A stylised model of environment-GDP trade-offs

In ECONS101 last week, we covered constrained optimisation models. I spent most of the week's lectures working through examples with the consumer choice model, then at the end I introduced the worker's labour-leisure trade-off model, and then briefly the constrained optimisation model for savers (deciding between consumption today and consumption in the future). I also mentioned that the constrained optimisation model can be used in lots of other situations.

Over on The Visible Hand of Economics blog (good to see that it is back in business), Matt Nolan wrote a spirited defence of GDP as a measure of production, where at one point he wrote:

GDP is surprisingly nifty for these things … as long as we don’t always start with the unconditional prior “more GDPs are good”.  We need to ask “what is the trade-off that creates this GDP”.

Since trade-offs can often be described using the constrained optimisation models we covered in ECONS101 last week, it got me thinking about what such a model might look like. This is a stylised model so I'm making a number of un-stated assumptions, and there are a number of obvious criticisms of the model that you can read more about if you scroll to the notes at the bottom of the post.

First, we need two goods that our decision-maker has to choose between. In this case, let's consider a model of the trade-off between GDP and environmental quality. Our decision-maker in this case is a benevolent social planner (or the government, you choose). [*]

Next, we need to think about what constrains our decision-maker's decision about the quantity of GDP and quantity of environmental quality. In this case, it's resources. Resources (land, labour, capital variously defined) can be applied to produce stuff (increase GDP) or to increase environmental quality. This sets the trade-off as being more GDP (Y) or more environmental quality (Q) - if we want more production (higher GDP), then we end up with lower environmental quality, and if we want higher environmental quality, we end up with less production (lower GDP).

The constraint can be represented as a straight line on the diagram below. [**] To draw a straight line of course, we only need two points. One is the point where we use all of our resources on environmental quality and have no production at all (the point QMAX on the diagram); the other is the point where we use all of our resources on production and have no environmental quality at all (the point YMAX on the diagram). The trade-off between environmental quality and production is represented by the slope of the constraint - this is the opportunity cost of environmental quality (it is the amount of production we would have to give up to get one more unit of environmental quality, however it was measured). The opportunity cost of environmental quality is related to productivity, which is a point we will come back to a bit later in the post.

Lastly, we need to represent the decision-maker's preferences. As with other constrained optimisation models, we do this with indifference curves. Just like indifference curves in the consumer choice model (and other constrained optimisation models) the decision-maker is trying to get to the highest possible indifference curve, which is up to the right. [***] On the diagram below, the highest possible indifference curve is I0, and the optimal bundle of production and environmental quality is E0 (which includes Y0 production or GDP, and Q0 environmental quality). So, we can see that the optimum point contains some positive level of production, and some positive level of environmental quality.

Now we have constructed our model, we can do something useful with it. Let's consider the question: what happens when productivity increases? An increase in productivity means that we can produce more using the same amount of resources. In that case, the constraint in our model pivots outwards and becomes steeper, as in the diagram below. That's because, if we applied all of our resources to production, we could now produce more (so the constraint moves up on the y-axis from YMAX to YMAX1). The productivity increase doesn't affect the maximum environmental quality we could obtain (which stays at QMAX). The steeper constraint means that the opportunity cost of environmental quality has increased - we now need to give up more production for each unit of environmental quality than we did before.

What will the decision-maker do? They could keep operating at the previous optimum (E0), because it is in the feasible set. But they won't, because they can get to a higher indifference curve (remember that the decision-maker is trying to maximise utility by getting to the highest possible indifference curve). That highest possible indifference curve is now I1, and their optimum is the bundle E1 (which includes Y1 production or GDP, and Q1 environmental quality).

This result is interesting. First, notice that the opportunity cost of environmental quality has gone up. Usually, when the cost of something goes up, we would expect people to want less of it, but our decision-maker wants more (notice that Q1 is bigger than Q0). [****] That's because, when you change the relative price of two 'goods' (in this case, the relative price of environmental quality and production), two effects are simultaneously occurring: (1) a substitution effect; and (2) an income effect.

The substitution effect suggests that the decision-maker will want less environmental quality, because it has now become relatively more expensive (higher opportunity cost). Given that the quantity of environmental quality has actually increased, that tells us that the income effect is working in the opposite direction, and is bigger than the substitution effect. In this case, the income effect works like this: the pivoting outwards of the constraint increases the 'purchasing power' of the decision-maker's resources, in the sense that they can purchase more production and more environmental quality (we can refer to this as an increase in their real income). Both production and environmental quality are normal goods (goods that we want to consume more of when our incomes increase), so because the decision-makers real income has increased, they want to consume more environmental quality as a result of this income effect.

So, that's a simple stylised model of environment-GDP trade-offs, with some illustration of how it can be used to explain decisions. It's an example of a constrained optimisation model, but in quite a different context from how it is usually presented in class.

*****

[*] Aggregating the preferences for many people into a single set of aggregate preferences leads to the problem described in Arrow's Impossibility Theorem. To avoid that problem (or to embrace it), we'll just assume there is a single decision-maker (who Kenneth Arrow labelled a 'dictator'), whose preferences (their indifference curves) represent those of society as a whole.

[**] A more realistic model would recognise that resources are not equally useful for production as for increasing environmental quality, and so the opportunity cost is not constant. That means that the constraint is not a straight line, but a curved line (bowed out away from the origin). Some of you would recognise that as a production possibilities frontier, and you would be right - this is a model of constrained production, not constrained consumption. Simplifying the model by assuming that the opportunity cost of environmental quality is constant just makes the model a bit easier to draw, but is otherwise not consequential in terms of the qualitative results we get from the model.

[***] The indifference curves here are curved because of diminishing marginal utility (just as they are in most constrained optimisation models). We gain extra utility (satisfaction) from more production, and from more environmental quality. However, the extra utility we get from one more unit (of production, or of environmental quality) diminishes as we get more of it. Maybe you would argue that there is not diminishing marginal utility of environmental quality (at least, some people would). However, you only need diminishing marginal utility of one of the goods in order for the indifference curves to be curved.

[****] Of course, it is also possible to draw the diagram in such a way that the quantity of environmental quality decreases after the change. The difference is purely a result of differences in the shape of the indifference curves. The case where environmental quality increases is more interesting, which is why I focus on it here.