Following up on yesterday’s post on the Heritage Foundation’s assessment of the Ryan plan, I thought it would be useful to see how the labor and capital supply elasticities that are implied in the simulations compare with the literature, for the benefit of my macroeconomics class. Unfortunately, I come up with some really odd numbers, so I must either be making a mistake somewhere, or the simulation is very odd. Update 4/10, 4:50pm Pacific: I added two graphs illustrating exactly how odd these numbers are.
Figure 1: Private nonfarm payroll employment (blue), baseline (dark blue), and simulated under Ryan plan (red). Actual data average of monthly data. Source: BLS via FREDII, and Heritage Foundation, Appendix 3: Simulation Results.
Figure 2: Real equipment investment, bn. Ch.05$ (blue), baseline (dark blue), and simulated under Ryan plan (red). Source: BEA, and Heritage Foundation, Appendix 3: Simulation Results.
Labor supply elasticities
First consider the labor supply elasticity. This elasticity is given by the following formula:
η = (∂N/∂ω) × (ω/N)
Where N is labor supply, and ω is the after tax real wage.
From Appendix Table 3 of the Heritage document (as revised 11am 4/6), we have average baseline private nonfarm payrolls at 119.9 million, simulated at 121.5 million; the log difference is 0.0132. The average baseline personal tax rate is 0.184, and the simulated is 0.181. The log difference is 0.0025. I don’t have demand side elasticities, but assuming a perfectly elastic demand for labor gives me the minimum figure for the implied supply elasticity. Hence, substituting these figures into the formula leads to:
5.28 = (0.0132/0.0025)
This figure is somewhat higher than the authors of the report indicate they are using (a value of 2), which is in turn in the mid-range of the estimates reported by Rogerson and Allenius. This disjuncture must mean something else is going on in the model (perhaps capital and labor are complements, or employment differs substantially from hours worked).
For reference, here are the elasticities used by CBO (reported earlier in this post):
Source: CBO, “The Effect of Tax Changes on Labor Supply in CBO’s Microsimulation Tax Model,” Background Paper (April 2007).
Now consider the implied behavior of capital investment.
The standard rental cost of capital approach to modeling investment goes back to Dale Jorgenson’s classic 1960′s paper. The key parameter is the elasticity of the investment-to-capital ratio with respect to the rental cost:
ln(I/K) = γ rK
Where the rental cost is given by:
(1-u)rK = (i – πK – d)(1-z)PK
Where u is the corporate tax rate, rK is the rental cost of capital, i is the interest rate on corporate bonds, πK is the inflation rate for capital goods, d is the economic depreciation rate, z is the present discounted value of tax credits and accelerated depreciation allowances, and PK is the price of capital goods. I’ll assume πK, d, and z equals zero (which is okay since they don’t change in the simulation), and the relative price of capital goods at unity. After solving for the rental cost of capital, this leads to:
rK = (i)/(1-u)
The baseline 10 year interest rate is 0.04972, the alternative 0.04596. The current statutory corporate tax rate is 0.35, the rate under the Ryan plan is 0.25. Then the initial rental cost of capital is:
0.07649 = 0.04972/0.65
and under the Ryan plan:
0.06128 = 0.04596/0.75
(I’m assuming a percentage point for percentage point change in the effective corporate tax rate, so there’s some slippage here). So the change in the rental cost of capital is 0.01521.
Now consider the change in equipment investment relative to the capital stock. Under the baseline, average equipment investment is 1.611 trillion Ch.05$, under the Ryan simulation, it is 1.827 trillion Ch.05$. I don’t have numbers for the real capital stock over this period, but for the sake of argument I’ll use the 2009 current cost (nominal) capital stock of 5.611 trillion. (Since the price index for equipment investment goods in 2010Q4 is about 0.975 for 2005=1, this is pretty close to a real magnitude, expressed in Ch.05$.)
Then the change in the investment-to-capital ratio is = 0.0385, the average investment-to-capital ratio over the next ten years is 0.2918 (assuming the net capital stock grows by 1% per year). Solving out for the elasticity yields:
γ = (0.0385/0.2918)/(-0.01521) = -8.67.
This figure is somewhat(!) above (in absolute value terms) the short run elasticity of 0.50 cited by, for instance, Gilchrist and Zakrajsek (2007) (their long run elasticity is about unity).
Now, I know that elasticities are supposed to be essentially partial derivatives, rather than total derivatives. But when the magnitudes are so far off, either I’m making a math mistake, or something really interesting is going on in the Heritage CDA’s combined microsimulation-plus-IHS Global Insight model.
So, if anybody can help me out, I’d be much obliged.
Update, 4/9 8:25am: Some readers might wonder why I spend time on the Heritage analysis. It’s because there are individuals out there who cite the projections from the Ryan plan (validated by the Heritage CDA analysis) as if they were serious, and compare them against CBO baselines. Take for instance Keith Hennessey’s 4/6/2011 post (which provided my first big guffaw of the day). By taking at face value the projections, he is taking at face value these implied elasticities.
Personally, people who have master’s degrees in policy analysis should know better. And that’s what I’m going to teach my students on Monday.
Footnote: I know it’s hard to keep up, but I wonder why Hennessey’s post of 4/6 made no mention of the CBO analysis of the Ryan plan, when it had been circulated on 4/5 (or why he made no update to the post to mention what was arguably an important document).