Homework Mathematical Programming Model And Shadow Price

Homework- Mathematical programming model and shadow price
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Rose et al. (1998) (Rose, A. et al. “International Equity and Differentiation in Global Warming Policy,” Environmental and Resource Economics, No. 2, 1998.)
Explain which features of the mathematical programming model or non-linear, and why they are non-linear. Why does non-linearity pose problems for the fundamental theorem of linear programming? Explain the meaning of the shadow price (dual variable) of either of the programming formulations in the paper. 
a] Explain which features of the mathematical programming model or non-linear, and why they are non-linear. Why does non-linearity pose problems for the fundamental theorem of linear programming?
Rose et al proposed a nonlinear programming model to determine burden sharing for greenhouse emissions based on an allocation and outcome based rules. The researchers proposed that the rules governing burden sharing in climate change policies can be allocation based, outcome based or process based rules that focus on equity and fair decision making.
In the allocation based model the researchers proposed that countries are permitted to reach certain emission targets and costs are then determined based on this. There are individual permits emission issued targeting the countries where the abatement costs where carbon trades are taken into consideration. This approach assumed that it was possible to allocate the emission costs, but ignored that small changes in the climate system have far reaching consequences in other parts of the world.
The allocation model further highlights abatement costs as a component of the non-linear model, where there are permit trade balances that are identified. The permits are tradeable and this reflects the costs and benefits to the countries that have agreed to trading permits. The marginal abatement costs are non linear since here is no proportional relationship between pollution and damage caused. The rise of emissions may result in modest increase of the marginal abatement costs, the costs many also increase continuously, or they may fall and then rise again. Generally, the marginal abatement costs increase over time.
In both the allocation and outcome based approaches, the article highlights the challenge of achieving equity at the international stage through agreements. For instance horizontal equity is criterion to distribute permits, with emphasis on the total population and per capital GDP as the reference base. Equalizing the net welfare changes for the different nations is a non linear relationship between emissions and costs/benefits cannot be linearly calculated.
In non linear models there are nonlinear constraint functions and in optimizing the functions this presents a problem in linear programming. For instance, minimize the nonlinear functions is more complex that the equations cannot be used in linear functions. At the same time, nonlinear programming does not necessarily distinguish the maximums, while it may be necessary to determine the minimum and maximums. This is especially when non linearity requires the derivatives of functions to identify either the minimums or the maximums.
Linear programming is ...