## 15 December 2015

### Shut Up About Ricardian Equivalence

Economists that are both opposed to and in favor of fiscal stimulus frequently cite Ricardian equivalence as a reason that, in models with perfect credit markets, it doesn't matter whether stimulus is funded through increased taxes or through deficits. The problem with this analysis is that it assumes lump sum taxation. That is, taxes are not collected from things like consumption expenditures, which are effectively no different than deficits because 1) they don't discourage people from working, consuming, investing, etc and 2) they are expected to rise at some point in the future to retire the current debt, so the present value of taxes goes up with government spending. In reality, the argument is about distortionary taxes vs. deficits (= lump sum taxes, as per Ricardian Equivalence).

Distortionary taxes are different than their lump sum counterparts since they directly act to disincentivize working (in the specific case of income taxes, which I will limit my analysis to from now on) and can thus either partially or fully negate the effects of a fiscal stimulus. So, when John Cochrane says something like "'Ricardian Equivalence,' which is the theorem that stimulus does not work in a well-functioning economy," [1] he's clearly confusing the two types of taxation as well as ignoring the fact that neoclassical economics predicts a positive multiplier on government spending [2]. To illustrate this, I wrote down a standard Real Business Cycle model and ran two simulations: one in which a temporary fiscal expansion was financed entirely with an income tax and another in which the same fiscal stimulus was financed partially by deficits (see the appendix for a derivation of the model).
 Figure 1: Impulse Response Function of Output to the Stimulus
 Figure 2: Impulse Response Function of the Income Tax Rate to the Stimulus
 Figure 3: Government Spending in both simulations; Government Debt in the second simulation

The Ricardian Equivalence argument would be irrelevant if 1) the stimulus had a positive effect on output and 2) the tax funded stimulus was initially less effective than the partially deficit funded one. As you can see in figure 1, both of these are true; the stimulus positively impacted output in each simulation and the stimulus was initially more effective when taxes were not increased to fully finance the stimulus on impact. The effectiveness of the stimulus is slightly less sound of a result, though. The fiscal multiplier in neoclassical models is highly dependent on calibration (see, e.g., [2]) and can range anywhere from zero to one, without distortionary taxation, depending on the specific calibration used. Regardless, the most important part of this argument is sound; the Ricardian Equivalence argument against deficit funded stimulus is wrong and should be ignored completely as it applies to a form of taxation that doesn't actually exist.

References:

[1] John Cochrane, 2011. "Krugman on Stimulus" The Grumpy Economist.

[2] Woodford, Michael. 2011. "Simple Analytics of the Government Expenditure Multiplier." American Economic Journal: Macroeconomics, 3(1): 1-35.

Appendix:

The following is a derivation of model that I used to generate the impulse response functions in figures 1, 2, and 3.

Household:

There is a representative household who maximizes the utility function $U = E_0 \sum^\infty_{t=0} \beta^t\left(\frac{c_t^{1-\sigma}}{1-\sigma} - \frac{n_t^{1+\phi}}{1 + \phi}\right)$ where $E_t$ is the rational expectations operator given information available in period $t$, $c_t$ is the household's consumption, $n_t$ is the labor supply, and $\beta$ is the household's discount factor - the rate at which future utility is discounted relative to current utility. The household can use net-of-taxes income from labor ($(1-\tau^w_t)w_t n_t$, where $w_t$ is the real wage), government bonds carried from last period ($R_{t-1} B_{t-1}$, where $R_t$ is the interest rate that bonds maturing in period $t$ - $B_t$ - pay), and net-of-depreciation income capital, $(1 + r_{t-1} - \delta)k_{t-1}$ to purchase consumption, new government bonds, or new capital. The budget constraint can be written as

$$(1.1)\: (1-\tau^w_t)w_t n_t + R_{t-1} B_{t-1} + (1 + r_{t-1} - \delta)k_{t-1} = c_t + B_t + k_t$$

The household maximizes $U$ subject to $1.1$  in order to determine its behavior:

$$(1.2)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} (1 + r_t - \delta)$$
$$(1.3)\: R_t = 1 + r_t - \delta$$
$$(1.4)\: (1-\tau^w_t)w_t = c_t^\sigma n_t^\phi$$

Additionally, it is useful to define investment, $i_t$ as the instrument of capital accumulation:

$$(1.5)\: k_t = (1-\delta)k_{t-1} + i_t$$

Firm:

The firm bundles capital carried from last period and labor using a Cobb-Douglas production function to form output, $y_t$

$$(2.1)\: y_t = k_{t-1}^\alpha n_t^{1-\alpha}$$

The firm maximizes profits, $y_t - w_t n_t - r_{t-1}k_{t-1}$ subject to $2.1$ in order to determine labor and capital demand

$$(2.2)\: w_t = (1 - \alpha)\frac{y_t}{n_t}$$
$$(2.3)\: r_t = \alpha E_t\frac{y_{t+1}}{k_t}$$

Government:

The government issues new government bonds and collects tax revenue to pay for both government spending and interest on government bonds carried from last period. The government budget constraint can be written as

$$(3.1)\: B_t + \tau^w_t w_t n_t = g_t + R_{t-1} B_{t-1}$$

In the first simulation, it is assumed that the government ensures $B_t = 0\: \forall t$, so government spending is simply financed by taxes

$$(3.2)\: \tau^w_t w_t n_t = g_t$$

In the second simulation, the government sets the tax rate as a function of the tax rate consistent with the long run level of government spending, $\tau^w_{SS}$ and the level of government debt issued in the previous period, $B_t$. The rule for the tax rate in the second simulation is

$$(3.3)\: \tau^w_t = \tau^w_{SS} + \phi_b B_{t-1}$$

In both simulations, government spending follows an autoregressive process and returns to its long run trend trend at decay factor $\rho$. Government spending follows

$$(3.4)\: g_t = (1 - \rho)g_{SS} + \rho g_{t-1} + \eta_t$$

Where $\eta_t$ also follows an autoregressive process with the same decay factor an is hit with with the shock $\epsilon^g_t$

$$(3.5)\: \eta_t = \rho \eta_{t-1} + \epsilon^g_t$$

Equilibrium:

Combining $1.1$, $1.5$, and $3.1$ yields the resource constraint for the economy

$$(1)\: y_t = c_t + i_t + g_t$$

Equations $1.2$-$3.5$ can be used to determine the equilibrium for the rest of the endogenous variables:

$$(2)\: c_t^{-\sigma} = \beta E_t c_{t+1}^{-\sigma} (1 + r_t - \delta)$$
$$(3)\: R_t = 1 + r_t - \delta$$
$$(4)\: (1-\tau^w_t)w_t = c_t^\sigma n_t^\phi$$
$$(5)\: k_t = (1-\delta)k_{t-1} + i_t$$
$$(6)\: y_t = k_{t-1}^\alpha n_t^{1-\alpha}$$
$$(7)\: w_t = (1 - \alpha)\frac{y_t}{n_t}$$
$$(8)\: r_t = \alpha E_t\frac{y_{t+1}}{k_t}$$
$$(9)\: B_t + \tau^w_t w_t n_t = g_t + R_{t-1} B_{t-1}$$
$$(10a)\: \tau^w_t w_t n_t = g_t\: \mbox{in simulation 1}$$
$$(10b)\: \tau^w_t = \tau^w_{SS} + \phi_b B_{t-1}\: \mbox{in simulation 2}$$
$$(11)\: g_t = (1 - \rho)g_{SS} + \rho g_{t-1} + \eta_t$$
$$(12)\: \eta_t = \rho \eta_{t-1} + \epsilon^g_t$$