Skip to main content

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


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 %.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3


  1. See Brandt (2005).

  2. In fisheries, for instance, there is an extensive literature on the relationship between tradability of ITQs and consolidation. See, for instance, Grafton et al. (2000), Fox et al. (2003), Kompas and Nu (2005) among others. There is also literature arguing that efficiency gains will be captured only by larger producers. See, for example Libecap (2007) or Olson (2011).

  3. The Kolmogorov–Fokker–Planck equation is widely used to describe population dynamics in ecology, biology, and finance, among other sciences. It has been used in economics by Merton (1975) in neoclassical growth models, by Dixit and Pindyck (1984) in a renewable resources model and by Da-Rocha and Pujolas (2011b) in fisheries. The use of Kolmogorov–Fokker–Planck equation to characterize the distribution of firms was suggested by Dixit and Pindyck (1994). There is a growing literature on general equilibrium models with heterogeneous firms that uses the Kolmogorov–Fokker–Planck equation to characterize the equilibrium invariant distribution of firms as in Luttmer (2007), Da-Rocha and Pujolas (2011a), Luttmer (2011), Luttmer (2012), Impullitti et al. (2013), Gourio and Roys (2014), Da-Rocha et al. (2014a, b, 2015), among others. Two good surveys are Gabaix (2009) and Luttmer (2010).

  4. Those changes are reported in Measure 10 of the Magnuson–Stevens Fishery Conservation and Management Act Provisions; Fisheries of the Northeastern United States; Northeast (NE) Multispecies Fishery; Amendment 16; Final Rule.

  5. We do not consider many other types of distortion or other issues that could appear when regulating the environment with output permits. Examples of such distortions include imperfect enforcement as in Chavez and Salgado Cabrera (2005) and Hansen et al. (2014); international price externalities as in Burguet and Sempere (2010); transboundary resources as Garza-Gil (1998); distributional deadweight losses as in Thompson (2013); market power in intertemporal settings as in Armstrong (2008); and joint-ownership fishing exploitation as Escapa and Prellezo (2003).

  6. This is a standard assumption in models with firm dynamics. See Hopenhayn (1992), Hopenhayn and Rogerson (1993) or Restuccia and Rogerson (2008).

  7. Our technology is in accordance with the fifty–fifty rule, i.e. 50 % of net revenues are accounted for by payments to crew members.

  8. See Dixit and Pindyck (1994) Chapter 4 for a formal definition and justification of these conditions.

  9. The Kolmogorov–Fokker–Planck equation is obtained by applying a simple Markov principle to the transition density function of the continuous stochastic process. Kolmogorov in the 1930’s and Feller at the end of the 40’s characterized the Kolmogorov–Fokker–Planck equation in such a way. For a formal characterization of the forward Kolmogorov equation and its relationship with the Markov stochastic process, see Mangel (2006).

  10. Laplace transforms are given by \(\mathscr {L}[f'(x)] = s \mathscr {L}[f(x)] - f(0)\) and \(\mathscr {L}[f''(x)] = s^2 \mathscr {L}[f(x)] -sf(0)-f'(0)\).

  11. See “Appendix 2”. Dixit and Pindyck (1994) offers an intuitive argument.

  12. With an exogenous reflecting barrier, the solution would be a bounded Pareto distribution. In fact, if it is assumed that \( c \subset [\underline{c},c_*]\) with an exogenous reflecting barrier at \(\underline{c}\), the solution can be obtained by solving a two-boundary value problem. However this solution would depend on the (exogenous) reflecting barrier imposed.

  13. See Kitts et al. (2011).

  14. Infinite support is standard in wealth studies. Heterogeneity in wages is assumed to be Log-normal distributed (with infinite support). Other empirical studies use statistics based on distributions with infinite supports. For example, Zwip’s low (applied to cities, landscape, etc.) is based on GBM with infinite supports.

  15. That is, from a legal standpoint an individual fishing quota is simply a fishing license with a certain tuple of stipulations.

  16. We are indebted to an anonymous referee for suggesting us this exposition.

  17. Those who cease activity cannot lease quota.

  18. Amendment 16 to the Northeast Multispecies Fishery Management Plan (FMP).

  19. Final ACT: Federal Register / Vol. 75, No. 68 / Friday, April 9, 2010 / Rules and Regulations.

  20. See, Measure 10. Magnuson–Stevens Fishery Conservation and Management Act Provisions; Fisheries of the Northeastern United States; Northeast (NE) Multispecies Fishery; Amendment 16; Final Rule.

  21. The rationality of this measure is explained as Removing the cap will facilitate effective use of the leasing program and will provide the ability for some vessels to acquire enough DAS to be profitable. See p 127 of the FINAL Amendment 16 To the Northeast Multispecies Fishery Management Plan. Northeast Multispecies FMP Amendment 16. October 16, 2009

  22. Those changes are similar to the ones reported by Brandt (2005) in the Atlantic Surf Clam and Ocean Quahog Fishery.

  23. Gini coefficients were 0.663 in 2007; 0.678 in 2008: 0.684 in 2009 and 0.76 in 2010. See Kitts et al. (2011) Table 36, page 63 and Figure 21 page 96.

  24. See Kitts et al. (2011) second paragraph of p. 22.

  25. CDF of revenue are characterized in “Appendix 4”.

  26. “Appendix 7” shows the relationship between Lorenz curves and CDF’s via a simple example.

  27. That is, we want to choose \(\xi \) and \(y_*\) so that the revenue Lorenz curve generated by our F(y) is the closest to the revenue Lorenz curve generated by the data.

  28. Note that a Lorenz curve always starts at (0,0) and ends at (1,1), independently of the support of the variable.

  29. Weninger and Just (2002) assumed that c is the unit operating cost. This unit operating cost is assumed to be distributed over a bounded support \([\underline{c}, c_*)\). Therefore, the GBM process (\(\mu \) and \(\sigma \)) and the support, \(\underline{c}\), can be estimated using average variable cost data. They use a sample of 22 vessels from the Mid-Atlantic surf clam and ocean quahog fishery to estimate \(\mu =0.04\), \(\sigma ^2=0.16\) and \(c_*0=.62\).

  30. The support of the distribution is given by the size of \(c_*\), which is endogenous in the model. They find that the abandonment cost threshold is \(c_*=5.57\), 278 % greater than the cost of the most efficient vessel. They set the price at 7.60. Therefore, the size of the revenue, \(p-c\), support is equal to \((7.60-0.62)/(7.60-5.57) \simeq 344\,\%\).

  31. The 2007 and 2010 Lorenz curves were obtained using data from 658 and 450 vessels, respectively.

  32. This is consistent with the empirical evidence. For example Morrison Paul et al. (2009) found significant growth in economic productivity after a property rights-based management reform.

  33. The formulas used for \(c_f\) and \(c_e\) are given in “Appendix 5”, and the formulas used to compute the model equilibrium are given in “Appendix 6”.

  34. See, for instance Restuccia and Rogerson (2008).

  35. See Kitts et al. (2011) Table 26, page 55.

  36. See Kitts et al. (2011) Table 15, page 47 and Table 20, page 51.

  37. Note that, as in Weninger and Just (2002), firms are operating one unit of capital.

  38. It is clear that a firm which is fishing the “minimum” to be considered active \(y(c_*)\rightarrow 0\), finds it optimal to lease its quota without paying the idle cost, rather than paying the idling cost to lease the quota.

  39. We compute the wealth distribution by using W(c) and f(c) for each economy.

  40. “Appendix 7” describes the Brown Formula used to compute the Gini coefficient.

  41. See Diaz-Gimenez et al. (2011).

  42. This ratio for the total US Economy (i.e. including households) is 8.1631

  43. This is a well known result. See Weninger and Just (2002).

  44. We computed wealth distribution in 2010 by using

    $$\begin{aligned} W = \left\{ \begin{array}{lll} p_q q &{} \hbox { if } &{} c \in (c_*^{2010}, c_*^{2007}] \\ W^{2010}(c) &{} \hbox { if } &{} c \in (0, c_*^{2007}] \\ \end{array}\right. {.} \end{aligned}$$
  45. We are indebted to an anonymous referee for this suggestion.

  46. For instance, in Da-Rocha and Pujolas (2011a) heterogeneity comes by differences in the species composition of vessel catches.

  47. Veracierto (2001) founds that capital is not important for understanding the stationary equilibrium. For this reason, the literature refrains from considering capital, or introduces it as a static decision.

  48. See Lai (2007) and Hannesson (1996).


  • Armstrong C (2008) Using history dependence to design a dynamic tradeable quota system under market imperfections. Environ Resour Econ 39(4):447–457

    Article  Google Scholar 

  • Arnason R (2002) A review of international experiences with ITQs: an annex to Future options for UK fish quota management. In: CEMARE

  • Brandt S (2005) The equity debate: distributional impacts of individual transferable quotas. Ocean Coast Manag 48:15–30

    Article  Google Scholar 

  • Brandt S (2007) Evaluating tradable property rights for natural resources: the role of strategic entry and exit. J Econ Behav Organ 63:158–176

    Article  Google Scholar 

  • Burguet R, Sempere J (2010) Trade of permits for greenhouse gas emissions: bilateral trade need not be the answer. Environ Resour Econ 46(4):495–509

    Article  Google Scholar 

  • Chavez C, Salgado Cabrera H (2005) Individual transferable quota markets under illegal fishing. Environ Resour Econ 31(3):303–324

    Article  Google Scholar 

  • Da-Rocha JM, Pujolas P (2011a) Policy distortions and aggregate productivity: the role of idiosyncratic shocks. BE J Macroecon 11(1)

  • Da-Rocha JM, Pujolas P (2011b) Vessel dynamics, itq’s and endogenous vessels distributions. In: Simmons J et al (ed) Impact assessment of multi-annual plans for Southern hake, angler fish and Nephrops (STECF-11-06). EUR—Scientific and Technical Research series—ISSN 1831-9424 (online), ISSN 1018-5593 (print)

  • Da-Rocha JM, Pujolas P, Sempere J (2014a) Eficiencia y equilibrio en un modelo de formación de derechos de propiedad. Trimest Econ 323(3):579–593

    Article  Google Scholar 

  • Da-Rocha J-M, Tavares MM, Restuccia D (2014b) Policy distortions and aggregate productivity with endogenous establishment-level productivity. Working Paper, University of Toronto

  • Da-Rocha J-M, Tavares MM, Restuccia D (2015) Firing costs, misallocation, and aggregate productivity. Working Paper, University of Toronto

  • Diaz-Gimenez J, Glover A, Rios-Rull J (2011) Facts on the distributions of earnings, income, and wealth in the United States: 2007 update. Fed Reserve Bank Minneap Q Rev 34(1):2–31

    Google Scholar 

  • Dixit AK, Pindyck RS (1984) Uncertainty in the theory of renewable resource markets. Rev Econ Stud 51(2):289–303

    Article  Google Scholar 

  • Dixit AK, Pindyck RS (1994) Investment under uncertainty. Princeton University Press, Princeton, NJ

    Google Scholar 

  • Escapa M, Prellezo R (2003) Fishing technology and optimal distribution of harvest rates. Environ Resour Econ 25(3):377–394

    Article  Google Scholar 

  • Fox KJ, Grafton R, Kirkley J, Squires D (2003) Property rights in a fishery: regulatory change and firm performancee. J Environ Econ Manag 46:156–177

    Article  Google Scholar 

  • Gabaix X (2009) Power laws in economics and finance. Annu Rev Econ 1:255–293

    Article  Google Scholar 

  • Garza-Gil MD (1998) ITQ systems in multifleet fisheries. Environ Resour Econ 11(1):79–92

    Article  Google Scholar 

  • Gomez-Lobo A, Peña-Torres J, Barría P (2011) ITQs in Chile: measuring the economic benefits of reform. Environ Resour Econ 48(4):651–678

    Article  Google Scholar 

  • Gourio F, Roys N (2014) Size-dependent regulations, firm size distribution, and reallocation. Quant Econ 5:377–416

    Article  Google Scholar 

  • Grafton R, Squires D, Fox K (2000) Private property and economic efficiency: a study of a common-pool resource. J Law Econ 42:679–713

    Article  Google Scholar 

  • Grainger C, Costello C (2014) Capitalizing property rights insecurity in natural resource asset. J Environ Econ Manag 67:224–240

    Article  Google Scholar 

  • Grainger C, Costello C (2015) The value of secure property rights: evidence from global fisheries. Working Paper 17019, NBER Working Papers Series

  • Hannesson R (1996) Long-term industrial equilibrium in an ITQ managed fishery. Environ Resour Econ 8(1):63–74

    Google Scholar 

  • Hansen L, Jensen F, Nøstbakken L (2014) Quota enforcement in resource industries: self-reporting and differentiated inspections. Environ Resour Econ 58(4):539–562

    Article  Google Scholar 

  • Hopenhayn H (1992) Entry, exit, and firm dynamics in long run equilibrium. Econometrica 60:1127–50

    Article  Google Scholar 

  • Hopenhayn H, Rogerson R (1993) Job turnover and policy evaluation: a general equilibrium analysis. J Polit Econ 101(5):915–938

    Article  Google Scholar 

  • Impullitti G, Irrazabal A, Opromolla L (2013) A theory of entry into and exit from export markets. J Int Econ 90:75–90

    Article  Google Scholar 

  • Kitts A, Bing-Sawyer E, Walden J, Demarest C, McPherson M, Christman P, Steinback S, Olson J, Clay P (2011) Final report on the performance of the northeast multispecies (groundfish) fishery. US Department of Commerce, Northeast Fish Science Center

  • Kompas T, Nu C (2005) Efficiency gains and cost reductions from individual transferable quotas: a stochastic cost frontier for the Australian south east fishery. J Prod Anal 23:285–307

    Article  Google Scholar 

  • Lai Y-B (2007) The optimal distribution of pollution rights in the presence of political distortions. Environ Resour Econ 36(3):367–388

    Article  Google Scholar 

  • Libecap G (2007) Assigning property rights in the common pool: implications of the prevalence of first-possession rules for ITQs in fisheries. Mar Resour Econ 22:407–423

    Article  Google Scholar 

  • Luttmer E (2007) Selection, growth, and the size distribution of firms. Q J Econ 122(3):1103–1144

    Article  Google Scholar 

  • Luttmer E (2010) Models of growth and firm heterogeneity. Annu Rev Econ 2:547–576

    Article  Google Scholar 

  • Luttmer E (2011) On the mechanics of firm growth. Rev Econ Stud 78(3):1042–1068

    Article  Google Scholar 

  • Luttmer E (2012) Technology diffusion and growth. J Econ Theory 147:602–622

    Article  Google Scholar 

  • Mangel M (2006) The theoretical biologists toolbox. Quantitative methods for ecology and evolutionary biology. Cambridge University Press, New York, NY

    Book  Google Scholar 

  • Merton R (1975) An asymptotic theory of growth under uncertainty. Rev Econ Stud 42(3):289–303

    Article  Google Scholar 

  • Morrison Paul C, Torres MO, Felthoven R (2009) Fishing revenue, productivity and product choice in the Alaskan pollock fishery. Environ Resour Econ 44(4):457–474

    Article  Google Scholar 

  • Olson J (2011) Understanding and contextualizing social impacts from the privatization of fisheries: an overview. Ocean Coast Manag 54:353–363

    Article  Google Scholar 

  • Restuccia D, Rogerson R (2008) Policy distortions and aggregate productivity with heterogeneous establishments. Rev Econ Dyn 11(4):707–720

    Article  Google Scholar 

  • Thompson H (2013) Resource rights and markets in a general equilibrium model of production. Environ Resour Econ 56(1):131–139

    Article  Google Scholar 

  • Veracierto M (2001) Employment flows, capital mobility, and policy analysis. Int Econ Rev 42(3):571–595

    Article  Google Scholar 

  • Weninger Q, Just R (2002) Firm dynamics with tradable output prices. Am J Agric Econ 84:562–584

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to José-María Da-Rocha.

Additional information

Helpful comments and suggestions from David Finnoff, three anonymous referees, and conference participants at the World Congress of Environmental and Resource Economics (Istanbul, Turkey) and the 2013 ICES Annual Science Conference (Reikiavik, Iceland) are gratefully acknowledged. Jose Maria Da Rocha gratefully acknowledges financial support from the European Commission (MYFISH, FP7-KBBE-2011-5, N 289257 and MINOUW, H2020-SFS-2014-2, Number 634495), the Spanish Ministry of Economy and Competitiveness (ECO2012-39098-C06-00) and Xunta de Galicia (ref. GRC 2015/014 and ECOBAS). A preliminary version of this paper was circulated as Da-Rocha, Mato-Amboage, R. and J.Sempere (2014) Wealth distribution in markets with output permits: Does transferability increase inequality?


Appendix 1: Proof of Proposition 1

Let J(xt) 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

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

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Da-Rocha, JM., Sempere, J. ITQs, Firm Dynamics and Wealth Distribution: Does Full Tradability Increase Inequality?. Environ Resource Econ 68, 249–273 (2017).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • ITQ
  • Wealth distribution
  • Firm dynamics
  • Inequality
  • Permit markets