \max _{d \in \left\{ stay,exit\right\} }\left\{ \max _{u} \pi (c,u)+\mu c W'(c) +\frac{\sigma ^2 c^2}{2}W''(c), p_q q\right\} . \end{aligned}$$

Assume that $c_*$ is the optimal exit point such that

$$\begin{aligned} d=\left\{ \begin{array}{lll} \hbox {stay } &{} \hbox { if } &{} c\le c_* \\ \hbox {exit } &{} \hbox { if } &{} c > c_*, \end{array} \right. \end{aligned}$$

and $\displaystyle \max _{u} \pi (c,u)=\pi (w,r_q) c^{-1}$. Then the optimal policy satisfies

$$\begin{aligned} \rho W(c) = \pi (w,r_q) c^{-1}+(r_q q-c_f)+\mu c W'(c)+\frac{\sigma ^2 c^2}{2} W''(c), \end{aligned}$$

subject to the boundary conditions $W(c_*)=p_qq$ and $W'(c_*)=0$. Guessing that $W(c) =A_1 c^{\beta }+ A_2 c^{-1} + A_3$, the JHB equation becomes equal to:

$$\begin{aligned} \rho \left( A_1 c^{\beta }+ A_2 c^{-1} + A_3 \right)= & {} \pi (w,r_q) c^{-1}+(r_q q-c_f) \\&+\,\mu c \left( \beta A_1 c^{\beta -1}- A_2 c^{-2}\right) \\&+\,\frac{(\sigma c)^2}{2} \left( \beta (\beta -1)A_1 c^{\beta -2}+2 A_2 c^{-3}\right) . \end{aligned}$$

Rearranging terms we have

$$\begin{aligned} 0= & {} \beta ^2-\left( \frac{1}{2}-\frac{\mu }{\sigma ^{2}}\right) \beta +\frac{2\rho }{\sigma ^{2}}, \\ A_2= & {} \frac{\pi (w,r_q)}{\rho + \mu - \sigma ^2 }, \\ A_3= & {} (r_q q-c_f). \end{aligned}$$

Finally we use the boundary conditions

$$\begin{aligned} W(c_*)=\left. A_{1}c^\beta + \frac{\pi (w,r_q) c^{-1} }{\rho + \mu - \sigma ^2 } + (r_q q-c_f) \right| _{c=c_*}= & {} p_q q, \\ c_* W'(c_*)=\left. \beta A_{1}c^\beta + \frac{\pi (w,r_q) c^{-1} }{\rho + \mu - \sigma ^2 } \right| _{c=c_*}= & {} 0, \end{aligned}$$

to obtain $A_1$ and $c_*$. That is

$$\begin{aligned} c_*= & {} \frac{(1+\beta )}{\beta } \frac{\rho }{(\rho +\mu -\sigma ^2)}\left( \frac{\pi (w,r_q)}{\rho p_q q+ c_f -r_q q}\right) , \\ A_1= & {} \left( p_qq-\frac{(r_qq-c_f)}{\rho }\right) \frac{\beta }{1+\beta }\left( \frac{1}{c_*}\right) ^\beta . \end{aligned}$$

Hence, the value function of an individual is

$$\begin{aligned} W(c)=\left( p_qq-\frac{(r_qq-c_f)}{\rho }\right) \frac{\beta }{1+\beta }\left( \frac{c}{c_*}\right) ^\beta + \frac{ \pi (w,r_q) c^{-1} }{\rho +\mu -\sigma ^2}-\left( \frac{c_f-r_q q }{\rho }\right) . \end{aligned}$$

Appendix 2: Proof of Proposition 2

Applying Laplace transforms in $f''(x) - \gamma _1 f'(x) + \gamma _2 f(x) =0$, gives:

$$\begin{aligned} (s^2 -\gamma _1 s +\gamma _2) \mathscr {L}[f(x)]- (s -\gamma _1) f(0)- f'(0)= 0. \end{aligned}$$

Using the boundary condition $f(0)=0$ we find:

$$\begin{aligned} \mathscr {L}[f(x)]= \frac{f'(0) }{(s^2 -\gamma _1 s +\gamma _2)}. \end{aligned}$$

Note that $\gamma _1 = \frac{2 \hat{\mu } }{\sigma ^{2}}>0$ and $\gamma _2 = \frac{2\varepsilon }{\sigma ^2}>0$ implies that only solutions with positive roots can exist. The solution depends on the number of (positive) roots of the equation $s^2 -\gamma _1 s +\gamma _2=0$. It is clear that from this equation, the solution must satisfy

$$\begin{aligned} r_i = \frac{\gamma _1 \pm \sqrt{\gamma _1^2 - 4 \gamma _2} }{2} \ \ \ \forall i=1,2. \end{aligned}$$

(1)

There are then two possible solutions. One implies $r_1\ne r_2=r$ and the other $r_1=r_2=r$. We prove that the first possibility cannot be a solution of our problem by contradiction. Consider a solution with two different roots, so that the discriminant $\gamma _1^2 - 4 \gamma _2$ does not vanish, implying that $\gamma _1^2 \ne 4 \gamma _2$. With two different (positive) roots the solution of the second order differential equation becomes:

$$\begin{aligned} \mathscr {L}[f(x)]=\frac{f'(0)}{(s-r_1)(s-r_2)}. \end{aligned}$$

We obtain the solution by solving the Laplace inverses given by:

$$\begin{aligned} f(x)=\mathscr {L}^{-1}\left[ \frac{f'(0)}{(s-r_1)(s-r_2)}\right] =\frac{f'(0)}{(r_1-r_2)}\left( e^{r_1x} - e^{r_2x}\right) . \end{aligned}$$

Note that $f'(0)=r_1 r_2 \ne 0$, and this implies a contradiction.

Therefore, the solution satisfies $r=r_1=r_2$. This implies that $\gamma _1^2 - 4 \gamma _2=0$, and that $r_i = \frac{\gamma _1 }{2} \ \ \ \forall i=1,2$. This solution gives us

$$\begin{aligned} f(x)=\mathscr {L}^{-1}\left[ \frac{f'(0)}{(s-r)^2}\right] = f'(0)x e^{r x}. \end{aligned}$$

Appendix 3: Proof of Proposition 4

First note that $\displaystyle \int _0^{c_*}c^a g(c)dc=\left( \displaystyle \frac{1+\xi }{1+\xi +a}\right) ^2c_*^a$. Taking expectations, and using the value of $c_*$, we have that

$$\begin{aligned} W^e= & {} \int _{0}^{c_*} W(c)g(c) dc -w c_e- p_q q \\= & {} \frac{(\xi +1)^2}{(1+\beta )(\xi +1+\beta )^2} \left( p_q q -\frac{(r_qq-c_f)}{\rho }\right) +\frac{(\xi +1)^2}{\xi ^2}\left( \frac{(p-r_q)^2}{4 w}\frac{ c_*^{-1} }{\rho +\mu -\sigma ^2}\right) \\&+\,\frac{(r_qq-c_f)}{\rho }-w c_e- p_q q \\= & {} \left( p_q q -\frac{(r_qq-c_f)}{\rho }\right) \left[ \frac{(1+\xi )^2}{(1+\beta +\xi ^2)(1+\beta )}+\frac{\beta (1+\xi )^2}{(1+\beta )\xi ^2}- 1\right] -w c_e. \end{aligned}$$

Then, from the f.o.c. of the entering firm’s problem we have

$$\begin{aligned} \left( p_q -\frac{r_q}{\rho }\right) \left[ \frac{(1+\xi )^2}{(1+\beta +\xi ^2)(1+\beta )}+\frac{\beta (1+\xi )^2}{(1+\beta )\xi ^2}- 1\right] =0 \Rightarrow p_q=\frac{r_q}{\rho }. \end{aligned}$$

Appendix 4: Cumulative Distribution Function

Revenue $y(w,r_q,c) =\displaystyle \left( \frac{1-r_q}{2w}\right) c^{-1}$ is non linear in c. However, the invariant distribution of revenue is a simple change in the power of the invariant cost distribution. That is,

$$\begin{aligned} f(y)= -\frac{(\alpha -1)^2}{ y_*} \left( \frac{y_*}{y}\right) ^{\xi +2}\ln (y_*/y). \end{aligned}$$

We calculate

$$\begin{aligned} F(y)=\displaystyle \int _{y_*}^{y}f(y)dy=\displaystyle \int _{y_*}^{y}-\frac{(\alpha -1)^2}{ y_*} \left( \frac{y_*}{y}\right) ^{\xi +2}\ln (y_*/y)dy. \end{aligned}$$

Trivial manipulation implies that the cumulative distribution function is

$$\begin{aligned} F(y)=\displaystyle \int _{y_*}^{y}f(y)dy= 1- \left( \frac{y_*}{y}\right) ^{\xi +1} \left[ (-\xi -1)\, \ln \left( \frac{y_*}{y}\right) + 1\right] . \end{aligned}$$

Appendix 5: Calibration

We proceed as follows. Given $\mu$ and $\sigma$, first we set $M=1-\Delta \text {fleet}_{2010}$ and $p-r_q= \text {margin}_{2010}$, and we compute

$$\begin{aligned} c_f= & {} \frac{\text {margin}_{2010}}{2 (1-\Delta \text {fleet}_{2010})^2} \left( \frac{\xi }{\xi +1}\right) ^2 \frac{(1+\beta )}{\beta } \left( \frac{\rho }{(\rho +\mu -\sigma ^2)}\right) ,\\ c_e= & {} \frac{1}{\text {entry}}\left[ \frac{1}{(1-\Delta \text {fleet}_{2010})}-2 c_f \left( \frac{\xi +1}{\xi }\right) ^2 \frac{\beta }{(1+\beta )} \left( \frac{(\rho +\mu -\sigma ^2)}{\rho }\right) \right] ,\\ w= & {} \frac{c_f }{\rho c_e}\left[ \frac{(1+\xi )^2}{(1+\beta +\xi )^2(1+\beta )}+\frac{\beta (1+\xi )^2}{(1+\beta )\xi ^2}-1\right] ,\\ c_*= & {} \frac{ (\text {margin}_{2010})^2}{4w}\frac{1}{c_f}\frac{(1+\beta )}{\beta } \left( \frac{\rho }{(\rho +\mu -\sigma ^2)}\right) . \end{aligned}$$

Finally we compute $c_*$ in 2007 to match $\Delta \text {fleet}_{2010}$. That is

$$\begin{aligned} \Delta \text {fleet}_{2010}=\int _{c_*}^{c_*^{2007}}-\frac{(1+\xi _{2007})^2}{c_*^{2007}} \log (x/c_*^{2007}) \left( \frac{x}{c_*^{2007} }\right) ^{\xi _{2007}} dx. \end{aligned}$$

Appendix 6: Solving for the Equilibrium

Given $\mu$ and $\sigma$, the equilibrium, w, $r_q$ $p_q$, M, and $c_*$, are given by the following set of five equations. First, entry condition

$$\begin{aligned} w=\frac{1}{c_e}\left( \frac{c_f}{\rho }\right) \left[ \frac{(1+\xi )^2}{(1+\beta +\xi )^2(1+\beta )}+\frac{\beta (1+\xi )^2}{(1+\beta )\xi ^2}-1\right] . \end{aligned}$$

From the labour market condition, we can obtain the mass of firms M,

$$\begin{aligned} 1 -M \varepsilon \times c_e = M \int _{0}^{c_*} l(c) g(c) dc=M \left( \displaystyle \frac{(p-r_q)}{2 w}\right) ^2 \left( \frac{\xi +1}{\xi }\right) ^2 c^{-1}_*. \end{aligned}$$

From the output market, we have

$$\begin{aligned} 1= M \overline{q} = M^2 \int _{0}^{c_*}y(c) g(c) dc=M^2 \frac{(p-r_q)}{2 w} \left( \frac{\xi +1}{\xi }\right) ^2 c^{-1}_*. \end{aligned}$$

and the maximum cost $c_*$, is

$$\begin{aligned} c_*=\frac{(1+\beta )}{\beta } \frac{(p-r_q)^2}{4 w}\frac{1}{(\rho +\mu -\sigma ^2)}\left( \frac{\rho }{c_f}\right) , \end{aligned}$$

and $p_q$ is such that $p_q=\displaystyle \frac{r_q}{\rho }$. Simple manipulation allows us to find the close-form solution:

$$\begin{aligned} w= & {} \frac{c_f}{\rho c_e} \left[ \frac{(1+\xi )^2}{(1+\beta +\xi )^2(1+\beta )}+\frac{\beta (1+\xi )^2}{(1+\beta )\xi ^2}-1\right] ,\\ \frac{1}{M}= & {} c_e \varepsilon + 2 c_f \left( \frac{\xi +1}{\xi }\right) ^2 \frac{\beta }{(1+\beta )} \left( \frac{(\rho +\mu -\sigma ^2)}{\rho }\right) ,\\ (p-r_q)= & {} 2 c_f M^2 \left( \frac{\xi +1}{\xi }\right) ^2 \frac{\beta }{(1+\beta )} \left( \frac{(\rho +\mu -\sigma ^2)}{\rho }\right) ,\\ c_*= & {} \frac{(p-r_q)^2}{4 c_f w } \frac{(1+\beta )}{\beta } \left( \frac{\rho }{(\rho +\mu -\sigma ^2)}\right) . \end{aligned}$$

Appendix 7: The Computation of Gini Coefficients and Lorenz Curve

In order to compute the Gini coefficients in our calibrations we use the approximation by trapezoids known as Brown’s formula. Formally, define p(n) as the density and P(n) as the accumulated proportion of the population variable, for$n = 0$, with N being the types of individuals differentiated by wealth (and ordered from least to greatest wealth), with $P(0) = 0$ and $P(N) = 1$. Define as $w = 0{\ldots }W$ the different wealth levels (where wealth is ordered in a non decreasing fashion) and let f (w) be the density and F(w) be the cumulative proportion of the wealth variable. The Gini coefficient can then be defined as

$$\begin{aligned} Gini=1-\sum _i (P(i)-P(i-1))(F(i)+F(i-1)) \end{aligned}$$

An application that measures the effect of a lump sum transfer on the Gini coefficient is presented in Table 9. Column 1 is the amount transferred. Column 2 is the proportion of the population in each wealth level. Columns 3 and 4 are the wealth levels before and after the transfer, respectively. Column 5 represents the cumulaivte distribution of people and columns 6 and 7 the cumulative distribution of wealth before and after the transfer. The rest of the columns are helpful in computing Brown’s formula. It is immediately apparent by straightforward application of the formula that the Gini coefficient is 0.44 before the transfer and 0.22 after it.

Table 9 Gini Index: impact of a transfer

The Lorenz curve plots the cumulative proportion of wealth as a function of the cumulative proportion of the population. Table 10 shows the calculation and the effect on the Lorenz curve of the transfer discussed in Table 9. As before, Column 1 is the amount transferred. Column 2 is the proportion of the population in each wealth level. Columns 3 and 4 are the wealth levels before and after the transfer, respectively. Column 5 and 6 represent the proportion of wealth belonging to each type of agent before and after the transfer. The Lorenz curve corresponding to the case without the subsidy plots column 7 in the horizontal axis and column 8 in the vertical axis (the Lorenz curve corresponding to the economy with the subsidy is symmetrically defined using column 9).

Table 10 Lorenz curve: impact of a transfer