Roy Model

Roy Model is the foundational microeconomic and econometric framework of occupational sorting, comparative advantage, and self-selection across multiple sectors, originally formulated by British economist Andrew D. Roy in his 1951 Oxford Economic Papers article, "Some Thoughts on the Distribution of Earnings." Seeking to explain why observed empirical earnings distributions are positively skewed even when underlying human biological or cognitive traits are symmetrically distributed, Roy modeled a population of heterogeneous workers who possess two distinct sector-specific skill endowments—illustrated parabolically as "hunting" and "fishing"—and who rationally select the occupation that maximizes their individual income. Formalized mathematically with bivariate log-normal skill distributions by Gary S. Maddala (1983), James J. Heckman and Bo E. Honoré (1990), and George J. Borjas (1987), the Roy model serves as the workhorse paradigm in labor economics, international migration theory, the economics of education, and structural program evaluation.

Consider an economy with two distinct sectors (or occupations, schooling levels, or countries), indexed by j ∈ {0, 1}. Each individual i in the population is endowed with a pair of sector-specific log-skill (or human capital) residuals (u_0i, u_1i), which are known to the individual at the time of occupational choice. Potential log earnings in Sector 0 (the baseline or source sector) and Sector 1 (the alternative or destination sector) are given by:

ln w_0i = μ₀ + u_0i

ln w_1i = μ₁ + u_1i

where μ₀ and μ₁ denote the population mean log earnings in Sectors 0 and 1, respectively, and the skill innovations (u_0i, u_1i) follow a zero-mean bivariate normal distribution in the population:

(u₀, u₁) ~ N(0, 0, σ₀², σ₁², ρ)

Here, σ₀² = Var(u₀) and σ₁² = Var(u₁) measure the dispersion (or return to skill inequality) in each sector, σ₀₁ = Cov(u₀, u₁) represents the covariance of skills across sectors, and ρ = σ₀₁ / (σ₀σ₁) ∈ [−1, 1] is the correlation coefficient between the two sector-specific abilities.

Assuming workers are income maximizers and switching sectors entails a proportional time-equivalent or psychological cost C (which is zero in a frictionless occupational choice model and positive in a migration or college-attendance model), individual i chooses Sector 1 if and only if the net log wage gain I_i* exceeds zero:

I_i* = ln w_1i − ln w_0i − C = (μ₁ − μ₀ − C) + (u_1i − u_0i) > 0

Define the composite selection error v_i = u_1i − u_0i, which has mean zero and variance σ_v² = σ₁² + σ₀² − 2σ₀₁. Letting z = −(μ₁ − μ₀ − C) / σ_v and defining the standard normal variable ε_i = v_i / σ_v ~ N(0, 1), the probability that a randomly drawn worker selects Sector 1 (D_i = 1) is:

Pr(D_i = 1) = Pr(ε_i > z) = 1 − Φ(z) = Φ(−z)

where Φ(·) denotes the standard normal cumulative distribution function and φ(·) denotes the standard normal probability density function.

Because individuals sort non-randomly based on their private comparative advantage v_i = u_1i − u_0i, the subpopulation observed working in Sector 1 (D_i = 1) and the subpopulation observed in Sector 0 (D_i = 0) are no longer random samples of the general population. Using the properties of the truncated bivariate normal distribution, the expected log earnings of individuals who self-select into Sector 1 and Sector 0 are:

E[ln w₁ | D = 1] = μ₁ + E[u₁ | ε > z] = μ₁ + (σ₀σ₁ / σ_v)(σ₁ / σ₀ − ρ) λ(z)

E[ln w₀ | D = 0] = μ₀ + E[u₀ | ε ≤ z] = μ₀ + (σ₀σ₁ / σ_v)(σ₀ / σ₁ − ρ) [−φ(z) / Φ(z)]

where λ(z) = φ(z) / [1 − Φ(z)] > 0 is the inverse Mills ratio (the hazard rate of the standard normal distribution). Similarly, the counterfactual expected earnings that Sector 1 entrants would have earned had they remained in Sector 0 is:

E[ln w₀ | D = 1] = μ₀ + (σ₀σ₁ / σ_v)(ρ − σ₀ / σ₁) λ(z)

Define Q₀ = E[u₀ | D = 1] and Q₁ = E[u₁ | D = 1] as the average skill selectivity of Sector 1 entrants in Sector 0 and Sector 1 skills, respectively. Because λ(z) > 0 and σ₀σ₁ / σ_v > 0, the signs of Q₀ and Q₁ are governed by the relative dispersion ratio σ₁ / σ₀ and the skill correlation ρ:

Q₁ = (σ₀σ₁ / σ_v)(σ₁ / σ₀ − ρ) λ(z)

Q₀ = (σ₀σ₁ / σ_v)(ρ − σ₀ / σ₁) λ(z)

Notice that Q₁ < 0 and Q₀ > 0 can never occur simultaneously: multiplying σ₁ / σ₀ < ρ and σ₀ / σ₁ < ρ would imply 1 < ρ², contradicting the Cauchy–Schwarz inequality |ρ| ≤ 1. Economically, rational self-selection can never cause workers who are above-average in Sector 0 to systematically abandon their advantage in favor of a sector where they are below-average.

One of the most influential empirical applications of the Roy model is George J. Borjas's (1987) analysis of the skill composition and earnings assimilation of international immigrants. In Borjas's formulation, Sector 0 represents the worker's country of origin (source country) and Sector 1 represents the host country (such as the United States). The correlation coefficient ρ reflects the international transferability of skills: between advanced industrial economies with shared languages and educational credentials, ρ is positive and high; during political upheavals or expropriations where entrepreneurial talent is penalized at home, ρ may be small or negative.

Borjas's Roy model yields three sharp predictions regarding the composition of immigrant flows:

1. Positive Selection from Egalitarian Source Countries (σ₁ > σ₀): When a host country has a more dispersed earnings distribution and higher returns to skill than the source country (for example, migration from a compressed Nordic welfare state to the United States) and skills are transferable (ρ > σ₀ / σ₁), high-skill workers face a strong tax on their talent at home and a high reward abroad. Consequently, migrants are positively selected from the upper tail of both the home and host skill distributions (Q₀ > 0, Q₁ > 0).
2. Negative Selection from High-Inequality Source Countries (σ₀ > σ₁): When the source country exhibits much higher income inequality and higher returns to elite skill than the host country (for example, a developing economy with high skill premia and limited social safety nets), high-skill individuals are heavily rewarded for staying home, while lower-skill workers have the strongest incentive to migrate toward the compressed wage floor of the host country (Q₀ < 0, Q₁ < 0).
3. Refugee Sorting (ρ < min(σ₁/σ₀, σ₀/σ₁)): When a regime change confiscates the wealth or penalizes the specific human capital of high-performing entrepreneurs in the home country, ρ becomes low or negative. Emigrants perform poorly under the new regime in the home distribution (Q₀ < 0) but excel in the market-oriented host economy (Q₁ > 0).

In empirical econometrics, the Roy model highlights the pervasive problem of sample-selection bias: an econometrician never observes both potential outcomes (w_0i, w_1i) for the same individual, observing instead only the realized outcome w_i = D_i w_1i + (1 − D_i)w_0i. For instance, when estimating the Mincer earnings function for college graduates (D = 1) versus high school graduates (D = 0), or for labor force participants versus non-participants, simple ordinary least squares (OLS) comparison of observed average wages confounds the true causal treatment effect (μ₁ − μ₀) with endogenous sorting:

E[ln w₁ | D = 1] − E[ln w₀ | D = 0] = (μ₁ − μ₀) + {E[u₁ | D = 1] − E[u₀ | D = 0]}

Building directly on the structure of the normal Roy model, James J. Heckman (1976, 1979)—recipient of the 2000 Nobel Memorial Prize in Economic Sciences—developed the Heckman two-step sample-selection correction (or Heckit estimator). Suppose individual outcome equations and the latent selection index depend on observed covariates X_i and an exclusion restriction Z_i (an instrumental variable that shifts the cost of sector entry C without directly affecting potential wages w₀ or w₁):

1. Step 1 (Probit Selection Equation): Estimate the binary sector-choice probability Pr(D_i = 1 | X_i, Z_i) = Φ(X_iγ + Z_iδ) via maximum likelihood Probit to obtain consistent estimates of the index (X_iγ̂ + Z_iδ̂) and compute each observation's estimated inverse Mills ratio λ̂_1i = φ(X_iγ̂ + Z_iδ̂) / Φ(X_iγ̂ + Z_iδ̂) for Sector 1 and λ̂_0i = −φ(X_iγ̂ + Z_iδ̂) / [1 − Φ(X_iγ̂ + Z_iδ̂)] for Sector 0.
2. Step 2 (Augmented Outcome Regressions): Estimate the sector-specific log wage equations by OLS including the estimated inverse Mills ratio as an additional control regressor:

ln w_1i = X_iβ₁ + θ₁ λ̂_1i + η_1i (for D_i = 1) ln w_0i = X_iβ₀ + θ₀ λ̂_0i + η_0i (for D_i = 0)

Heckman and Honoré (1990) later proved that without an exclusion restriction Z_i or strong parametric functional-form assumptions, the joint distribution of (u₀, u₁) in the cross-sectional Roy model is non-parametrically under-identified from the distribution of realized maximum earnings max(w₀, w₁) alone. Modern generalized Roy models overcome this identification challenge by introducing continuous instruments Z_i to estimate the Marginal Treatment Effect (MTE) (Heckman & Vytlacil, 2005), which traces out the treatment gain E[ln w₁ − ln w₀ | X, U_D = p] across the entire continuum of unobserved resistance to treatment p ∈ (0, 1).

How does the Roy model explain positive skewness in aggregate earnings distributions?

Even if underlying abilities in every individual occupation follow symmetric normal distributions, the Roy model shows that workers truncate the lower tail of their potential earnings by abandoning the occupation in which they perform poorly and selecting the occupation where their realization is highest: ln w_i = max(ln w_0i, ln w_1i). Taking the upper envelope of two normal distributions shifts probability mass toward the right tail, reinforced further when log skills are exponentiated into dollar earnings w_i = exp(ln w_i).

Why is an exclusion restriction important when estimating a Roy model with the Heckman two-step procedure?

If the identical covariates X_i appear in both the first-stage Probit selection equation and the second-stage outcome equation without any excluded instrument Z_i that affects sector choice C independently of potential wages, identification rests only on the non-linearity of the inverse Mills ratio λ(·). Because λ(·) is nearly linear over the middle range of its domain, omitting an exclusion restriction causes strong multicollinearity and fragile parameter estimates.

What is the difference between absolute advantage and comparative advantage in the Roy model?

An individual has an absolute advantage in Sector 1 if their skill rank in Sector 1 exceeds the population median (u_1i > 0) and exceeds their own performance in Sector 0. By contrast, an individual has a comparative advantage in Sector 1 whenever their relative productivity in Sector 1 exceeds that in Sector 0 (u_1i − u_0i > μ₀ − μ₁ + C), even if they are below the population average in both sectors (u_0i < 0 and u_1i < 0). In the Roy model, occupational choice is governed by comparative advantage.

1. Roy, A. D. (1951). Some thoughts on the distribution of earnings. Oxford Economic Papers, 3(2), 135–146. https://doi.org/10.1093/oxfordjournals.oep.a041827
2. Heckman, J. J. (1979). Sample selection bias as a specification error. Econometrica, 47(1), 153–161. https://doi.org/10.2307/1912352
3. Borjas, G. J. (1987). Self-selection and the earnings of immigrants. The American Economic Review, 77(4), 531–553. https://doi.org/10.3386/w2248
4. Heckman, J. J., & Honoré, B. E. (1990). The empirical content of the Roy model. Econometrica, 58(5), 1121–1149. https://doi.org/10.2307/2938303
5. Heckman, J. J., & Vytlacil, E. (2005). Structural equations, treatment effects, and econometric policy evaluation. Econometrica, 73(3), 669–738. https://doi.org/10.1111/j.1468-0262.2005.00594.x