John Reid and Wilson Cabral de Sousa Jr. write in Conservation Biology:
"Why are public roads public? Economists generally espouse the premise that goods and services can most efficiently be provided by private enterprise, although public investment or involvement in economic activities can sometimes be justified--in, for example, curbing the abuses of monopolists. Roads, pipelines, and power lines all represent natural monopolies, industries for which it makes practical sense to have just one firm supplying the service in a given place. Government must either be that firm or regulate it so that the firm does not underproduce, maintain high prices, and collect excess profits called "monopoly rents." Public investment can also distribute resources more equitably. Transportation and energy are [...] regarded as basic economic rights [...] [and] can be either universally subsidized or specifically subsidized through special low-income rates to ensure that all citizens receive them. Business losses are then covered by taxpayers [...]. Managing risks that otherwise suppress private investment is another justification for public investment [the authors offer the example of dam construction]."
Then Reid and Cabral de Sousa review the costs and benefits, especially in the context of biological conservation, of several public roads, dams and river diversions built (perpetrated, I would say) by the Brazilian government, and point out their questionable economic efficiency and equity, and their severe environmental impacts.
Although they do not say so explicitly, I infer that the Brazilian government has done large-scale public works in the wrong places, in the wrong way and, above all, too many. This is not surprising. Politicians spend not their own money but the taxpayers'. They gather more popular support by "doing things" than by (judiciously) not doing them. And diverting a river offers a convenient avenue for diverting money.
Ironically, Reid and Cabral defend government intervention to avoid "underproduction" by private monopolies. It is quite likely that government intervention in natural monopolies creates worse problems than the underproduction it tries to solve. But, on top of that, in the particular case of gigantic dams, river diversions and roads crossing Brazilian forests, I do not see "underproduction" as a problem at all. I do prefer underproduction. And I also prefer "underproduction" in transportation and energy to government subsidies.
In the case of public roads across the Amazon, the Atlantic forests and other wild areas, there is an additional distortion. The roads are toll-free. If roads were privately produced people would have to pay for their use. If trucks that carry soya beans from deep inside the Amazon basin to the trade centers had to pay road tolls, there would be less incentive to produce soya so far away.
So I suggest that private enterprise would provide roads, dams and river diversions more efficiently and prudently than governments. As a result there would probably be more forests.