EGA MMGRATON, EMPOYER SANCTONS AND EQUBRUM Chisato YOSHDA aculty of Economics Okayama University 7th June, 2003 Abstract We extend the seminal Bond and Chen 987 two-country model of illegal immigration using a constant elasticity of substitution CES production function. The objectives of this paper are to prove the existence of equilibrium both when capital is mobile and when it is immobile between two countries, and to re-evaluate the effects of restrictive internal inspections on domestic firms who intentionally hire illegal foreign labor. We find equilibrium under both capital mobility and immobility. When capital is mobile, if the equilibrium is saddle point stable, then an increase in inspections increases illegal immigration and decreases capital exports. JE classification No.: 2, 22 eywords: illegal immigration, employer sanctions, CES production function, equilibrium, saddle point stable Correspondence to: Dr. Chisato Yoshida aculty of Economics Okayama University 3-- Tsushima-Naka Okayama 700-8530, Japan Tel/ax: 8-86-25-7525 E-mail: cyoshida@e.okayama-u.ac.jp would like to extend my deep appreciation to Professor Alan D. Woodland for his profound observations, and my gratitude to Mark Melatos and ishti Mohan Sen for their insightful comments. also thank the University of Sydney for hosting me from May 3, 2000 to May 30, 200. Earlier drafts of this paper were presented at several seminars and conferences, and benefited from the comments and discussions of seminar and conference participants. n particular, would like to thank Professors Takao ujimoto, Caremen Herrero, ernando Vega Redondo, Ramon auli Oller, and Pepe, as well as other seminar participants at the Economics Seminar in the Departamento de undamentos del Análisis Económico, Universidad de Alicante, Espania, on September 24, 200; participants at the Seminar of nternational Capital Movement at ansai University, Osaka, Japan on December, 200; Professors Mary E. Edwards, ostas Axarloglou, and Börje Johansson for their helpful comments at the 53rd nternational Atlantic Economic Conference held in Paris on March 3-7, 2002; and Professor Alessandra Venturini for her penetrating comments at the V Convegno Nazionale di Economia del avoro, Salerno, talia, on September 27, 2002. The present version is to be presented at the 2003 Congress of the European Economic Association, to be held in Stockholm for 20-24 August, 2003. The financial support of a Grant-in-Aid for Scientific Research No.473005 is gratefully acknowledged. Any remaining errors are, of course, mine.
EGA MMGRATON, EMPOYER SANCTONS AND EQUBRUM. ntroduction The inequalities that exist between rich and poor countries in both wages and in employment opportunities have caused a mass migration of residents from poorer countries into wealthier ones, though wealthier countries have tended to restrict this inflow of foreign migrants. This has caused illegal immigration from poorer, less developed countries to wealthier, developed countries. llegal immigration is usually considered the problem of the labor-importing or host country. The governments of developed countries deal with the problem of illegal foreign workers mainly by using two enforcement tools: border patrol and internal enforcement. Border patrol involves the prevention of attempted illegal migration at the border, while internal enforcement, also called employer sanctions, involves catching foreign workers who are working illegally in domestic firms. Sanctions against employers have been introduced in the United States U.S., Japan, and several European countries, including Germany, rance, Austria, the Netherlands, Sweden, Norway, taly, and Hungary see Brochmann and Hammar 999 and rlenkaueuser 200. Ethier 986a, b broke new ground by analyzing the effects of border and internal enforcement policies in a one-country model using a crime-theoretic analysis Becker 968. n his model, Ethier examined how a small country could use domestic border controls and internal enforcement policies to achieve domestic policy objectives with respect to illegal immigration and income distribution. He also demonstrated that the wages of legal domestic workers might rise with stronger internal enforcement. Bond and Chen 987 extended the Ethier model by constructing a standard, two-country, one-good, two-factor model. They found that when capital is immobile, internal inspections increase domestic wages, while foreign wages decrease since
enforcement reduces the level of illegal immigration. They also showed that the imposition of enforcement policy by the labor-importing country's government might increase the host country s welfare if the host country is large enough to influence foreign wages and if the marginal costs of enforcement are sufficiently low. However, they demonstrated that when capital is mobile between the two countries, stricter immigration policies cause both domestic and foreign wages to increase Yoshida 200 reintroduced the seminal Bond and Chen model using a Cobb-Douglas production function and a specification of the probability with which illegal foreign workers are detected by the host country s immigration authority. That paper, which assumed that capital was mobile between the two countries, had the same results as Bond and Chen with respect to the effects of severe employer sanctions on domestic and foreign wages. The three papers mentioned above implicitly assumed that a stable equilibrium existed within their respective model. The present paper extends the Bond and Chen model, substituting a constant elasticity of substitution CES production function. The objectives of this paper are to prove the existence and define the characteristics of the equilibrium both when capital is mobile and when it is immobile between two countries, and to verify the effects of restrictive internal inspections on domestic firms who intentionally hire illegal foreign workers. We discover that there is a unique equilibrium that is a stable node, when capital is internationally immobile. We also find that in the absence of capital mobility, higher levels of enforcement result in lower levels of illegal immigration resulting in benefits to domestic labor and losses to foreign labor in terms of their wage rates. This result is similar to that of Bond and Chen. We also ascertain that there are a variety of equilibria when capital is mobile between the two countries. n an equilibrium that is saddle point stable, a rise in internal 2
enforcement causes illegal immigration to rise and capital outflows to decline. This outcome is the inverse of that of Bond and Chen, in which the equilibrium is likely to be a stable node. We also know that stricter internal enforcement causes domestic and foreign wages to decrease when the equilibrium is saddle point stable. This result is also the opposite of that of Bond and Chen 987, in which both domestic and foreign workers gained due to stricter enforcement. Before presenting our model, we briefly review other important theoretical models of illegal immigration. Bucci and Tenorio 996 analyzed the effects of illegal immigration on a host country s income when an immigration reform, such as employer sanctions, is undertaken using alternative financing instruments. Their analysis utilized a one-small country model and incorporated a government budget constraint into an illegal immigration model similar to Ethier s. Bandyopadhyay and Bandyopadhyay 998 modeled the source country of illegal immigration and employer sanctions at length. n a recent paper, Gaytan-regoso and ahiri 2000 explored the effect of foreign aid on illegal immigration by developing a new model in which the decision to migrate is made by families, with no risks of apprehension by border patrol or employer sanctions for illegal immigrants. Gaytan-regoso and ahiri 200 also extended their two-country model to analyze a connection between commodity trade and illegal immigration, in which illegal immigration is a substitute for trade between countries. Agiomirgianakis and Zervoyianni 200 have introduced an open economy with heterogeous labor which imports foreign irregular unskilled workers. They showed that when skilled native wage setters and policymakers in the economy act as Nash players, illegal immigration causes the position of entire native labor force which involves both skilled and unskilled workers to be better off. On the other hand, they concluded that when the native skilled workers and the policymakers perform cooperatively, illegal immigration might make the entire native workforce worse off. 3
However, none of these papers checked the existence and characteristics of their equilibrium. This paper examines the existence of equilibrium in a model of illegal immigration and seeks to determine the characteristics of the equilibrium that exists. We are able to identify the existence of equilibrium and the peculiarities of the equilibrium in a model that uses a CES production function. The next section extends the Bond and Chen model of illegal immigration under capital immobility using a CES production function. This section verifies the existence and uniqueness of an equilibrium, and examines the impacts of enforcement on the domestic and foreign economies. n Section 3, we extend our model to the case of capital mobility, evaluate the existence of the equilibrium and the characteristics of the equilibrium, and assess how capital mobility alters the impacts on the two countries arising from stricter enforcement. The final section offers some concluding remarks. 2. The Basic Model We extend the Bond and Chen two-country, one good, two-factor model of illegal immigration, in which there are a labor-importing capital abundant country and a labor-exporting labor abundant country, substituting a CES production function for their production function. We consider the model under capital immobility, in which there is illegal labor movement between the two countries. We assume workers in both the host and the foreign countries are fully employed. ollowing Bond and Chen, we assume that the domestic country s government enforces employer sanctions against domestic firms who purposely hire illegal foreign-migrant workers, and the probability that an illegal immigrant worker is arrested increases with the effort that the authority devotes to internal inspection: / β p = E / R, p '> 0, p '' < 0, p 0 = 0, p, 4
where E 0 is the effort expended on internal enforcement by an immigration authority, R > 0 is the initially endowed national budget of the host country, and β is a parameter that is larger than unity. We assume that the domestic country's Congress allocates a part of the national budget, E, to the immigration authority, and hence, E R. When E = 0, the host country s government adopts laissez-faire on the inflows of illegal migrants, whilst a case in which E = R implies the probability of detection of the migrants is unity. The immigration authority behaves as a budget-maximizing government bureau see Davila et. al 999. 2 Both domestic and foreign firms have CES production functions, respectively: and / [ ] =, 2. / = A, 2.2 where variables and functions with a refer to the foreign country. n these equations, and are the levels of output in the domestic and foreign countries; and are the levels of domestic and foreign real capital; denotes the domestic employment of Because technologies are assumed to differ between the countries, there is a disparity in wages between the areas, even if E = 0. 2 Bandyopadhyay and Bandyopadhyay 998 assumed that enforcement expenditure, E, is a function of the percentage of illegal immigrants in the labor force of the host country. This implies that higher numbers of illegal immigrants result in higher levels of enforcement. However, since the aim of this paper is to re-examine the impacts of enforcement, we follow Bond and Chen and assume ormula holds. 5
both domestic labor and illegal foreign labor, while is the number of foreign workers employed in the foreign country; and are positive constants with < < and 0 < <, and and level contained in the interval 0,. are defined in a similar way; and A is the foreign technology ike Bond and Chen, we assume that a domestic firm is indifferent between hiring legal domestic and illegal foreign workers. This implies that the cost of employing a domestic worker is equal to that of employing an illegal foreign worker: w = p w p w z = w p z, 3 where w and w are the domestic wages of legal domestic and illegal foreign labor, respectively; and z is the penalty paid by a domestic firm for each arrested illegal alien. The number of illegal aliens employed in domestic firms is determined so that formula 3 is satisfied. We regard w and w as endogenous, since the level of border patrol is assumed to be fixed throughout the paper see Bond and Chen 987, p. 37 and Yoshida 2000, p. 37 for details. The first order conditions for cost-minimization by domestic and foreign firms are: w = /, 4..a r = /, 4..b w = A /, 4.2.a 6
r = A /, 4.2.b where we define the initial endowments of the host country as and ; the labor and capital endowments, respectively, of the foreign country as and ; the wages of domestic and foreign labor as w and w ; the rental price of domestic and foreign capital as r and r ; and the successful number of illegal immigrants as. 3 Equations system D, which consists of 3 and 4..a 4.2.b, simultaneously determines the equilibrium values of, w, r, w and r. To confirm the uniqueness of the equilibrium, we consider the following adjustment procedure: = w w p z α, 5 where is a time derivative, d / dt, and α > 0 is a constant. The number of illegal immigrants who are not caught increases over time when the home firm can hire the immigrants at a lower cost than that of the domestic worker, i.e. > 0 when w > w p z. Proposition : Given the adjustment procedure defined in 5, and capital immobility, there exists a unique equilibrium in which a positive number of immigrants is employed illegally by domestic firms, i.e. > 0. 3 The number of illegal immigrants is defined as the number of foreign workers who are employed illegally in domestic firms and not caught and deported as a result of internal inspections. 7
Proof: See Appendix. Hence, the dynamic adjustment of the number of illegal immigrants can be depicted in a phase diagram. O E igure We confirm the existence of a stable equilibrium, E, from the phase diagram in igure. 4 Note that = 0 at the point E. et be such that = 0, denoted by > 0, which is an interior solution to system 5. Once the number of illegal immigrants,, is determined, the levels of = and = are also determined, because we assume that the labor force located in each country is fully employed. Because capital in each country is also assumed to be fully utilized, = and =. We know from 2. and 2.2 that the levels of output, 8
and, are fixed. We can therefore verify from 4..a 4.2.b that there is an interior solution, w, r, w and r. Referring to 3 and 4..a 4.2.b, we define the number of illegal immigrants,, the wages of domestic and foreign labor, w and w, and the rental prices of domestic and foreign capital, r and r as functions of the level of employer sanctions, E : E, w we, r re, w w E, and r r E. 6 We can easily obtain the effects of stricter internal inspections on w, r, w, r and : d / de < 0, dw / de > 0, dr / de < 0, dw / de < 0, and dr / de > 0. 7 These comparative static results are similar to those of Bond and Chen. 5 We have verified, as a result, that in the presence of capital immobility we obtain the same impacts on the two countries arising from employer sanctions as those of Bond and Chen, even with the use of a CES production function. 3. Capital Mobility n this section we extend the Bond and Chen model by introducing capital mobility, and continue to use the CES production function of the previous section. Domestic capital migrates to the foreign country until the return, net of taxes, is equal to the return to 4 See ujimoto 989 for details with respect to the stability of equilibrium. 5 See pages 37-320 in Bond and Chen for the comparative static results. The reader can readily verify that they obtain the same results as those of Bond and Chen, even if the production function is Cobb-Douglas. 9
domestic capital employed in the domestic country: r = r t, 8 where t is the tax rate, contained in 0,. With capital mobility, the factor market equilibrium conditions in the domestic and foreign countries become: w = /, 9..a r /, 9..b = w = /, 9.2.a = / r, 9.2.b where production technologies are assumed to be the same in both countries. The six unknown variables,,, w, r, w and r are endogenously determined from equations 3, 8 and 9..a - 9.2.b. We first ascertain whether there is an interior solution, and then consider a dynamic adjustment process: = w w p z c, 0.a 0
= t r r c2, 0.b where c > 0, i =, 2 is a constant. 0.a is essentially identical to 5. i We now obtain the phase diagram illustrated in igure 2, from which the global stability of the point E,, can be easily seen see Appendix 2 for particulars of igure 2. '= 0 E 2 = 0 E '= 0 = 0 O igure 2 rom igure 2, we see a global change in the level of illegal immigration and capital exports resulting from a large increase in the level of enforcement. This figure demonstrates that the equilibrium shifts from E to E 2 when enforcement is increased. Although it is impossible to stem the rise in illegal immigration caused by tighter enforcement in igure 2, we realize from system 0 that this cannot happen. Stronger
enforcement causes illegal immigration to decrease, causing an increase in the labor-force in the foreign country, which is further increased by the deported illegal aliens. The increase in the foreign labor-capital ratio causes the foreign rental price of capital to rise, which results in domestic capital moving to the foreign country. Considering the previous continuous-time dynamical system, the roots of a characteristic equation are given by 2 trj ± trj 4det J λ,2 =, 2 where λ, 2 are eigenvalues see Appendix 2. Referring to orenz 993, we show various equilibria in the model because we cannot determine the signs of η i, i =,2, 4 without any sufficient conditions, which are constituents of the Jacobian matrix J. Note also that the sign of η 3 is always positive. Proposition 2: There are all sorts of equilibria, a stable node, an unstable node, a saddle point, a stable focus, an unstable focus and center dynamics, when capital is allowed to be mobile between the two countries. Proof: See Appendix 3 for details on those equilibria. Next, we examine the effects of severe internal enforcement on the six endogenous variables under i.a or i.b.2, referring to Appendixes 3 and 4. irst we consider the former case, in which the equilibrium is a stable node. We find from A.7 and A.8 that when employer sanctions are increased, the level of 2
illegal immigration decreases and capital exports increase see igure 2 and case.a in Appendix 3 on the effects of a global increase in enforcement, and also Appendix 4 for mathematical calculations on comparative static effects of stricter internal inspections. However, we obtain the opposite effects on illegal immigration and capital outflows when we assume that the equilibrium is saddle point stable see i.b.2 in Appendix 3 and Appendix 4. Under a saddle point stable equilibrium, the sign of 2 is positive, and an increase in the enforcement level of employer sanctions causes illegal immigration to rise and capital exports to decrease see igure 3 concerning the global impacts on those factor movements. ' = 0 ' = 0 E S E 3 S E 2 = 0 = 0 O igure 3 As shown in igure 3, an initial steady state is represented by the point E, and a new steady state is given by the point E 3. To interpret the figure, note first that an increase 3
in expenditures on employer sanctions, E, results in a higher number of arrests of illegal immigrants, who are then deported. This causes the [ = 0] locus to shift inward. The repatriation of illegal migrants decreases the foreign wage rate and increases the domestic wage rate, which causes more foreign workers to migrate. Therefore, in terms of 0.a, the level of illegal immigration must be increased to uphold the equality w = w p z. As a result, the [ = 0] locus moves outward to the [ ' = 0] locus. Similarly, higher levels of enforcement result in a decrease in illegal immigration and an increase in the outflow of domestic capital to the foreign country. This makes the [ = 0] locus shift outward, which forces the domestic rental price to rise and the foreign rental price to fall. Considering 0.b, then < 0, from which the [ = 0] locus shifts inward to the [ ' = 0] locus. The new equilibrium point E 3 is associated with more illegal immigration and less capital outflow owing to tougher employer sanctions. Capital exports immediately jump down to point E 2 on a new saddle-path SS and keep increasing at the same time that illegal immigration is rising see Shone 997 for a saddle-path. Proposition 3: When an equilibrium with capital exports is saddle point stable, tighter employer sanctions induce more illegal immigration and less capital exports. n the traditional literature on illegal immigration, such as Ethier 986, Bond and Chen 987, and Bucci and Tenorio 996, it was well accepted that austere internal enforcement reduces illegal immigration. nvestigating the characteristics of an equilibrium in the presence of capital mobility, we find that the equilibrium can be a 4
saddle point as well as a stable node. Although we find the same impact on illegal immigration as in earlier works when the equilibrium is a stable node and capital is mobile, we discover the reverse effect if the equilibrium is saddle point stable. t is impossible to determine the signs of A.9 through A.2 without knowing whether the equilibrium is a stable node or saddle point stable. We first consider the case of the stable node see Appendix 4. We know from A.9 through A.2 that factor prices in both countries, w, r, w and r, are higher or lower due to stiffer enforcement, according to the size of d / de and d / de, considering some sufficient conditions to ensure the local stability of equilibrium, i.e., η < 0, η 2 > 0 and η 4 > 0. 6 When stricter weaker enforcement increases decreases the domestic labor-capital ratio, /, i.e., d / de < d / de, the domestic wage, w, falls rises and the domestic rental price r rises falls. f the foreign labor-capital ratio, / is larger, i.e., d / de > d / de, owing to tighter internal inspection, the foreign wage, w is lower, and the foreign rental price, r is higher, and vice versa. Bond and Chen stated that a necessary condition for the equilibrium to be stationary is / / > 0. t is possible that the magnitudes of / and / become smaller or larger depending 6 We define the labor/capital ratios in both countries as The total differentiation of f. and f.2 is: / = /, f. / = /. f.2 d / / de = { d / de / d / de}/, f.3 d / / de = { d / de / d / de}/. f.4 We know from f.3 and f.4 that the signs of d / / de and d / / de are negative and positive, respectively, only if d / de is larger than d / de, from which tighter looser enforcement causes w and r to increase decrease and r and w to decrease increase. 5
upon the sizes of d / de and d / de owing to austere enforcement as long as / / > 0, because capital exports and the deportation of illegal immigrants rise simultaneously, and illegal immigration falls. We generally conceive from a dual theorem of cost-minimization that the labor-capital ratio determines both wages and the rental price of capital. We can also explain the impacts of stricter internal enforcement on factor prices in a similar way to the case of i.a, when the equilibrium is saddle point stable. We readily observe from A.7 and A.8 that the impacts on the prices in the case of i.b.2 are opposite to those in the case of i.a. As described in equations A.9 - A.2, the effects of stricter employer sanctions on all factor prices are decomposed into the impacts on illegal immigration and capital exports. Those effects change according to the size of d / de and d / de. Proposition 4: When there is capital mobility between a labor-importing host country and a labor exporting country, the equilibrium is a stable node when and are not both zero, domestic labor and foreign capital gain, and foreign labor and domestic capital lose, provided that d / de > d / de, and vice versa. Remark: f the equilibrium is saddle point stable, then the same conditions as before will result in effects on factor prices that are the reverse of those arising from a stable node. Bond and Chen 987 concluded that in the presence of capital mobility, severe internal enforcement causes domestic and foreign wages to increase significantly. This implies that once the labor-force in the labor-exporting country increases due to 6
repatriated illegal aliens, foreign wages decrease while the foreign rental price increases. The increase in the foreign rental price attracts domestic capital, which results in greater domestic capital exports to the foreign country, thereby reducing the foreign labor-capital ratio and increasing foreign wages and the welfare of foreign workers. When we use the CES production function so as to consider an elasticity of substitution between labor and capital, we are only able to determine whether the foreign wage rises or falls due to a change in enforcement level. As shown in our propositions, we can resolve the impacts of stiffer enforcement on factor prices, w, r, w and r into their relative effects on illegal immigration and capital exports, but we are unable to determine how factor prices change in terms of relative impacts when utilizing the general CES function also see footnote 6. However, we can verify the same comparative static effects as Bond and Chen when we assume = = 0, which reduces the CES function to the special case of a Cobb-Douglas production function. 4. Conclusions We extended the Bond and Chen 987 two-country model of illegal immigration, using a CES production function. We explored whether there is equilibrium, both when capital is mobile and when it is immobile between the two countries. n the presence of capital immobility, uniqueness of a stable equilibrium requires that the sign of µ µ 2 be negative. n the absence of capital mobility, stricter internal inspection causes illegal immigration to decrease, which is similar to the result of Bond and Chen. Capital mobility, however, complicates the outcome. We found that under capital mobility, there are six equilibria which are, in turn, stable node, unstable node, saddle point, stable focus, unstable focus and neutrally stable. We discovered this variety of equilibria using the dynamic adjustment procedure 0, which has not been analyzed in any preceding 7
research on illegal immigration. or the saddle point stable equilibrium under capital mobility, stricter internal enforcement results in an increase in illegal immigration and a decrease in capital outflows. This outcome is contrary to that of Bond and Chen, in which an increase in enforcement causes a decrease in illegal immigration and an increase in capital exports. urthermore, Proposition 2 stated that when the equilibrium is a stable node saddle point stable strengthening internal inspections increases decreases the price of domestic labor and foreign capital and decreases increases the price of domestic capital and foreign labor, in accordance with the size of illegal immigration and capital exports. These results differ from those of Bond and Chen, in which tougher employer sanctions result in increases in both domestic and foreign wages and decreases in the capital rental price in both countries. f we introduce the CES production function in the same framework as Bond and Chen, then we can resolve the effects of enforcement on each country s factor prices into those on the sizes of illegal immigration and capital inflows, unless the elasticity of substitution is unity. f we set the elasticity of substitution to one, then we obtain the same effects on factor prices as Bond and Chen. These results have not been noted in earlier papers on illegal immigration. 8
Appendix : Proof: or the line arising around the steady state, we obtain: = µ µ 2 d, A. where µ = / / { / }, µ = A 2 / / { A / }. f the sign of µ is negative, then there is a unique equilibrium value,. n other µ 2 words, this means the gradient of is negative in a space. What µ µ 2 0 means < the sum of a weighted output per worker in each country is less than unity. Appendix 2: Total differentiation of 9..a 9.2.b and the linearisation of 0.a and 0.b around the solutions, and yields, in matrix form: η η2 = d, A.2 η3 η4 d 9
20 where 4 3 2 η η η η J, and } / { / / = η } / { / /, η = 2 / / / { } / / /, η = 3 / / / 0 / / / > t, } / { / / [ 2 4 = η }] / { / / 2 t. rom A.2, we derive:
det J = η η4 η2 η3, A.3 and trj =η η 4. A.4 f trj < 0 and det J > 0, then there exists an equilibrium point, such that = 0 = is locally stable. The sign of trj is negative only if η < 0 and η 4 > 0, in which case det J 3 > is positive only if η 2 > 0, since η 0. Hence, there exists at least one equilibrium point,,, if these sufficient conditions are satisfied. Note that η 0, η 0 and η 0, < 2 > 4 > when both and are zero. Once there are interior solutions,,, we can verify the existence of other equilibrium solutions, w, r, 9..a 9.2.b. 7 We find from A.2 that w and r, referring to 2 and d d = η / η > 0, A.5 / 2 = 0 d d = η / η < 0. A.6 / 3 4 = 0 Note from A.5 and A.6 that the slope of the = 0 locus is positive, and the slope of 7 See ujimoto 989 concerning details on the stability of equilibrium. 2
the = 0 locus is negative. Hence, the two curves [ = 0] and [ = 0] intersect at least once in space. Appendix 3: We show equilibria in turn which occurs in our model: i Real roots: i.a These eigenvalues determine the dynamical behavior of the system 0. The 2 eigenvalues λ, 2 in are real when the discriminant, = trj 4det J, is positive or equal to zero. A positive determinant, det J > 0, implies that both eigenvalues have the same sign. f the trace of J is negative, both eigenvalues are negative, and the trajectory of the system monotonically approaches a finite point,. The point E in igure 2 is called a stable node. As a result, we can verify that there are other interior solutions for w, r, w and r from formulas 2 and 9..a 9.2.b. We should note that if we assume = = 0, then η 0, η 0 and η 0, the CES production function reduces to < 2 > a Cobb-Douglas production function, and the case of i.a results. 4 > i.b. System 0 also presents the case of an unstable node when the sign of trj is positive, which occurs when η > 0, and η 4 < 0, and hence det J > 0 only if η 0. 2 > i.b.2 A negative sign for det J can arise, if η η η η 3 0, and only if η 0, η 0 4 2 > < 2 < and η 4 > 0. t is therefore possible in the present system 0 for the sign of discriminant 22
to be positive, in which case the equilibrium is saddle point stable. This produces contrary effects on illegal immigration and capital mobility to those in case i.a in which the equilibrium is a stable node. ii Complex roots: ii.a The sign of det J is positive when the symbol for η is the reverse of that for η 4, and η 0, which brings about 0, only if trj 2 < 4det J. Thus, we are aware of the 2 > < existence of a stable or unstable focus in the system 0 when Re λ i is negative or positive. ii.b We can also consider the case when Re λ is zero in, which is called the center dynamics or neutrally stable. We have shown that there are all sorts of equilibria that have not been previously investigated, as in cases i.a ii.b. Appendix 4: Differentiation of equilibrium conditions 3, 8 and 9..a 9.2.b with respect to E produces the following effects on,, w, r, w and r : β / β / 4 2 d de = { z / R β E / R } η /, A.7 β / β / 3 2 d de = { z / R β E / R } η /, A.8 23
24 / / / 2 de dw = / } / { [ de d ] / / de d, A.9 2 / / / de dr = / / [ de d ] / } / { de d, A.0 2 / / / de dw = / } / { [ de d ] / / de d, A. 2 / / / de dr = / / [ de d
{ / } d / de], A.2 where. 2 = η η4 η2 η3 25
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