# How can I formulate the following optimization problem?

I want to set up an optmiization problem for global warming in which a planner determines how much carbon dioxide gas is emitted. Let's say we reduce this problem down to two periods, then I formulated the problem as follows:

$$\max_{C_0, C_1} \quad \beta^0L_0u\left(\frac{C_0}{L_0}\right)+\beta^1L_1u\left(\frac{C_1}{L_1}\right)$$

Where we have two time periods, $$u$$ are the utility functions in each time period, $$\beta_1, \beta_2$$ are the discounted utility, $$L_t$$ is the population size and $$C_t$$ is total consumption. I am confused how to include how the social planner determines how much carbon dioxide gas is emitted. I am assuming this is done through the constraints in the optimiztion problem, but how could one go about implementing such a model?

• Edinburgh course?... Commented Apr 1, 2022 at 23:19

If you want to determine how much carbon dioxide should be omitted by solving an optimization problem, then a constraint on the quantity of $$CO_2$$ isn't quite what you need. The normal constraint on an optimization model for a whole economy is the production function, so one way to include $$CO_2$$ would be to make it a variable within the production function. To illustrate this I make the simplifying assumptions that consumption in each period equals output, capital $$K$$ is constant, labour equals population, and that $$CO_2$$ emissions take place at the start of each period. Hence I write the first period constraint as:

$$C_0\leq f(K,L_0,aC_0)$$

where $$a$$ is $$CO_2$$ emissions per unit of output/consumption, and $$f$$ is increasing in $$K$$ and $$L$$ and decreasing in $$aC_0$$, the last point reflecting an assumed adverse effect of $$CO_2$$ on agricultural output. To allow for the accumulation of $$CO_2$$ in the atmosphere, I write the second period constraint as:

$$C_1\leq f(K,L_1,a(C_0+C_1))$$

On solving the optimization problem with these constraints and obtaining values of $$C_0$$ and $$C_1$$, the $$CO_2$$ emissions in each period can then be calculated as $$aC_0$$ and $$aC_1$$.

An alternative approach, focusing on the direct effects of warming due to $$CO_2$$ emissions on human health and welfare, would be to reflect these in the utility function. So the problem might be formulated as:

$$\max_{C_0,C_1}\beta^0L_0u\left(\frac{C_0}{L_0},aC_0\right)+\beta^1L_1u\left(\frac{C_1}{L_1},a(C_0+C_1)\right)$$

where $$u$$ is decreasing in the $$CO_2$$ terms. This could be subject to a simple production constraint:

$$C_i\leq f(K,L_i)$$

These alternatives could be combined, including $$CO_2$$ in both the objective function and the constraint, although the more complicated the problem becomes the more likely that its solution will be intractable.

• Thank you, that is very helpful! One question I do have, let's say the model wanted to allow for investments which reduce CO2 emissions. Could we then introduce an investment variable i_t which is a negative input to the utility function and a positive input to the production function? Or would such an i_t not be part of the utility functions you describe?
– user40917
Commented Apr 8, 2022 at 13:36
• In that case I would suggest: a) Drop my simplifying assumption of consumption equals output and replace by consumption = output minus investment; b) Distinguish at least two types of capital, one of which, call it $K_2$, would reduce emissions; c) Introduce an emissions function in terms of the two capitals and labour which would be decreasing in $K_2$. I don't think the utility function should need to change. Commented Apr 9, 2022 at 18:06
• Thank you very much for your help, it has definitely helped with formulating an interesting model. One question I did have, is the model you formulated with the effect on human health and welfare a convex problem? I have been trying to prove it is but not entirely sure if I have to make a bunch of assumptions. Commented Apr 18, 2022 at 12:23

There are various ways how you could do it but probably most strait forward would be to introduce 'carbon budget'.

Suppose that good $$C_0$$ emits $$b_0$$ units of $$CO_2$$ and $$C_1$$, $$b_1$$ units of $$CO_2$$. Suppose social planer wants to limit amount of all carbon emitted not to exceed $$Q$$ units of $$CO_2$$.

Then it is straightforward to see that your problem can be constrained by using constraint: $$C_0 b_0 + C_1 b_1 \leq Q$$ (you could simplify by making it hold with equality). Hence you will have constrained optimization problem:

$$\max_{C_0,C_1}\beta^0L_0u\left(\frac{C_0}{L_0}\right)+\beta^1L_1u\left(\frac{C_1}{L_1}\right) \quad st \quad C_0 b_0 + C_1 b_1 \leq Q$$

The amount of $$CO_2$$ permitted by social planner will be given by $$Q$$.

• Ah thanks, that makes a lot of sense. And what if I wanted to add some extra variable to the system, like let's say I also wated to account for investments in low-carbon technology. Would such a constraint be implemented in a similar way?
– user40867
Commented Apr 1, 2022 at 22:26
• @L.Johnson you don’t need to add extra constraint for that you could add another investment variable (which will also be choice variable) that you can subtract from consumption in first period (representing portion of first period output invested or something like that) that determines the size of $b_1$ in next period. Although if you would want to do it rigorously it would be better to have bigger general equilibrium model for such thing.
– 1muflon1
Commented Apr 1, 2022 at 22:37