Back Matter
  • 1 https://isni.org/isni/0000000404811396, International Monetary Fund

References

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Appendix I. Labor Supply in Balanced Growth Models

This appendix shows that under the log utility function, labor supply must be constant over time along the balanced growth path.

Consider a continuous time growth model. Let a utility function be:

(A1.1)log(C)+αlog(1L)

where C is consumption and L is labor supply and 1 is a normalized time endowment. Let a production function be:

(A1.2)Y=AKθ(L)1θ

The intramarginal condition (the first order condition for labor supply) is:

(A1.3)cα1L=A(1θ)(KL)θ

The resource constraint is:

(A1.4)C+K˙=AKθL1θ

By dividing this resource constraint by K, we obtain:

(A1.5)CK+K·K=A(LK)1θ

Along the balanced growth path, K·/K is constant over time at g. Equation (A1.5) implies that

(A1.6)C·C=K·K=g

Let X=1-L. From (A1.3), we obtain:

(A1.7)C·CX·X=A·A+θ(K·KL·L)

which is rewritten as:

(A1.8)K·KX·X=A·A+θ(K·KL·L)

since both C and L grow at the same rate.

From (A1.5), we have:

(A1.9)A·A+(1θ)(L·LK·K)=0

(A1.8) and (A1.9) imply that:

(A1.10)X·X=L·L

Since X+L=1, this implies that:

(A1.11)X·X=L·L=0

(A.11) says that labor supply is constant over time along the balanced growth path.

Appendix II. Analysis of Welfafre Effects of Different Government Size

This appendix describes the construction of the measure used in a counter-factual thought experiment to analyze an economy’s welfare in response to different sizes of government.

We measure the welfare effect of the change in tax rate from t0 to t1 in the following way. Let c(t) and h(t) be consumption and hours worked associated with tax rate t. Suppose that tax rate is reduced from t0 to t1, where t0 > t1 > 0. The welfare improvement in consumption equivalent from this change in tax rate, which is denoted by x, is given by:

(A2.1)log[c(t1)(1x)]+αlog[100h(t1)]=log[c(t0)]+αlog[100h(t0)]

Note that while we can measure the welfare effect in terms of consumption, this measure explicitly accounts for the positive partial welfare effect from variations in leisure.

Appendix III. Introducing Risk Aversion and Capital

This appendix relates the consumption output ratio in the formula for predicted hours worked relates to deeper parameters in an intertemporal framework.

Consider a continuous time growth model. Let a production function be:

(A3.1)Y=Kθ(AH)1θ

where Y is output, K is capital stock, A is technology and H is total hours worked. Since we consider a balanced growth path, the technology improvement A needs to be labor-augmenting, as above.

The resource constraint is:

(A3.2)C+K+δK=Kθ(AH)1θ

This resource constraint implies that along the balanced growth path,

(A3.3)g=AA=CC=KK

where g>0 is the exogenous technological growth rate, and H is constant.

The household’s utility is:

(A3.4)U(C,H)=eρt[Ct(NtHt)α]1γ1γdt

where C is consumption and N is working age population, which is equal to the total time endowment, assuming that each person is endowed with one unit of time. Observe that Prescott’s (2002) specification occurs when γ goes to 1.

The household’s budget constraint is:

(A3.5)(1+τc)Ct+Kt+δKt=(1τh)wtHt+(1τk)(rtδ)Kt+Tt

The inter-temporal condition (the Euler equation) is:

(A3.6)g=CC=(1τK)(rtδ)ργ

Since the factor market equilibrium implies that

(A3.7)r=K[Kθ(AH)1θ]=θ(AHK)1θ

the Euler equation above is rewritten as:

(A3.8)CC=g=(1τK)[θ(AHK)1θδ]ργ

Observe that the rise in γ (the more risk averse) decreases (K/AH), which is intuitive.

Observe also that the preference parameter γ does not affect directly the leisure consumption static condition. But it affects indirectly through its effect on K/AH or K/Y. This is seen as follows.

The intra-temporal condition (the first order condition for labor supply) is:

(A3.9)αCtNtHt=(1τh)wt1τc

Let h=H/N be hours worked per working age population. Then, the intra-temporal condition above is rewritten as:

(A3.10)1ht=(1τc)αCt/Nt(1τh)wt

The factor market equilibrium implies:

(A3.11)w=H[Kθ(AH)1θ]=A(1θ)(KAH)θ

From the resource constraint,

(A3.12)CAH=(KAH)θ(g+δ)(KAH)

Hence, we obtain:

(A3.13)Cw=H[(K/AH)θ(g+δ)(K/AH)](1θ)(K/AH)θ

which is decreasing in (K/AH).

By plugging this expression for C/w into the equation for 1-h, we obtain:

(A3.14)1ht1=1+τc1+τhα1θ[1(g+δ)(KAH)1θ]

It is seen from this equation that h is decreasing in capital tax τK.

Also recall that (K/AH) is decreasing in the preference parameter γ. Hence, the rise in γ (the more risk averse) decreases hours worked h through its negative effect on (K/AH).

Appendix IV. Calibrating Labor Market Search Frictions for European Countries Using a Search Model

This appendix describes a model used in the section III.B., which examines the extent to which cross-country variation in labor market frictions may be relevant in explaining hours worked across countries.

A. Model

We consider a variant of the textbook search and matching model of Diamond (1982), Mortensen (1982) and Pissarides (1985, 2000). Time is continuous.

Production

A match of an employed worker and a firm produce 2wh per period, where h is hours worked per period, and w is a parameter of labor productivity.

Wage Determination

Assume that a (flow) wage is determined by splitting equally a (flow) output 2wh between a worker and a firm. While this differs from the standard wage determination rule of Nash bargaining, it greatly simplifies an exposition and can also be justified by an alternating-offer bargaining setup.

Choice of Hours Worked during Employment

Each employed worker chooses how many hours to work (h) by maximizing:

(A4.1)(1τ)wh+αlog(Th)

where τ is a wage income tax rate, T is a time endowment. T-h is leisure. The first order condition for optimal hours worked is given by:

(A4.2)h=Tα(1τ)w
Matching Technology

The labor market is characterized by search frictions. Each unemployed worker contacts a vacant job at a rate μθ 0.5, where θv/u is the vacancy unemployment ratio, with v the number of vacant jobs and u the number of job-seekers. Each vacant job contacts a job-seeker at a rate μθ−0.5.

Value of Unemployment and Employment

The value of job-seeking for each unemployed worker, U, satisfies:

(A4.3)rU=z+μθ0.5[WU]

The value of unemployment comes from (i) the utility during unemployment z, and (ii) an expected “capital” gain from changing a status from unemployment to employment.

The value of employment for the employed, W, satisfies:

(A4.4)rW=(1τ)wh+αlog(Th)+s(UW)

It comprises (i) utility during employment, and (ii) an expected capital gain from changing a state from employment to unemployment, which happens with an exogenous rate s>0.

Free Entry of Vacancies

The free entry of vacant firms implies that:

(A4.5)kμθ0.5=whr+s

which equates a vacancy cost (job opening, recruiting, etc) k with an expected value of a filled job. Recall that each filled job, employing a worker, receives a profit wh per period.

Steady State Accounting

The steady state accounting implies that a flow into unemployment (1-u)s must be equal to a flow out of unemployment pool uμ θ0.5, implying that:

(A4.6)u=ss+μθ0.5
Definition of an Equilibrium:

A steady state equilibrium for the economy described above is a set {U, W, h, θ, u} such that (A4.2), (A4.3), (A4.4), (A4.5) and (A4.6) are satisfied.

B. Calibration Procedure:
Assumptions

  • (1) α, s and μ is constant across countries.

  • (2) k and w differ across countries.

Parameter Choice

We choose one period to be one year.

s=0.3r=0.048
Step 1:

We choose France as a benchmark country.

Using (A4.6) and the observed unemployment rate u, we obtain μ. Normalize θ for France to be unity, without loss of generality.

Then, using (A4.5) and observed hours worked (h), we obtain k. Here, we normalize w for France to equal 1.

Finally, using (A4.2), we pick α.

Step 2:

We compute w(j), k(j) for each country j other than France.

First, using the steady state accounting equation (A4.6), we compute θ(j) for each country j. Recall the assumption that μ and s are constant across countries.

Next, using (A4.2), we compute w(j) for each country j. Note that we are assuming that T and α are constant across countries.

Then, using the free entry equation (A4.5), we compute k(j) for each country j.

C. Calibration Results

Now, we have a set of

{k(j),w(j)}

for each country j.

For France (the benchmark country):

For a benchmark country, France, we have:

μ=s(1u)uθ0.5=0.3(10.11)0.11×1=2.43

Observe that we normalize theta for France to be unity.

α=(1τ)w(Th)=(10.642)×1×(100×5229.17×52)=1319

where we assumed that an hourly wage for France is normalized to unity and yearly time endowment is 100*52 hours.

Using (A4.5), we have:

k(FRA)=μθ0.5×whr+s=2.427×1×1×29.17×520.048+0.3=10578.65

For France, the set is:

k(FRA)=10579,w(FRA)=1,θ(FRA)=1
For the other countries:

Using parameters α and μ common across countries calibrated for a benchmark France above, we calibrate k(j) and w(j) for other countries j other than France.

First, using a steady state accounting equation (A4.6), we compute θ(j) for each country j:

θ(j)=[s[1u(j)]u(j)μ]2

Next, using an equation (A4.2), we compute an hourly wage w(j) for each country j as follows:

w(j)=α[1τ(j)][Th(j)]

Recall that we are assuming that T and α are constant across countries.

Finally, using a free entry equation (A4.5), we compute a flow vacancy cost k(j) for each country j as follows:

k(j)=μ[θ(j)]0.5×w(j)h(j)r+s

Recall that we are assuming that r, s, and μ are constant across countries. Table A4.1. summarizes the calibrated values for each country:

Table A4.1.

Summary of Calibrated Parameter Values

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Source: Authors’ calculations.
1

Nori Tawara was an intern at the IMF when the paper was written. We thank participants in the IMF-EUR seminar, Javier Arze, Robert Arezki, and Werner Schule for very helpful comments and suggestions. All remaining errors are ours.

2

For example, work in the early 19th century posited a positive relation between per capita income and public expenditures (the so-called “Wagner’ Law”).

3

With time endowment T, T-h measures leisure. The representative agent’s time endowment may be thought of as the total amount of time in a given period (say, a week) minus time needed to meet the basic human biological requirements for rest and nutrition. For example, Prescott assumes the weekly time endowment to equal to 100 hours, so that a work week of 40 hours would imply 60 hours of leisure.

4
This latter result is due to the specification of the utility function. It is well known that a balanced growth path is consistent with utility functions of the form:
U=c1γ1γ

of which the above log-utility function is a special case (in which γ → 1). This utility function implies that the income and substitution effects exactly offset each other (see Appendix I). Models have adopted this assumption in order to reconcile the historical patterns of (sharply) rising real wages and not very varying per-capita hours worked (see Hall, 1997). With this utility function underpinning most equilibrium macro models, it is unsurprising that so far no formalization of the Blanchard argument, that stresses the importance of income over substitution effects, has been forthcoming.

5

The current analysis is concerned with long-run equilibrium and therefore adopts a steady-state specification in which deficits will not be possible.

6

In an alternative quantification, Prescott (2004) subtracts military expenditure from current expenditure.

7

This is the fraction of additional labor income that is claimed in taxes, if investment is held fixed. It is important to note that the tax wedge is thus not limited to income taxes alone, as consumption taxes also affect the labor-leisure choice, a fact at times overlooked in discussions of employment-friendly tax reforms.

8

There is some debate as to whether PAYGo pension contributions instead constitute savings. If there is some actuarial component to the contributions, it should indeed be perfectly substitutable to private savings. Disney (2004) estimates internal rates of return of PAYGo systems between minus ½ percent for Switzerland and positive 3.6 percent for Spain, with the OECD average at only 1.2 percent, thus adding support for the treatment of social security contributions as a tax, albeit not in all countries.

9

Prescott (2004) chose this number so that an average of predicted hours worked over his sample matches its observed counterpart. Note, though, that the analysis in the current paper adopts a different definition of consumption from Prescott’s, excluding military expenditure from government consumption (to broaden the sample to countries, where data on defense spending are not available). Thus, it can be argued that a different value for the leisure preference parameter should have been selected for the current analysis, as with higher government (and by, virtue of perfect substitutability, lower private) consumption, labor supply would be lower. On the other hand, choosing a different parameter would have made a comparison of our results with Prescott’s harder. To gage the magnitude of the difference, it is, of course, possible to compare our predictions to Prescott’s for any country that is present in both samples, i.e., France. In the event, predicted hours worked for France in our sample are only slightly below Prescott’s.

10

The improving fit of the model over the 1980s for the U.S. appears somewhat at odds with Mulligan’s (2002) analysis, which, to the contrary, finds labor supply during the 1980s to have become too high to be explained by taxes. This points to a problem underlying such calibration exercises, where structural parameters—in the current context, notably the leisure preference parameter and the labor share—are chosen to fit different samples (in our case, OECD countries 1970-2000, in Mulligan’s more than a century of US data).

11

In addition to the model’s steady state nature, it abstracts from other dynamic aspects, such as the impact of capital taxes and also assumes fairly quick transmission and adoption of the world-production possibility frontier. By the same token, it does not allow for country specific tax-raising technologies (which, would manifest themselves in differential marginal tax multipliers).

12

The general coefficient of determination R2 is defined as: R2RSSTSS, where RSS = residual sum of squares, and TSS = total sum of squares. For a linear estimator, such as an OLS regression, TSS = RSS + ESS, where ESS is the explained sum of squares, such that R2=RSSTSS. For assessing the fit of a non-estimated, calibrated model, the general definition is required.

13

This suggests caution in using this class of model as a benchmark in U.S. economic policy evaluations, e.g., Presscott’s (2004) favorable analysis of privatizing social security.

14

Another option would be to introduce country-specific α parameters, but this would subsume most of the relevant observations in hours worked and not yield interesting policy advise.

15

There are no observations for Canada, Denmark, Greece, Italy and Norway.

16

Similar to the selection of the α parameter in the baseline model, the depreciation rate δ (9.5 percent) was selected so as arrive at the observed sample averages for hours worked and the capital stock.

17

In the context of the debate over the respective merits of different European social policy “models” (e.g., Sapir, 2006), it is interesting that the model underpredicts more often for European countries outside the euro zone (U.K., Sweden, Denmark, and Switzerland), and only two for euro zone countries in the group (Belgium and Finland).

18

If the model were correct, then this implied tax rate would correspond to the actually observed one.

19

This does not mean that the analysis only applies to countries where the basic model under-predicts labor supply. Much rather, there may well be general efficiency-enhancing government activities, but not all countries may be pursuing them. Alternatively, some countries are better at implementing such activities than others, with the latter then exhibiting less labor supply than predicted by the basic model.

20

A different channel could model the government’s role in income redistribution, as was discussed in Knappe (1980). In the current framework, this could be modeled as a (minimum) level of other economic agents’ utility, that enters the aggregate utility function. This would be a public good whose provision will have to rely on mandatory taxes, and that, if achieved, lifts everybody’s utility.

21

Mulligan (2002) introduces the same idea to the analysis of the U.S. labor supply over the 20th century.

22

Of course, Germany, but also likely the Netherlands, have been adversely affected by the German unification shock over much of the sample, which in the current calibration may be picked up in the “productivity” estimate.

23

On the other hand, Fang and Rogerson (2007) pointed out that product market regulation affects labor supply in the same way as taxes, and that a variation of regulation has negligible effects on labor supply to the market. They show that only income transfers to households matter, which usually do not keep pace with regulatory costs, and are in any event, part of the basic model’s government (and thus household) consumption. Our analysis is not subject to these caveats as labor market regulation is shown to have important effects that go beyond transfer payments to households.