# Who cares how fast GDP grows?

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

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

$\displaystyle U = u(C) + v(H) \ \ \ \ \ (1)$

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

$\displaystyle u'(xR) = v'[(1-x)R], \ \ \ \ \ (2)$

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

$\displaystyle \frac{1}{xR} = \theta, \ \ \ \ \ (3)$

which solves out to

$\displaystyle x = \frac{1}{\theta R}. \ \ \ \ \ (4)$

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.

## 4 thoughts on “Who cares how fast GDP grows?”

1. You make a good point by calling attention to the tradeoff between consumer goods and intangibles, leading to the conclusion that increasing the rate of GDP growth, or even having any growth of GDP at all, is not automatically optimal. It depends on the tradeoff and the utility functions.

I would just add a couple of points as refinements to your approach:

First, if, by C, you mean the personal consumption component of GDP and if, by H, you mean output of intangibles as measured by something like the Social Progress Index or its components, you have to be careful of double counting. In some cases, for example, healthcare, the elements of C, which include doctors’ services, hospital services, drugs, etc. are really best thought of as inputs to the health component of H, which includes things like infant mortality and longevity. Logically, to avoid double counting of both inputs and outputs, C should be adjusted downward to reflect that. For other components, say education, the components of the SPI are things like enrollment in primary and secondary education. Those seem to me to be input-type concepts, for which the corresponding output would be some measure of knowledge that is not easily available. In that case, it seems to me the proper adjustment would be to leave schooling inputs in C and take education out of the measure of H. Still other things, like personal freedoms, are pure H that do not require any explicit material inputs.

Second, your focus on optimality seems to me to direct our attention to finding the best point along the C-H frontier. That is all well and good, but the actual data as reported in my post suggest that the US is at present well inside the frontier. That being the case, we should be looking for policies that include both C and H, or at least that increase H without decreasing C.

• Ed – thanks for reading. You’re right on that in my toy model, I make a too-stark distinction between C and H. Some fraction of H would show up in GDP as well (some health care costs, federal spending to build a new national park, environmental abatement, etc..). So you could get some substitution from C towards H without affecting total GDP, for sure. The model was there to illustrate that even if H didn’t have any affect on GDP, we still might prefer to have it rather than C.

And you’re right on that just because I assumed you’d pick C/H optimally in the model, that does not mean that we are currently at the optimal C/H ratio. From the perspective of the curves you drew, that’s like saying that the marginal utility of H is currently higher than the marginal utility of C. We could trade-off some C (GDP) to increase our H (health, freedom, etc..) and actually increase total welfare/utility.

2. Dear Sir,
I’d love to comment on the conclusion of your and Ed Dolan’s posta, namely :

“The ultimate point of Ed Dolan’s post, and this one, is that there is nothing inherently desirable about rising GDP.”

I find it a confusing conclusion in the light of 1 bln unemployed (rough estimate) all over the world In USA alone about 20 mln people are willing and capable to work full time.

I find SPI and GDP comparison as a very attractive source of indication how to re-shape economy structure toward better fit with social expectations. Anyway it seems to me that your articles conclusion unintentionally promotes unemployment. I hope you find my comment valuable.