# Who cares how fast GDP grows?

Posted by {"login"=>"dvollrath", "email"=>"[email protected]", "display_name"=>"dvollrath", "first_name"=>"", "last_name"=>""} on May 30, 2014 · 8 mins read

I came across an interesting post by Ed Dolan, on what we should do about slowing growth in the U.S. His answer is "Nothing", and he gives a very capable explanation of why this is the case. His argument is that while GDP and human welfare (the general concept, not the government program) are correlated, once you are very rich the correlation drops enough that it is quite possible to raise human welfare without having GDP go up.

This is a really interesting point, and it relates to the marginal utility of consumption goods (which are goods and services that get counted as part of GDP) as compared to the marginal utility of what I'll call intangibles. Intangibles are things like good health, or a clean environment, that we might value in and of themselves, but they are not necessarily tied to the production of real goods and services that are counted in GDP.

Very simply, let overall utility be

$$U = u(C) + v(H)$$

where ${C}$ is consumption of tangible goods and services and ${H}$ is the consumption of intangibles. We've got some stock of resources (labor, capital, natural resources, etc..) that we can use to produce things with. Consumption goods are ${C = xR}$, where ${x}$ is the share of the resources we use in producing consumption. It's a really simple model with only consumption goods, so GDP is just equal to ${C}$, so that ${Y = xR}$.

Intangible goods are ${H = (1-x)R}$, or they "use" the remaining share of resources. Note that I don't necessarily mean that we have to use up resources to produce intangibles - you can think of ${(1-x)}$ as the fraction of resources that we idle, or leave pristine, or shut down in order to enjoy better health, a nicer environment, or more free time.

Maximizing utility involves picking the optimal value for ${x}$, what share of our resources to commit to consumption. Before throwing some math at it, think of ${R}$ as the total potential number of donuts I could produce using all available resources. The trade-off I face is how many donuts to actually produce. I'll produce some (${C}$), because donuts are yummy. But I'll hold off on producing all the possible donuts because I want to be healthy enough to shoot baskets with my kids in the driveway (${H}$). What is the optimal split of ${R}$ into donuts and "health"? And will that split ever change?

The first-order condition here is

$$u'(xR) = v'[(1-x)R],$$

which just says that the marginal utility of consumption goods should be equal to the marginal utility of intangible goods. If they weren't equal, then you could fiddle with the value of ${x}$ and get higher overall utility.

What happens as ${R}$ goes up? The marginal utility of both types of goods falls. If I already have lots of consumption goods (donuts, cars, iPhones) then the marginal utility of another one gets small. Similar for intangible goods - if I've got great health and lots of beautiful national parks to visit, then it's hard to feel much better or visit an additional park.

The key is going to be how fast these marginal utilities fall. That is, how quickly does an extra donut get old and boring, versus how quickly better health gets old and boring. We often use log utility to describe consumption, of ${u(C) = \ln{C}}$, which means that the marginal utility of consumption is ${u'(C) = 1/C}$, or in terms of resources, ${u'(C) = 1/xR}$. As Chad Jones will tell you, log utility is "very curved", meaning that the marginal utility quickly runs down towards zero as you load up on more donuts. [Aside: log utility, though, is less curved than other typical utility functions for consumption, so I'm probably understating how fast utility falls with more consumption].

What's the utility function for intangible goods? I don't know that there is any kind of consensus about what this looks like. But let me use a very simple utility function that will demonstrate the logic of not caring if GDP grows. Let's have ${v(H) = \theta H}$, so that ${v'(H) = \theta}$. This function is linear in ${H}$, so that the marginal utility of intangible goods doesn't depend on how much ${H}$ you consume - you can never be too healthy, so to speak. The most important part here is that marginal utility falls more slowly for ${H}$ than for consumption goods.

Back to our optimal choice of ${x}$. Using the assumed utility functions, I get that my first-order condition is

$$\frac{1}{xR} = \theta,$$

which solves out to

$$x = \frac{1}{\theta R}.$$

That is, the optimal fraction of resources to spend on consumption goods falls as ${R}$ rises. As we get more resources (labor, capital, technology) we use fewer of them on actually producing consumption goods. The payoff in terms of utility is just too low compared to the payoff in utility from having more intangible goods.

Remember that GDP is just ${Y = xR}$, and under our optimal assumption for ${x}$ this is just ${Y = 1/\theta}$. In other words, it would be optimal in this model for GDP to stay constant at ${1/\theta}$, even as the available resources ${R}$ are increasing. We would willingly sacrifice additional GDP because it only enhances consumption goods without increasing intangibles. No growth in GDP is utility-maximizing.

By fiddling with the exact utility function for intangibles you could get a different answer. Perhaps GDP optimally rises very slowly (if intangible goods have a declining marginal utility), or GDP optimally falls over time (if intangible goods have an increasing marginal utility as you use them - think of enjoying national parks more if you are healthy enough to hike through them).

The ultimate point of Ed Dolan's post, and this one, is that there is nothing inherently desirable about rising GDP. It is simply a statistical construct capturing the total value of currently produced goods and services. If we prefer things that are not currently produced goods and services, then who cares if GDP rises or falls?

Something that I didn't address here is how we adapt to a lower fraction ${x}$. If ${x}$ falls, this implies that we are idling resources, like labor. If I'm going to consume fewer donuts, I'm going to put some bakers out of business. If you're lucky, the bakers don't mind because they would have chosen to go backpacking through Yosemite anyway. If you're not, then these unemployed bakers are looking for something to do. As usual in these kinds of questions, seeing the different equilibrium outcomes is a lot easier than seeing how to transition from one to the other.