The unemployment gap-the distance between actual and ef cient unemployment rates-is a key statistic for macroeconomic policy. In practice, many governments are mandated to reduce the unemployment gap to zero. For example, in the United States, the Humphrey-Hawkins Full Employment Act of mandates the government to maintain the economy at "full employment." Since achieving zero unemployment is physically impossible, full employment should not be interpreted as zero unemployment but rather as an ef cient amount of unemployment;
the mandate of US policymakers therefore is to close the unemployment gap. In theory, many optimal policies also depend on the distance from labor-market ef ciency, measured by the has since been observed in many countries (Jackman, Pissarides, and Savouri ; Nickell et al.
side, more vacancies are required to sustain lower unemployment, as described by the Beveridge curve, which forces rms to allocate more workers to recruiting, thus reducing production. We obtain our formula from the condition that when unemployment is ef cient, the pluses and minuses balance out.
Next, we apply our unemployment-gap formula to the United States. We nd that the ef cient unemployment rate started around % in the s, steadily climbed to almost % in the s, fell just below % in the early s, and remained at that level until . These variations are caused by changes in the level and elasticity of the Beveridge curve. Hence, the US unemployment gap is almost always positive-indicating that the labor market does not generally operate ef ciently, but tends to be inef ciently slack. The unemployment gap is especially high in slumps: as high as percentage points in , . points in , and . points in . Thus, it would be bene cial to implement stabilization policies that reduce unemployment in bad times.
Of the three statistics in our formula, the most uncertain is the nonpecuniary value of unemployment. Our mid-range estimate of the nonpecuniary value of unemployment relative to employment is . (see Borgschulte and Martorell ) , suggesting that unemployed workers derive a small value from unemployment-from added leisure or home production. Yet, survey evidence suggests that the nonpecuniary value of unemployment could be quite negative, possibly due to lower mental health. Using such survey calibration, the ef cient unemployment rate is a bit lower, around %. At the other end of the range of available estimates, some macro studies argue that the nonpecuniary value of unemployment could be almost as high as labor productivity (see Hagedorn and Manovskii ) . Under such calibration the ef cient unemployment rate is much higher, above %, so that unemployment is always inef ciently low, even at the peak of the Great Recession. This result seems implausible, suggesting that such macro calibration overstates the nonpecuniary value of unemployment.
Conceptually, our measure of the unemployment gap is quite different from the two common measures in the literature (see Crump et al. ) . The rst common measure is the gap between actual unemployment and its secular trend. This measure and ours differ because trend unemployment is separate from ef cient unemployment. Indeed, unemployment is generally not ef cient on average (Pissarides , chap. 
We consider a labor market with both unemployed workers and vacant jobs. The unemployment rate u is the number of unemployed workers divided by size of the labor force. The vacancy rate is the number of vacancies divided by size of the labor force. The labor market tightness is the ratio of vacancy rate to unemployment rate: θ = /u. Unemployment rate and vacancy rate are related by a Beveridge curve. Formally, the vacancy rate is given by the function (u), which is strictly decreasing and convex. A key statistic in the measure of the unemployment gap is the Beveridge elasticity:
D . The Beveridge elasticity is the elasticity of the vacancy rate with respect to the unemployment rate along the Beveridge curve, normalized to be positive: The Beveridge curve appears in scatter plots of unemployment and vacancy rates (panels C and D of gure ; for readability, we separately plot the Q -Q and Q -Q periods).
Abraham ( ) was concerned that the Conference Board index might have been a biased proxy for vacancies in the s and s, because at that time, the structure of the newspaper industry signi cantly changed, and business changed how they used help-wanted advertising in response to antidiscrimation laws. However, Zagorsky ( , p. ) nds that such bias is minimal, and that the Conference Board index tracks vacancies well until . The conversion of the index into a vacancy rate is not in Barnichon's article but is implemented in the version of the vacancy proxy, available at https://sites.google.com/site/regisbarnichon/research. Log vacancy rate 1951-1959 1959-1971 1971-1975 1975-1987 C. Beveridge Barnichon ( ) from the Conference Board help-wanted advertising index; for Q -Q , the vacancy rate is the number of job openings measured by the BLS in JOLTS, divided by the civilian labor force constructed by the BLS from the CPS. All unemployment and vacancy rates are quarterly averages of seasonally adjusted monthly series. The shaded areas represent recessions, as identi ed by the National Bureau of Economic Research (NBER). Panels C & D: The Beveridge curve is a scatter plot of log unemployment rate (from panel A) vs. log vacancy rate (from panel B). Panel C depicts the Q -Q period, and panel D the Q -Q period. The panels highlight in color the subperiods during which the Beveridge curve was stable:
In-between quarters, during which the Beveridge curve shi s, are depicted in gray.
The Beveridge curve is stable over long periods of time, and shi s outward or inward every so o en.
The Beveridge curve was stable for seven subperiods, during which unemployment and vacancies moved up and down along a clearly de ned curve: 
The Beveridge curve determines the tradeoff between unemployment and vacancies. Both have welfare costs, which we describe here.
Vacancies. The welfare cost of vacancies arises because lling a vacancy requires labor: ρ > workers per unit time. Hence, vacancies divert labor away from production and toward recruiting-In many models with a matching function, the unemployment rate follows a law of motion, and the Beveridge curve is de ned as the locus of unemployment and vacancy rates that are consistent with a stable level of unemployment in the absence of shocks. Technically, therefore, unemployment and vacancy rates may not be on the Beveridge curve if unemployment moves very slowly over time. However, as noted by Pissarides ( a, p. ), "Perhaps surprisingly at rst, but on re ection not so surprisingly, we get a good approximation to the dynamics of unemployment if we treat unemployment as if it were always on the Beveridge curve." The reason is that labor market ows are so large that a er a shock, the unemployment rate adjusts very rapidly to its new stable level, where in ows into unemployment equal out ows from unemployment (Pissarides ; Hall b; Pissarides b; Elsby, Michaels, and Solon ; Shimer ).
an activity that does not directly contribute to welfare.
The share of the labor force devoted to lling vacancies at any point in time is ρ . Recruiters spend time and effort nding appropriate workers for their rm. As a result, the recruiters do not have time to produce the goods and services sold by rms to consumers. In contrast, a fraction n of the labor force is devoted to the production of goods or services eventually consumed by rms'
customers. Since all workers are either producers or recruiters, and since the employment rate is − u, the share of producers in the labor force is
The number of producers is below the size of the labor force because some workers do not nd jobs (u > ), and some workers are allocated to recruiting instead of producing (ρ > ).
Unemployment. Welfare is determined by the number of producers and unemployed workers.
Unemployed workers contribute positively to welfare if they enjoy additional utility from leisure or through home production; they contribute negatively if they suffer mental and physical health cost or loss of human capital. Since the size of the labor force is taken as given, we assume that welfare is given by a function W(n, u), where u is the unemployment rate, n is the share of the labor force that is employed and devoted to production of consumption goods and services, and the function W is strictly increasing in both arguments.
The welfare cost of unemployment arises because unemployed workers contribute less to social welfare than production workers (if not, it would be optimal to have everybody unemployed). A key statistic in the measure of the unemployment gap is the nonpecuniary value of unemployment relative to production: D . The nonpecuniary value of unemployment is the marginal rate of transformation between unemployed workers and production workers in the welfare function:
The statistic z captures the value of leisure and of home production, net of the psychological costs of being unemployed, relative to the value of market production. It measures the resource value/cost of unemployment and hence should not include unemployment bene ts, which are transfers from employed to unemployed workers. Accordingly, as in Hosios ( ), our analysis abstracts from the issue of insurance-which is covered by Landais, Michaillat, and Saez ( a,b).
The textbook Diamond-Mortensen-Pissarides model, with linear production and utility func-tions, provides a simple example of welfare function. Assume that the size of the labor force is ; the productivity of employed workers is a; and the productivity of unemployed workers in home production is a × h, where the presence of the factor a re ects the fact that technological advances bene t both rm and home production (appliances, computers, infrastructure, and so on), and h < is labor productivity at home relative to on the job. Assume also that the value of time is the same at home and on the job. The welfare function then is W(n, u) = an + ahu = (n + hu)a.
Here the nonpecuniary value of unemployment simply is z = h.
We rst provide a graphical representation of the ef cient unemployment rate and unemployment gap in a Beveridge plan. Then we develop a suf cient-statistic formula for the unemployment gap. The formulas apply to any labor market model in which unemployment and vacancy rates are related by a Beveridge curve.
. Social welfare is given by the following function of the unemployment rate u:
where W(n, u) is the welfare function, ρ is the recruiting cost, and (u) is the Beveridge curve. The e cient unemployment rate, denoted u * , maximizes social welfare. The e cient vacancy rate is * = (u * ), the e cient labor market tightness is θ * = u * / * , and the unemployment gap is u − u * .
As in Hosios ( ), we determine the unemployment rate that maximizes welfare at any point in time. We do not incorporate any dynamical elements to the analysis. This simpli cation allows us to obtain a simple graphical representation and a suf cient-statistic formula.
Since the ef cient unemployment rate maximizes ( ), it satis es the rst-order condition
A er dividing by ∂W/∂n and reshuf ing the terms, we obtain ( )
We assume that the maximization problem is well behaved: the function u → W( −u − ρ (u), u) admits a unique extremum, and the extremum is an interior maximum. Under this assumption, ( ) is a necessary and suf cient condition for optimality. This gives us a rst result: P . In a Beveridge plan, the e cient unemployment rate is found at the point where the Beveridge curve is tangent to a downward-sloping line with slope −( − z)/ρ, where z < is the nonpecuniary value of unemployment and ρ > is the recruiting cost. The condition says that at ef ciency, welfare costs and bene ts from moving one worker from unemployment to employment are equalized. When one worker moves from unemployment to employment, more welfare is produced since z < . Hence, the welfare contribution of an extra job is − z. At the same time, there is a tradeoff between unemployment and vacancies: having one less unemployed worker means having − (u) > more vacancies, which diverts an extra ρ[− (u)] workers away from production and toward recruiting. Hence, the welfare cost of an
The representation of the ef cient unemployment rate in the Beveridge plan offers several comparative-static results, illustrated in panels B, C, and D of gure :
. The e cient unemployment rate increases in the following cases
• when the recruiting cost increases;
• when the nonpecuniary value of unemployment increases;
• when the Beveridge elasticity increases;
• and when mismatch increases (such that the Beveridge curve becomes µ × (u) where µ > is the mismatch factor).
The intuitions are simple. When unemployment is more valuable (higher nonpecuniary value) or vacancies are more costly (higher recruiting cost), then the ef cient unemployment rate increases. When reducing unemployment requires more vacancies (higher Beveridge elasticity), then vacancy-unemployment tradeoff becomes less favorable to unemployment, and the ef cient unemployment rate increases. The same mechanism operates with an increase in mismatch. At any unemployment rate, the Beveridge curve becomes steeper (the derivative becomes µ (u) > (u)), so reducing unemployment requires more vacancies, which implies a higher ef cient unemployment rate.
Over the business cycle, the unemployment and vacancy rates move along the Beveridge curve ( gure , panels C and D). When ρ and z remain stable, such movements lead to uctuations in the unemployment gap. In slumps ( gure , panel E), the unemployment rate is too high, the vacancy rate is too low, and the unemployment gap is positive. In booms ( gure , panel F), the unemployment rate is too low, the vacancy rate is too high, and the unemployment gap is negative.
We rework the optimality condition ( ) to obtain a formula that we can use to measure the unemployment gap in real time. The issue with ( ) is that the slope of the Beveridge curve (u) changes over time, so it would be dif cult to estimate in real time. In contrast, the Beveridge elasticity ϵ = −(u/ ) (u) = − (u)/θ is stable over time, as showed by the linearity of the branches of the Beveridge curve ( gure , panels C and D). We therefore re-express ( ) with the Beveridge elasticity:
which gives at the optimum ϵθ = ( − z)/ρ. Hence the ef cient labor market tightness only depends on three suf cient statistics:
. The e cient labor market tightness only depends on the Beveridge elasticity (ϵ), recruiting cost (ρ), and nonpecuniary value of unemployment (z): Beveridge elasticity ϵ because it controls the tradeoff between unemployment and vacancies. It also features the two parameters measuring the welfare cost of unemployment and vacancies:
the nonpecuniary value of unemployment z < and the recruiting cost ρ > .
We expect the ef cient tightness to be fairly stable over time because it is unaffected by two prevalent labor market shocks. First, it is unaffected by labor-demand shocks, such as productivity shocks, wage shocks, or aggregate demand shocks. These shocks move the labor market along the Beveridge curve without affecting z or ρ. Second, it is unaffected by mismatch shocks, which shi the Beveridge curve inward or outward without affecting ϵ.
The ef cient labor market tightness is easy to visualize on the diagrams in gure : it is the slope of the origin line going through the ef cient point on the Beveridge curve.
With ( ) in hand, we can obtain a formula for the unemployment gap. Guided by the evidence from panels C and D in gure , we assume that the Beveridge curve is isoelastic:
with Beveridge elasticity ϵ > . This isoelastic expression implies that along the Beveridge curve, tightness is related to unemployment rate by θ = (u)/u = u −( +ϵ) and θ * = (u * ) −( +ϵ) . The link between ef cient tightness and ef cient unemployment rate implies that u * = (θ * / o ) − /( +ϵ) .
Hence, in addition to the factors affecting the ef cient tightness, the ef cient unemployment rate is also affected by shi s in the Beveridge curve (changes in ). This implies that in theory, the ef cient labor market tightness is more stable than the ef cient unemployment rate.
From the previous relationships, we link the unemployment gap to the tightness gap:
.
Hence the unemployment gap only depends on current unemployment and vacancy rates, and the same three suf cient statistics:
. The e cient unemployment rate and unemployment gap can be measured from current unemployment rate (u), current vacancy rate ( ), Beveridge elasticity (ϵ), recruiting cost (ρ), and nonpecuniary value of unemployment (z). The e cient unemployment rate is given by
from which the unemployment gap u − u * immediately follows.
The proposition gives an explicit formula for the unemployment gap, expressed in terms of observable suf cient statistics. It is valid in any Beveridgean labor market model, irrespective of rms' production function, workers' utility function, the wage mechanism, or the structure of the labor market. Another advantage of our formula is that we do not need to take a stand on or measure the shocks disturbing the labor market: productivity, wage, labor-force participation, matching function, job separations, and so on. The suf cient statistics are all we need to observe.
We apply formula ( ) to measure the unemployment gap in the United States over the period. The rst step is to measure the following statistics: Beveridge elasticity (ϵ), recruiting cost (ρ), and nonpecuniary value of unemployment (z). We also compare our unemployment-gap measure to other existing measures, and we describe its sensitivity to the nonpecuniary value of unemployment.
We estimate the Beveridge elasticity ϵ by OLS regression of log vacancy rate on log unemployment rate. Since the Beveridge curve shi s over time, we separately estimate the elasticity on the seven subperiods during which the Beveridge curve was stable:
One such regression is illustrated in gure , panel A; the regression results on each subsample are summarized in gure , panel B.
We nd that during the -period, the Beveridge elasticity uctuates between . and . . The Beveridge elasticity steadily increased from . in the s to . in the s, before suddenly dropping back below in , and dropping further to . in . Furthermore the t of the seven linear regressions is very good: the R varies between . and . . Such high R con rms that unemployment and vacancy travel on tightly de ned branches of the Beveridge curve, and that each branch is almost perfectly isoelastic.
To construct the recruiting cost ρ, we rely on the evidence from the National Employer Survey:
in this survey, the Census Bureau asked , establishments about their recruiting process, and found that rms spend . % of their labor costs on recruiting (Villena Roldan ). This means that in , ρ = . % × ( − u). Moreover, the average vacancy rate in is . % ( gure , The Beveridge elasticity is precisely estimated: across the seven regressions the standard error varies between . and . . panel B). At the same time, the average unemployment rate is . % ( gure , panel A). Hence, using these data, we nd that the recruiting cost is ρ = . % × ( − . %)/ . % = . .
States, we assume that the recruiting cost remains constant at its value.
The nonpecuniary value of unemployment z measures the well-being of an unemployed worker, without receiving any monetary transfers from the government or others, relative to the productivity of an employed worker. To measure it, we rely on the work of Borgschulte and Martorell ( ).
Using military administrative data for the -period, they study how servicemembers' choice between reenlisting and exiting the military is affected by the unemployment rate in the local labor market where they would enter. They are able to measure the dollar value of utility lost in the transition to civilian employment when unemployment is one percentage point higher, and compare this value to actual earnings losses for military leavers subject to different labor markets. Their main nding is that between % and % of the estimated earnings loss (the value of employment) is offset by leisure and home production (the nonpecuniary value of unemployment) as well as by public bene ts (the pecuniary value of unemployment)-giving a midpoint of %.
Since servicemembers' bene ts are not observed in the dataset, we abstract from bene ts and set the nonpecuniary value of unemployment to z = . . (Accounting for bene ts would reduce z further.) And since we have no evidence on the time variations of the nonpecuniary value of unemployment, we assume it to be constant.
Using formula ( ), the calibrated statistics, and the unemployment and vacancy rates from gure , we now measure the unemployment gap in the United States between and .
We begin by computing the ef cient labor market tightness using formula ( ) ( gure , panel C). Mirroring the movements of the Beveridge elasticity, the ef cient tightness falls between the s and the s, from . down to . ; it moves back to . in the s and . in the s;
and it climbs to . in the s. Compared to its ef cient level, actual tightness was almost always too low during the period. The only two episodes when tightness was inef ciently high were -, during the Korea war, and -, at the peak of the Vietnam war; tightness is virtually ef cient in . This implies that the US labor market is generally inef ciently slack.
Next, applying formula ( ), we compute the ef cient unemployment rate ( gure , panel To summarize, the US unemployment rate appears generally inef ciently high; this inefciency is exacerbated in slumps; and the inef ciency only disappears in deep booms. These results have implications for macro policies and macro models. First, given that the unemployment increases in recessions are inef cient, it is warranted to deploy scal and monetary policy in slumps to attempt to reduce unemployment. Second, given that the unemployment rate is almost always inef cient, and sometimes markedly so, it might not be productive to insist upon modeling the labor market as ef cient-either by assuming that the Hosios condition holds, or by assuming competitive search (Moen ).
For context, we compare our measure of the unemployment gap to other existing measures.
First, our unemployment gap is higher than the gap between actual unemployment and its secular trend, because ef cient unemployment is lower than trend unemployment. Indeed, Crump et al. ( , g. ) estimate that trend unemployment was around % in the s, rose to % in the mid-s, remained above % until , and slowly declined to . % over the period. Hence trend unemployment is always at least . percentage points above ef cient unemployment.
Second, our unemployment gap is higher than the gap between actual unemployment and the NAIRU, because ef cient unemployment is lower than the NAIRU. Crump et al. ( , g. ) estimate that the NAIRU uctuated around % in the s, climbed to % in the mid-s, remained above % until the mid-s, stayed in the %-% range until , before falling to % in -. Thus the NAIRU is always above ef cient unemployment, and follows a different pattern-but they seem to converge at the end of the sample.
Third, our unemployment gap is higher than the unemployment gap computed by the Congressional Budget Of ce (CBO)-which features prominently in scal and monetary policy discussions (Dickens , p. ) . The CBO compute the unemployment gap as the distance between actual unemployment and the "natural" rate of unemployment. They construct the natural rate by blending trend and NAIRU considerations (Shackleton , app. B). The CBO's natural unemployment rate started at . % in , consistently rose to . % in , and from there persistently declined to reach . % in . The CBO's natural unemployment rate is therefore always above our ef cient unemployment rate, although they were quite close in the s.
Given the uncertainty around the exact nonpecuniary value of unemployment, we consider alternative calibrations and explore their impact on the ef cient unemployment rate.
Our calibration (z = . ) implies that the nonpecuniary value of unemployment is much lower than labor productivity. In contrast, some macro-labor studies argue that unemployed workers derive signi cant utility from leisure and home production. A well-known calibration, due to Hagedorn and Manovskii ( ), is z = . . Such a calibration has a drastic impact on ef cient unemployment: it implies an ef cient unemployment rate always above %, and sometimes as high as %; it also implies an average unemployment gap of − . percentage points ( gure , panel E). Hence the calibration implies that actual unemployment is always much too low, even at the peak of the Great Recession-suggesting that such calibration might be implausible. To describe the effect of the nonpecuniary value of unemployment on the ef cient unemployment rate, we graph the ef cient unemployment rates obtained when the nonpecuniary value of unemployment spans a plausible range: z ∈ [− . , . ] ( gure , panel F). For each nonpecuniary value of unemployment, the ef cient unemployment rate is fairly stable over time, so we summarize its entire time series by its average value over -. When the nonpecuniary value of unemployment is negative, the ef cient unemployment rate is between % and . %. Then when the nonpecuniary value rises from to . , the ef cient unemployment rate increases slowly from . % to %. Last, when the nonpecuniary value is higher than . , the ef cient unemployment rate grows rapidly: % at z = . , % at z = . , % at z = . , and . % when z = . .
Finally, to provide further perspective, we infer from the calculations the nonpecuniary value of unemployment that would render US unemployment ef cient on average. We nd that to obtain an average ef cient unemployment rate of . %, which is average unemployment rate over the -period, the required nonpecuniary value of unemployment is z = . . We can also do the same exercise with the CBO's natural rate of unemployment. To obtain an average ef cient unemployment rate of . %, which is average value of the CBO's natural unemployment rate over the -period, the required nonpecuniary value of unemployment is z = . . Hence, if the nonpecuniary value of unemployment was around z = . , the US unemployment rate would uctuate around its ef cient value-so the average unemployment gap would be zero-and the CBO's natural unemployment rate would overlap with our ef cient unemployment rate.
This paper develops a method to measure the unemployment gap-the distance between actual and ef cient unemployment rates. We consider a labor-market model with only one structural el-ement: a Beveridge curve, which relates unemployment and vacancy rates. Our framework covers a broad set of modern labor market models, including the Diamond-Mortensen-Pissarides model.
In this framework, we express the unemployment gap as a function of actual unemployment and vacancy rates, and three suf cient statistics: elasticity of the Beveridge curve, nonpecuniary value of unemployment, and recruiting cost.
With our formula, we measure the unemployment gap in the United States for -.
We nd that the US unemployment gap is almost always positive. The gap becomes close to zero or slightly negative only in deep booms, and it is especially high in slumps. Hence, the US unemployment rate is generally inef ciently high, and such inef ciency is especially prominent in slumps. The implication is that scal and monetary policy could do more to stabilize the labor market over the business cycle.
We currently have good measures of unemployment and vacancies in the United States (provided by the BLS), from which we can estimate the Beveridge elasticity. To cement our measure of the unemployment gap, it would be valuable to obtain more evidence on the two other statistics.
Measuring the time variations of the recruiting cost would be possible by adding new questions into JOLTS, asking rms to report the number of man-hours devoted to recruiting each month, in addition to the number of vacancies. Obtaining more estimates of the nonpecuniary value of unemployment might be feasible by applying the revealed-preference approach of Borgschulte and Martorell ( ) to other populations (their estimate comes from military personnel). Evidence on the time variations of the nonpecuniary value of unemployment would also be helpful but is less urgent-because the nonpecuniary value of unemployment only has a small effect on the ef cient unemployment rate around our preferred calibration ( gure , panel F). Our methodology could be applied to other countries as well.