Partial privatization in an international mixed oligopoly under product differentiation

Abstract

We consider an international mixed market that comprises two countries, each of which owns one public firm and one private firm. As a benchmark case, we consider a single country that owns all four firms. In both cases, governments decide whether to partially privatize their public firms or not. Under an international mixed market privatization decisions are driven by strategic reasons, while in the benchmark case they are driven by efficiency reasons. We find that whether governments privatize more or less in the former case than in the latter depends on the type of goods produced by the firms (homogeneous, independent in demand, complements and substitutes). We also find that social welfare may be greater under an international mixed market than in the benchmark case. Finally, under an international mixed market there is more privatization when each public firm produces the same good as the domestic private firm than when they produce different goods.

Appendix

1.1 Case 1: an international mixed market

$$\begin{aligned} q_{i}^{kA} & = a(b^{2} \beta^{k} (12 - 5\beta^{l} ) - 5(6 - \beta^{k} )(6 - \beta^{l} ) + 2b(6 - 5\beta^{k} )(6 - \beta^{l} ))/G_{1} , \\ q_{i}^{kB} & = a(12b(6 - \beta^{l} ) - 5(6 - \beta^{k} )(6 - \beta^{l} ) - b^{2} \beta^{k} (12 - 5\beta^{l} ))/G_{1} , \\ \end{aligned}$$

$$\begin{aligned} W^{k} & = ((2a^{2} (75(6 - \beta^{k} )^{2} (6 - \beta^{l} )^{2} - b^{4} (\beta^{k} )^{2} (12 - 5\beta^{l} )^{2} - 10b(6 - \beta^{k} )^{2} (6 - \beta^{l} )(21 - 5\beta^{l} ) \\ & \quad + \;6b^{3} (12 - \beta^{k} (13 - 5\beta^{k} ))(6 - \beta^{l} )(12 - 5\beta^{l} ) - 6b^{2} (9(192 - \beta^{k} (244 - 67\beta^{k} )) \\ & \quad + \;15(60 - \beta^{k} (74 - 17\beta^{k} ))\beta^{l} - 5(15 - \beta^{k} (23 - 5\beta^{k} ))(\beta^{l} )^{2} )))/(G_{1} )^{2} , \\ & \quad {\text{where}}\;G_{1} = b^{2} (12 - 5\beta^{k} )(12 - 5\beta^{l} ) - 25(6 - \beta^{k} )(6 - \beta^{l} ),\; i \ne j;\;i,j = 1,2;\;k \ne l;\;k,l = I,II. \\ \end{aligned}$$

1.2 Case 1: a single country

$$q_{i}^{kA} = 3a(15 - 4\beta^{l} - b(6 - \beta^{l} ))/G_{1} ,\;q_{i}^{kB} = a(3 - \beta^{k} )(15 - 4\beta^{l} - b(6 - \beta^{l} )/G_{2} ,$$

$$\begin{aligned} W^{I} + W^{II} & = a^{2} (2b^{3} (6 - \beta^{k} )^{2} (6 - \beta^{l} )^{2} + (\beta^{k} )^{2} (2853 - 1464\beta^{l} + 160(\beta^{l} )^{2} ) \\ & \quad - \;6\beta^{k} (4185 - 2160\beta^{l} + 244(\beta^{l} )^{2} ) + 9(5400 - 2790\beta^{l} + 317(\beta^{l} )^{2} ) \\ & \quad - \;6b^{2} ((\beta^{k} )^{2} (30 - 10\beta^{l} + (\beta^{l} )^{2} ) + 6(144 - 48\beta^{l} + 5(\beta^{l} )^{2} ) - 2\beta^{k} (144 - 48\beta^{l} + 5(\beta^{l} )^{2} )) \\ & \quad - \;6b(\beta^{k} (897\beta^{l} - 1854 - 85(\beta^{l} )^{2} ) + (\beta^{k} )^{2} (174 - 85\beta^{l} + 8(\beta^{l} )^{2} ) + 6(630 - 309\beta^{l} + 29(\beta^{l} )^{2} )))/(2(G_{2} )^{2} ), \\ & \quad {\text{where}}\;G_{2} = b^{2} (6 - \beta^{k} )(6 - \beta^{l} ) - (15 - 4\beta^{k} )(15 - 4\beta^{l} ),\;i \ne j;\;i,j = 1,2;\;k \ne l;\;k,l = I,II. \\ \end{aligned}$$

1.3 Case 2: an international mixed market

$$\begin{aligned} q_{i}^{kA} & = a(3(6 + \beta^{k} )(10 - \beta^{l} ) - 7b^{2} \beta^{k} \beta^{l} + 2b^{3} \beta^{k} \beta^{l} - 4b(18 + \beta^{k} (3 - 2\beta^{l} )))/G_{3} , \\ q_{j}^{kB} & = a(3(10 - \beta^{k} )(6 - \beta^{l} ) - 3b^{2} \beta^{k} \beta^{l} + 2b^{3} \beta^{k} \beta^{l} - 2b(36 - (6 - \beta^{k} )\beta^{l} ))/G_{3} , \\ \end{aligned}$$

$$\begin{aligned} W^{k} & = (2a^{2} (18(72(5 - 2b)^{2} (3 + b) - 12(14 + b)(5 - 2b)\beta^{k} - (73 - 16(5 - b)b)(\beta^{k} )^{2} ) \\ & \quad - \;6(36(19 - b)(5 - 2b) + 12( - 45 + b(6 + b(48 - b(23 + 4(2 - b)b))))\beta^{k} \\ & \quad + \;(1 - b)(3 - 2b)(b(22 + b) - 14)(\beta^{k} )^{2} )\beta^{l} + (18(77 - 8(5 - b)b) - 6(1 - b)(3 - 2b)(13 + (15 - b)b)\beta^{k} \\ & \quad - \;(3 - 2b)^{2} (1 - b)^{2} (1 - b(3 + b(3 + b)))(\beta^{k} )^{2} )(\beta^{l} )^{2} ))/ (G_{3} )^{2} ,\;{\text{where}}\;G_{3} = 9(10 - \beta^{k} )(10 - \beta^{l} ) \\ & \quad + \;4b^{4} \beta^{k} \beta^{l} - b^{2} (144 + 13\beta^{k} \beta^{l} ),\;i \ne j;\;i,j = 1,2;\;k \ne l;\;k,l = I,II. \\ \end{aligned}$$

1.4 Case 3: an international mixed market

$$q_{i}^{kA} = a\left( {5 - 2b} \right)\left( {6 + \beta^{k} - \beta^{l} } \right)/3G_{4} ,\; q_{j}^{kB} = a\left( {10 - 4b - \beta^{k} - \beta ^{l} } \right)/G_{4} .$$

$$\begin{aligned} W^{k} & = (a^{2} (5400 + 288b^{3} - 73(\beta^{k} )^{2} - 1140\beta^{l} + 77(\beta^{l} )^{2} - 8^{k} (105 - 13\beta^{l} ) + 4b(20(\beta^{k} )^{2} - 630 \\ & \quad + \;\beta^{k} (69 - 10\beta^{l} ) + 129\beta^{l} - 10(\beta^{l} )^{2} ) - 8b^{2} (72 + 2(\beta^{k} )^{2} + 3\beta^{l} - (\beta^{l} )^{2} - \beta^{k} (3 + \beta^{l} ))))/(9(G_{4} )^{2} ), \\ & \quad {\text{where}}\;G_{4} = (5(10 - \beta^{k} - \beta ^{l} ) - 8b^{2} ),\;i \ne j;\;i, \, j = 1,2;\;k \ne l;\;k,l = I,II. \\ \end{aligned}$$

1.5 Case 3: a single country

$$q_{i}^{kA} = a\left( {5 - 2b} \right)\left( {3 - \beta^{l} } \right)/G_{5} ,\; q_{j}^{kB} = a\left( {15 - \beta^{k} \left( {4 - \beta^{l} } \right) - 4\beta^{l} - b\left( {6 - \beta^{k} - \beta^{l} } \right)} \right)/G_{5} ,$$

$$\begin{aligned} W^{I} + W^{II} & = (a^{2} (5400 - 2790\beta^{l} + 317(\beta^{l} )^{2} - 8b^{3} (6 - \beta^{k} - \beta^{l} )^{2} - 4b^{2} (144 + 5\left( {\beta^{k} } \right)^{2} \\ & \quad - \;6\beta^{k} (8 - \beta^{l} ) - 48\beta^{l} + 5(\beta^{l} )^{2} ) + (\beta^{k} )^{2} (317 - 146\beta^{l} + 12(\beta^{l} )^{2} ) - 2\beta^{k} (1395 - 697\beta^{l} + 73(\beta^{l} )^{2} ) \\ & \quad - \;4b(630 - 309\beta^{l} + 29(\beta^{l} )^{2} + (\beta^{k} )^{2} (29 - 11\beta^{l} ) - \beta^{k} (309 - 144\beta^{l} + 11(\beta^{l} )^{2} ))))/(2(G_{5} )^{2} ), \\ & \quad {\text{where}}\;G_{5}=75 - 12b^{2} + 5\beta^{k} \beta^{l} - 2\left( {10 - b^{2} } \right)\left( {\beta^{k} + \beta^{l} } \right),\;i \ne j;\;i,j = 1,2;k \ne l;k,l = I,II. \\ \end{aligned}$$