So this Tweet went viral, and I'm currently on Spring Break, so why not address it? Communists have been having a field day with all the toilet paper shortages and whatnot, but is this a failure of capitalism?

Let's assume "capitalism" means "non-communist", so I'll make my argument this: the general response to the coronavirus outbreak is exactly what a "capitalist model" would predict! "Capitalist model" here means a neoclassical growth model, since she's talking about growth.

Note: this R1 will probably be simple for trained economists, but on the technical side for non-economists/econ students.

So let's first set up the problem. People want to maximize their utility (u), which is a strictly increasing, concave function of their consumption (c), and satisfies (the asymptotic derivative portions of) the Inada conditions.

Now let's make this a constrained optimization problem. People have constraints on how much they can consume, relative to how much capital (K) they have. Specifically, people each have production functions (f) that also satisfy the Inada conditions, are strictly increasing in capital, and strictly concave. So we require that f(K) is greater than or equal to c.

Now let's make this a dynamic problem with an infinite horizon. People want to maximize their utility in the long run. However, future utility is exponentially discounted by a discount factor beta (fun fact: you can generally tell if an economist is a micro or macro person by how they write their discount factors. If they write betas, they're probably macro, and if they write deltas, they're probably micro.). Furthermore, capital depreciates over time by a constant depreciation factor delta (and this is why even though I'm more micro, I wrote the discount factor as beta). In addition, people have two things they can do with their f(K): they can consume, or they can invest (I) in each period. Investing adds to the capital one has in the next period.

So, adding time (t) subscripts as necessary, the problem looks like this. Now, we can solve this using different methods, e.g. Lagrangians. Let's instead break out our copies of SLP and use dynamic programming!

Let's first combine the budget constraint and the law of motion for capital, so we eliminate the investment variable and get this lovely inequality. Now let's think about the Kuhn-Tucker conditions here. Let's call f(K_t) a, and the right-hand side b. The marginal utility of additional capital (MUoK) will always be positive, so since an optimum is reached only if MUoK times (a-b) = 0, we require that a-b = 0, and so the above inequality is actually an equality when we optimize.

So we can rearrange this to isolate c_t like so. Now we can solve the dynamic programming problem. Let's write out the Bellman equation, which has how much capital we currently have as the state variable, and consumption as the control variable. But we can rearrange it to have capital in the next period as the control variable as follows from the previous equation.

Now we get the first-order condition and envelope condition (mistake in the EC, it should be (f'(K_t) (1-delta)), the two shouldn't be multiplied), and when we set the envelope condition one period forward, the two can be combined to get the Euler equation. For ease of notation, and also for concepts later, let's collapse the long expressions back in terms of consumption. Let's look at this intuitively. f'(K_{t 1}) - delta is the net marginal product of capital after depreciation (net MPK), so the Euler equation says that to balance marginal utility between two periods, we need the discount factor on marginal utility in the next period to be offset by 1 net MPK.

Now since we're talking about economic growth here, we need the concept of a steady state, which is the point at which consumption and capital levels remain unchanged from one period to the next. In technical terms, we require that K{t 1} - K_t = 0, and u'(c{t})/u'(c{t 1}) = 1, implying that c{t 1} = c_t (since because it is strictly increasing and strictly concave, u' is an injective function). So the Euler equation in steady state looks like this, and out combined budget constraint law of motion from above looks like that.

Time for some graphs! Well, only one. This is a phase diagram of how consumption and capital evolve at different stages. The red line is called a saddle path, and the intersection between everything is the steady state of our system. And this concludes our construction of this neoclassical growth model.

But then Coronavirus strikes, almost out of nowhere! Almost completely unanticipated by most Americans! People are suddenly afraid for what the future holds, so they buy out stores' supplies of toilet paper!

How does this translate to our model? Well, people suddenly care more about the future, meaning that they value the future more, meaning that they discount it less, meaning that beta goes up. Remember that beta only affects the Euler equation. So in steady state, this happens to our model. Specifically we get a NEW steady state corresponding to how the vertical consumption line moved to the right. We know that the vertical line cannot move beyond the peak of the change in capital = 0 locus because of the Inada condition on f(k), unless beta were greater than 1, which means you value the future more than the present, and I've never seen this in a model.

But look! We were still at the old steady state, so we dropped all the way down to where the new saddle path intersects the old vertical line. This means that immediately, following the Coronavirus shock, our "capitalist model" predicts a sudden decrease in consumption, which is what our communist Twitter friends are complaining about in real life. This is because people are saving more for the future, so along the saddle path, consumption goes up. And this means that capital savings are going up, so letting K = toilet paper or whatever people are stocking up on, this is exactly what's going on in real life.

More specifically, we can graph the impulse response functions like so.

So we can see that "capitalism" is going exactly as expected by our "capitalist model", and clearly, the Tweet's claim that it can't survive a 2-week slowdown is false, because this is absolutely normal for a shock like the Coronavirus.

TL;DR: haha Tweet wrong, neoclassical growth model go whrrrrr

EDIT: So there are a lot of critiques to this post, a lot of which are saying I missed the point, which is valid since COVID-19 has many effects and this is a simplified model. I'll address one though, that the virus affects the production side of things, which is true. So let's specialize f(K) to K^alpha (sorry, I'm on my phone so no Latex or graphs here), where alpha is between 0 and 1. The virus acts as a shock to alpha (economically, this acts as a decline in income), and let's say that the shock is anticipated a couple of periods in advance. This has two effects.

For the capital-side steady state condition, a decline in alpha in the future will bring down the capital locus, creating a new steady state below the current one. Graphically, the saddle path will be below the current one, so according to the phase diagram, people will start accumulating more capital in response to the anticipated decline in income, so they shift to the right. This is so that once the shock to alpha occurs, they will be on the new saddle path, and will head towards the new steady state. (Also, I've been assuming that shocks are permanent, as people don't know exactly when this will end, but temporary shocks can be incorporated without too much additional effort, but off the top of my head I don't think they change the qualitative analysis that much.)

The effect on the consumption-side is such that when alpha goes down, this leads to an ambiguous result about the level of capital in steady state. You can actually solve for the steady state level of capital by taking the consumption steady state equation where the ratio of marginal utilities equals 1 and isolating K_{t 1}. When you differentiate this, you get a beastly expression, and it tells you that for sufficiently high alpha, the level of capital will increase when alpha decreases (at a small enough amount), but for low alpha, capital will decrease when alpha increases (I think this is correct; I plugged it into a calculator). Intuitively, this means that if your rate of income (not the same as capital, in this model) is high enough, when you know that's about to go down, you'll start saving and accumulating more. But at low levels of income, a decline in future income prospects won't hurt as much, and you won't be as incentivized to save more for the future.

So a negative, anticipated shock to alpha in this specialized model leads to ambiguity, and we'd have to give values for parameters, at least alpha and the change in alpha, to see the effect.