The BenPorath Model.
Assume the Human Capital Production Function is \(I_t= \sqrt(2) \left(\theta_t H_t \right)^{(1/2)}\) where
\(\theta_t\) is the share of time spend on investing on human capital at t,
\(H_t\) is the human capital at t, and I_t represents new units of human capital produced at t. This unit of human capital can be used in the market for the first time in the next period. Assume that human capital depreciates at rate \(\delta\) and individuals discount the future at rate r. Further assume that the rental rate per unit of human capital is w and constant over time.
Assume individuals work until they are 60 years old.
 Show the exact expression of the value of a unit of human capital at age t, where t is assumed to be subsequent to schooling.All else equal, holding age fixed, the value of a unit of human capital is larger if an individual is still in school than if the individual has left schooling. Explain why.
 Value of human capital at age t
\[\begin{align}
V(age=60) &= 0 \\
V(age=59) &= \dfrac{1}{1+r} w_{60} \\
V(age=58) &= \dfrac{1}{1+r} \left[w_{59}+ \left(\dfrac{1\delta}{1+r} \right) w_{60}\right] \\
V(age=57) &= \dfrac{1}{1+r} \left[ w_{58} + \left(\dfrac{1\delta}{1+r} \right) w_{59}+\left(\dfrac{1\delta}{1+r} \right)^2 w_{60}\right] \\
V(age=56) &= \dfrac{1}{1+r} \left[ w_{57} + \left(\dfrac{1\delta}{1+r} \right) w_{58} + \left(\dfrac{1\delta}{1+r} \right)^2 w_{59}+\left(\dfrac{1\delta}{1+r} \right)^3 w_{60}\right] \\
. & \\
. & \\
V(age=t) &= \dfrac{1}{1+r} \left[ w_{t+1} + \left(\dfrac{1\delta}{1+r} \right) w_{t+2}+\left(\dfrac{1\delta}{1+r} \right)^2 w_{t+3} +... +\left(\dfrac{1\delta}{1+r} \right)^{60t1} w_{60} \right]
\end{align}\]
If the wage is constant over time :
\(w_t=w\)
\[\begin{align}
V(age=t) &= \dfrac{w}{1+r} \left[ 1 + \left(\dfrac{1\delta}{1+r} \right) +\left(\dfrac{1\delta}{1+r} \right)^2 +... +\left(\dfrac{1\delta}{1+r} \right)^{60t1} \right] \\
& = \dfrac{w}{1+r} \left[ \dfrac{1 \left(\dfrac{1\delta}{1+r} \right)^{60t}}{1 \left(\dfrac{1\delta}{1+r} \right)} \right]
\end{align}\]
 Derive the cost function of investing into human capital.
\[\begin{align}
C(I_t) & = w_t \theta H_t \\
& = w_t \dfrac{I_t^2}{2}
\end{align}\]
The marginal cost of investing in human capital is the following:
\[\begin{align}
\dfrac{\partial C(I_t) }{\partial I_t}& = w_t * I_{t}
\end{align}\]
For the next few question, assume that the depreciation rate is \(\delta\)=3% and the discount rate is r=2%.
 Assume that an individual graduates from school at age 20. How much human capital does he/she have at that age.
At the equilibrium, the marginal cost of investing is equal to value of human capital. With this equilibrium, we can find the desired investement.
 At the equilibrium, we have the following .
\[\begin{align}
V_t & = MC(I_t)\\
\dfrac{w}{1+r} \left[ \dfrac{1 \left(\dfrac{1\delta}{1+r} \right)^{60t}}{1 \left(\dfrac{1\delta}{1+r} \right)} \right] & = w * I_t \\
\end{align}\]
 Let assume he invest in education at full time \(\theta=1\). THe human capital accumulation at age 20.
## [1] 17.32149
 Use Excel, R, or Stata to fill in the following table if the rental rate per unit of human capital is $5,000 to produce a graph that shows investment \(I_t\) , Potential (or Gross) Earnings, and Realized (or Net) Earnings as a function of age from age 20 to 60. Label the axes correctly and indicate the ages at which Potential and Realized Earnings respectively are highest.
Approach 1:
At each moment, it is possible to work or study but not to do both at the same time.
 Find the desired investment at each time t.
\[I_{t}^d=V_{t}/w \]
 If the value of human capital is superior to the marginal cost \(V_t > w * I_t =\sqrt2(\theta_t H_t)^{1/2}\), invest full time on human capital \(\theta_t=1\). If the value of human capital is \(V_t > MC(I_t)\), then invest only to maintain the equilibrium. The realized investment :
\[I^*_t = \min(I_t^d,\sqrt2(H_t)^{1/2})\]
 Using the law of motion of human capital where \(\delta\) represents the depreciation rate, find the human capital accumluation at every period.
\[H_{t+1}= H_{t}(1\delta) + I_{t}\]
Approach 2:
 Step 1 : Find the desired time to spend on human capital from the equilibrium
\[\begin{align}
V_{t} & = w* \sqrt2( \theta_t^d H_t)^{1/2} \\
V_{t}/w & = \sqrt2( \theta_t^d H_t)^{1/2} \\
V_{t}/w & = \sqrt2( \theta_t^d H_t)^{1/2} \\
(\dfrac{V_{t}}{\sqrt2 w})^2* \dfrac{1}{H_t } & = \theta_t^d
\end{align}\]
 Step 2 : Find the optimal share of time to spend on investing on human capital knowing the full time is 1.
\[ \theta^*_t = \min (1, \theta_t^d)\] Do this iteration until age of retirement, which is 60 in this exercise.
Table 1: human capital accumulation
20 
5 
86.60747 
17.32149 
5.885830 
86.60747 
0 
21 
5 
85.91714 
22.68768 
6.736123 
113.43840 
0 
22 
5 
85.19122 
28.74317 
7.581975 
143.71586 
0 
23 
5 
84.42788 
35.46285 
8.421740 
177.31426 
0 
24 
5 
83.62519 
42.82071 
9.254265 
214.10353 
0 
25 
5 
82.78113 
50.79035 
10.078725 
253.95175 
0 
26 
5 
81.89356 
59.34536 
10.894528 
296.72682 
0 
27 
5 
80.96024 
68.45953 
11.701242 
342.29766 
0 
28 
5 
79.97880 
78.10699 
12.498559 
390.53494 
0 
29 
5 
78.94678 
88.26234 
13.286259 
441.31168 
0 
 Value of HC (in thousands), Investment in HC, Human capital accumulation and Earnings (in thousands).
Application: Return to schooling, wage gap decomposition.
Consider the representative sample extract Can_Oaxaca based on the July 2018 wave of the Canadian Labor Force Survey collected by StatCan. The purpose of this exercise is to decompose the mean wage gap two groups: Native and foreign born (peoples born outside Canada). Each group should chose one province as indicated in table 1.
24 
Quebec 
35 
Ontario 
46 
Manitoba 
48 
Alberta 
59 
British Columbia 
 Calculate the average hourly wage (hrlyearn) for Native and foreignborn. Are foreign born workers earning less than Native (use the variable foreign_born which take yes if the worker was born outside Canada)?
Table 1: Avg wage by country of birth
Alberta 
31.137 
27.288 
BritishColumbia 
26.719 
25.391 
Manitoba 
24.775 
21.051 
Ontario 
27.039 
27.095 
Quebec 
25.048 
24.177 
 What fraction of workers has college degree (use the variable educ) in each group (native and foreignborn)? What is the average experience (exper_year) in each group? Discuss briefly what you observe? Are foreignborn workers more educated than Native? Are foreignborn workers have more experience than native?
Table 2: Endowments differences between Native and foreign
Alberta 
0.637 
0.736 
8.168 
7.188 
BritishColumbia 
0.662 
0.738 
8.027 
7.992 
Manitoba 
0.622 
0.748 
8.900 
6.966 
Ontario 
0.692 
0.763 
8.791 
8.909 
Quebec 
0.730 
0.836 
9.049 
6.880 
 Regress separately the log usual hourly wages (hrlyearn) on level education (educ) and experience (exper_year). Interpret the parameters \(\beta_{1,g}\) in sentences. Is having college degree more valued for native than foreignborn in the Canadian labour market?
The \(h_{i,g}\) is he hourly wage of individual I belonging to the group g. There are two group g={ N=Native, F= Foreignborn}. The \(s_{i,g}\) and \(E_{i,g}\) represents the level of education (educ in the database) and the number of years of experience of individual i belonging in group g , respectively.
Table 3: Wage Equations
Province 
Alberta 

Intercept 
3.248 
3.08 
LHS 
0.278 
0.252 
Experience 
0.024 
0.026 
Province 
BritishColumbia 

Intercept 
3.066 
3.045 
LHS 
0.234 
0.257 
Experience 
0.026 
0.02 
Province 
Manitoba 

Intercept 
2.967 
2.848 
LHS 
0.254 
0.209 
Experience 
0.027 
0.023 
Province 
Ontario 

Intercept 
3.072 
3.102 
LHS 
0.291 
0.321 
Experience 
0.025 
0.018 
Province 
Quebec 

Intercept 
3 
3.018 
LHS 
0.291 
0.336 
Experience 
0.023 
0.018 
 Now, let decompose and identify what part of the wage gap between these two groups’ stems from education and what is left unexplained. According to Oaxacadecomposition, the wage gap can be decomposed in two parts.
 We want to decompose the wage gap by taking the native as the reference group. The main determinants of wage in this labor market are education and experience. We assume also the relation is linear.
\[\begin{align}
\large \overline{\log w}_f & = \alpha_f + \overline{S_f}\hat{\beta}_f + \overline{E}_f \hat{\gamma}_f \\
\large \overline{\log w}_n & = \alpha_n + \overline{S_n}\hat{\beta}_n + \overline{E}_n \hat{\gamma}_n
\end{align}\]
\[\begin{align}
\Large \overline{\log w}_n  \large \overline{\log w}_f = & \hat{\alpha}_n \hat{\alpha}_f + \overline{S}_n\hat{\beta}_n + \underbrace{\overline{S}_f\hat{\beta}_n  \overline{S}_f\hat{\beta}_n}_{ = 0} \overline{S}_f\hat{\beta}_f \\ & + \overline{E}_n \hat{\gamma}_n + \underbrace{ \overline{E}_f \hat{\gamma}_n  \overline{E}_f \hat{\gamma}_n}_{ = 0}  \overline{E}_f \hat{\gamma}_f \\
& \\
\Large = & \underbrace{\left(\bar{S_n}\bar{S_f} \right)\beta_n + \left(\bar{E_n}\bar{E_f} \right)\hat{\gamma}_{n} }_{\text{Explained component}} \\
\large & + \underbrace{ (\hat{\alpha}_n \hat{\alpha}_f) + \left(\hat{\beta}_n\hat{\beta}_f\right)\overline{S}_f + \left(\hat{\gamma}_n\hat{\gamma}_f\right)\overline{E}_f }_{\text{Unxplained component }}
\end{align}\]
Approach 1: By hand
calculate explained and unexplained components using table 2 and 3. Let take Quebec for this illustration.
Explained education: \(\left( \bar{S}_n  \bar{S}_f \right) \hat{\beta}_n\) = (0.7300.836)*0.291 = 0.030846.
Explained experience: \(\left(\bar{E_n}\bar{E_f} \right) \hat{\gamma}_n\)= (9.0496.880)*0.023= 0.049887
Explained part =Explained education + Explained experience= 0.019041.
Do the same for the unexplained component.
Approach 2: Software
 See the command oaxaca in stata or R. You observe a little difference between decomposition from software and decomposition from what you did by hand. It is normal because in the software they did many regressions with different subsample to take the average of the coefficient and get also the s.e of the coefficient (Don’t worry about). The example is to shows it is possible to calculate this decomposition by hand if you have endowments and cofficients from the regressions.
Table 4: Wage gap decomposition with native as reference group
Province: 
Alberta 



Total gap 
0.13626 
1 
0.01069 
0.14695 
Intercept 

1 
0 
0.16727 
Education 

1 
0.02805 
0.00677 
Experience 

1 
0.01736 
0.01355 
Province: 
BritishColumbia 



Total gap 
0.05265 
1 
0.02025 
0.0729 
Intercept 

1 
0 
0.02134 
Education 

1 
0.02027 
0.00615 
Experience 

1 
2e05 
0.04541 
Province: 
Manitoba 



Total gap 
0.14788 
1 
0.01014 
0.13774 
Intercept 

1 
0 
0.11862 
Education 

1 
0.03253 
0.011 
Experience 

1 
0.04267 
0.03012 
Province: 
Ontario 



Total gap 
0.00312 
1 
0.02486 
0.02799 
Intercept 

1 
0 
0.03085 
Education 

1 
0.02222 
0.00708 
Experience 

1 
0.00264 
0.05176 
Province: 
Quebec 



Total gap 
0.03688 
1 
0.01554 
0.02134 
Intercept 

1 
0 
0.01884 
Education 

1 
0.03082 
0.00737 
Experience 

1 
0.04636 
0.03282 