From 7928fbecb124d233627d2cbea679864f11388ecc Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Mon, 22 Jul 2024 12:38:57 +1000 Subject: [PATCH 1/8] [equalizing_difference] Add an exercise --- lectures/equalizing_difference.md | 177 ++++++++++++++++-------------- 1 file changed, 97 insertions(+), 80 deletions(-) diff --git a/lectures/equalizing_difference.md b/lectures/equalizing_difference.md index 6a756af9..4858ae42 100644 --- a/lectures/equalizing_difference.md +++ b/lectures/equalizing_difference.md @@ -276,86 +276,6 @@ plt.ylabel(r'wage gap') plt.show() ``` -## Entrepreneur-worker interpretation - -We can add a parameter and reinterpret variables to get a model of entrepreneurs versus workers. - -We now let $h$ be the present value of a "worker". - -We define the present value of an entrepreneur to be - -$$ -c_0 = \pi \sum_{t=4}^T R^{-t} w_t^c -$$ - -where $\pi \in (0,1) $ is the probability that an entrepreneur's "project" succeeds. - -For our model of workers and firms, we'll interpret $D$ as the cost of becoming an entrepreneur. - -This cost might include costs of hiring workers, office space, and lawyers. - -What we used to call the college, high school wage gap $\phi$ now becomes the ratio -of a successful entrepreneur's earnings to a worker's earnings. - -We'll find that as $\pi$ decreases, $\phi$ increases, indicating that the riskier it is to -be an entrepreuner, the higher must be the reward for a successful project. - -Now let's adopt the entrepreneur-worker interpretation of our model - -```{code-cell} ipython3 -# Define a model of entrepreneur-worker interpretation -EqDiffModel = namedtuple('EqDiffModel', 'R T γ_h γ_c w_h0 D π') - -def create_edm_π(R=1.05, # gross rate of return - T=40, # time horizon - γ_h=1.01, # high-school wage growth - γ_c=1.01, # college wage growth - w_h0=1, # initial wage (high school) - D=10, # cost for college - π=0 # chance of business success - ): - - return EqDiffModel(R, T, γ_h, γ_c, w_h0, D, π) - - -def compute_gap(model): - R, T, γ_h, γ_c, w_h0, D, π = model - - A_h = (1 - (γ_h/R)**(T+1)) / (1 - γ_h/R) - A_c = (1 - (γ_c/R)**(T-3)) / (1 - γ_c/R) * (γ_c/R)**4 - - # Incorprate chance of success - A_c = π * A_c - - ϕ = A_h / A_c + D / (w_h0 * A_c) - return ϕ -``` - -If the probability that a new business succeeds is $0.2$, let's compute the initial wage premium for successful entrepreneurs. - -```{code-cell} ipython3 -ex3 = create_edm_π(π=0.2) -gap3 = compute_gap(ex3) - -gap3 -``` - -Now let's study how the initial wage premium for successful entrepreneurs depend on the success probability. - -```{code-cell} ipython3 -π_arr = np.linspace(0.2, 1, 50) -models = [create_edm_π(π=π) for π in π_arr] -gaps = [compute_gap(model) for model in models] - -plt.plot(π_arr, gaps) -plt.ylabel(r'wage gap') -plt.xlabel(r'$\pi$') -plt.show() -``` - -Does the graph make sense to you? - - ## An application of calculus @@ -488,3 +408,100 @@ Let's compute $\frac{\partial \phi}{\partial R}$ and evaluate it numerically at ``` We find that raising the gross interest rate $R$ increases the initial college wage premium $\phi$, in line with our earlier graphical analysis. + + +## Exercise +```{exercise-start} +:label: edm_ex1 +``` +In this exercise, We can add a parameter and reinterpret variables to get a model of entrepreneurs versus workers. + +We now let $h$ be the present value of a "worker". + +We define the present value of an entrepreneur to be + +$$ +c_0 = \pi \sum_{t=4}^T R^{-t} w_t^c +$$ + +where $\pi \in (0,1) $ is the probability that an entrepreneur's "project" succeeds. + +For our model of workers and firms, we'll interpret $D$ as the cost of becoming an entrepreneur. + +This cost might include costs of hiring workers, office space, and lawyers. + +What we used to call the college, high school wage gap $\phi$ now becomes the ratio +of a successful entrepreneur's earnings to a worker's earnings. + +We'll find that as $\pi$ decreases, $\phi$ increases, indicating that the riskier it is to +be an entrepreuner, the higher must be the reward for a successful project. + +Now adopt the entrepreneur-worker interpretation of our model. + +```{exercise-end} +``` + +```{solution-start} edm_ex1 +:class: dropdown +``` + +Here is one solution + +```{code-cell} ipython3 +# Define a model of entrepreneur-worker interpretation +EqDiffModel = namedtuple('EqDiffModel', 'R T γ_h γ_c w_h0 D π') + +def create_edm_π(R=1.05, # gross rate of return + T=40, # time horizon + γ_h=1.01, # high-school wage growth + γ_c=1.01, # college wage growth + w_h0=1, # initial wage (high school) + D=10, # cost for college + π=0 # chance of business success + ): + + return EqDiffModel(R, T, γ_h, γ_c, w_h0, D, π) + + +def compute_gap(model): + R, T, γ_h, γ_c, w_h0, D, π = model + + A_h = (1 - (γ_h/R)**(T+1)) / (1 - γ_h/R) + A_c = (1 - (γ_c/R)**(T-3)) / (1 - γ_c/R) * (γ_c/R)**4 + + # Incorprate chance of success + A_c = π * A_c + + ϕ = A_h / A_c + D / (w_h0 * A_c) + return ϕ +``` + +If the probability that a new business succeeds is $0.2$, let's compute the initial wage premium for successful entrepreneurs. + +```{code-cell} ipython3 +ex3 = create_edm_π(π=0.2) +gap3 = compute_gap(ex3) + +gap3ex3 = create_edm_π(π=0.2) +gap3 = compute_gap(ex3) + +gap3 +``` + +Now let's study how the initial wage premium for successful entrepreneurs depend on the success probability. + +```{code-cell} ipython3 +π_arr = np.linspace(0.2, 1, 50) +models = [create_edm_π(π=π) for π in π_arr] +gaps = [compute_gap(model) for model in models] + +plt.plot(π_arr, gaps) +plt.ylabel(r'wage gap') +plt.xlabel(r'$\pi$') +plt.show() +``` + +Does the graph make sense to you? + +```{solution-end} +``` From 328508f1625d056c96f7fb46253d04137c421ecd Mon Sep 17 00:00:00 2001 From: JingkunZhao <155940781+SylviaZhaooo@users.noreply.github.com> Date: Mon, 22 Jul 2024 16:34:21 +1000 Subject: [PATCH 2/8] Update lectures/equalizing_difference.md Co-authored-by: Matt McKay --- lectures/equalizing_difference.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lectures/equalizing_difference.md b/lectures/equalizing_difference.md index 4858ae42..546eaf28 100644 --- a/lectures/equalizing_difference.md +++ b/lectures/equalizing_difference.md @@ -410,7 +410,7 @@ Let's compute $\frac{\partial \phi}{\partial R}$ and evaluate it numerically at We find that raising the gross interest rate $R$ increases the initial college wage premium $\phi$, in line with our earlier graphical analysis. -## Exercise +## Exercises ```{exercise-start} :label: edm_ex1 ``` From f9854a840f120d1c2ce0ee97397e3e899926d235 Mon Sep 17 00:00:00 2001 From: mmcky Date: Tue, 23 Jul 2024 11:19:39 +1000 Subject: [PATCH 3/8] minor spelling and spacing; --- lectures/equalizing_difference.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/lectures/equalizing_difference.md b/lectures/equalizing_difference.md index 546eaf28..8bb85a88 100644 --- a/lectures/equalizing_difference.md +++ b/lectures/equalizing_difference.md @@ -411,6 +411,7 @@ We find that raising the gross interest rate $R$ increases the initial college w ## Exercises + ```{exercise-start} :label: edm_ex1 ``` @@ -434,7 +435,7 @@ What we used to call the college, high school wage gap $\phi$ now becomes the ra of a successful entrepreneur's earnings to a worker's earnings. We'll find that as $\pi$ decreases, $\phi$ increases, indicating that the riskier it is to -be an entrepreuner, the higher must be the reward for a successful project. +be an entrepreneur, the higher must be the reward for a successful project. Now adopt the entrepreneur-worker interpretation of our model. From 27756a8d5b4c7f1e78fcd19363f159532d83458a Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Tue, 23 Jul 2024 22:20:34 +1000 Subject: [PATCH 4/8] Update equalizing_difference.md --- lectures/equalizing_difference.md | 52 ++++++++++++++++++++++++++++--- 1 file changed, 47 insertions(+), 5 deletions(-) diff --git a/lectures/equalizing_difference.md b/lectures/equalizing_difference.md index 8bb85a88..4186bd12 100644 --- a/lectures/equalizing_difference.md +++ b/lectures/equalizing_difference.md @@ -415,7 +415,7 @@ We find that raising the gross interest rate $R$ increases the initial college w ```{exercise-start} :label: edm_ex1 ``` -In this exercise, We can add a parameter and reinterpret variables to get a model of entrepreneurs versus workers. +In this exercise, we add a parameter and reinterpret variables to get a model of entrepreneurs versus workers and do some computations like before. We now let $h$ be the present value of a "worker". @@ -437,7 +437,19 @@ of a successful entrepreneur's earnings to a worker's earnings. We'll find that as $\pi$ decreases, $\phi$ increases, indicating that the riskier it is to be an entrepreneur, the higher must be the reward for a successful project. -Now adopt the entrepreneur-worker interpretation of our model. +Now define `create_edm_π` and `compute_gap` following the previous *Computations* section, adopting the entrepreneur-worker interpretation of our model. + +Given: + +```{code-cell} ipython3 +R=1.05, # gross rate of return +T=40, # time horizon +γ_h=1.01, # high-school wage growth +γ_c=1.01, # college wage growth +w_h0=1, # initial wage (high school) +D=10, # cost for college +π=0 # chance of business success +``` ```{exercise-end} ``` @@ -446,8 +458,6 @@ Now adopt the entrepreneur-worker interpretation of our model. :class: dropdown ``` -Here is one solution - ```{code-cell} ipython3 # Define a model of entrepreneur-worker interpretation EqDiffModel = namedtuple('EqDiffModel', 'R T γ_h γ_c w_h0 D π') @@ -476,8 +486,20 @@ def compute_gap(model): ϕ = A_h / A_c + D / (w_h0 * A_c) return ϕ ``` +```{solution-end} +``` + +```{exercise-start} +:label: edm_ex2 +``` +If the probability that a new business succeeds is $0.2$, what is the initial wage premium for successful entrepreneurs? + +```{exercise-end} +``` -If the probability that a new business succeeds is $0.2$, let's compute the initial wage premium for successful entrepreneurs. +```{solution-start} edm_ex2 +:class: dropdown +``` ```{code-cell} ipython3 ex3 = create_edm_π(π=0.2) @@ -489,8 +511,28 @@ gap3 = compute_gap(ex3) gap3 ``` +```{solution-end} +``` + +```{exercise-start} +:label: edm_ex3 +``` Now let's study how the initial wage premium for successful entrepreneurs depend on the success probability. +Given + +```{code-cell} ipython3 +π_arr = np.linspace(0.2, 1, 50) +``` + +Plot the relationship between the wage gap and the values of $\pi$. + +```{exercise-end} +``` + +```{solution-start} edm_ex3 +:class: dropdown +``` ```{code-cell} ipython3 π_arr = np.linspace(0.2, 1, 50) models = [create_edm_π(π=π) for π in π_arr] From c206d180b7abfb4b1e7611680c736ca41a28f070 Mon Sep 17 00:00:00 2001 From: Humphrey Yang Date: Wed, 24 Jul 2024 11:30:15 +1000 Subject: [PATCH 5/8] update sympy code and exercise --- lectures/equalizing_difference.md | 111 +++++++++++++++++++----------- 1 file changed, 69 insertions(+), 42 deletions(-) diff --git a/lectures/equalizing_difference.md b/lectures/equalizing_difference.md index dc553725..4396d072 100644 --- a/lectures/equalizing_difference.md +++ b/lectures/equalizing_difference.md @@ -51,7 +51,9 @@ As usual, we'll start by importing some Python modules. import numpy as np import matplotlib.pyplot as plt from collections import namedtuple -from sympy import Symbol, Lambda, symbols +from sympy import Symbol, Lambda, symbols, refine, \ + Sum, simplify, Eq, solve, Lambda,\ + lambdify, And ``` ## The indifference condition @@ -276,7 +278,6 @@ plt.ylabel(r'wage gap') plt.show() ``` - ## An application of calculus So far, we have used only linear algebra and it has been a good enough tool for us to figure out how our model works. @@ -294,28 +295,41 @@ We'll use the Python module 'sympy' to compute partial derivatives of $\phi$ wit Define symbols ```{code-cell} ipython3 -γ_h, γ_c, w_h0, D = symbols('\gamma_h, \gamma_c, w_0^h, D', real=True) -R, T = Symbol('R', real=True), Symbol('T', integer=True) +R, w_h0, w_c0, γ_c, γ_h, ϕ, D, t, T = symbols( + 'R w^h_0 w^c_0 gamma_c gamma_h phi D t T', positive=True) + +refine(γ_c, γ_c>1) +refine(γ_h, γ_h>1) +refine(R, R>1) + +# Define the wage for college +# and high school graduates at time t +w_ct = w_c0 * γ_c**t +w_ht = w_h0 * γ_h**t ``` Define function $A_h$ ```{code-cell} ipython3 -A_h = Lambda((γ_h, R, T), (1 - (γ_h/R)**(T+1)) / (1 - γ_h/R)) +h_0 = Sum(R**-t * w_ht, (t, 0, T)) +A_h = simplify(h_0.doit() / w_h0) +A_h = simplify(A_h.args[1][0]) A_h ``` Define function $A_c$ ```{code-cell} ipython3 -A_c = Lambda((γ_c, R, T), (1 - (γ_c/R)**(T-3)) / (1 - γ_c/R) * (γ_c/R)**4) +c_0 = Sum(R**-t * w_ct, (t, 4, T)) +A_c = simplify(c_0.doit() / w_c0) +A_c = simplify(A_c.args[1][0]) A_c ``` Now, define $\phi$ ```{code-cell} ipython3 -ϕ = Lambda((D, γ_h, γ_c, R, T, w_h0), A_h(γ_h, R, T)/A_c(γ_c, R, T) + D/(w_h0*A_c(γ_c, R, T))) +ϕ = A_h/A_c + D/(w_h0*A_c) ``` ```{code-cell} ipython3 @@ -330,19 +344,27 @@ T_value = 40 γ_h_value, γ_c_value = 1.01, 1.01 w_h0_value = 1 D_value = 10 + +symbol_subs = {D: D_value, + γ_h: γ_h_value, + γ_c: γ_c_value, + R: R_value, + T: T_value, + w_h0: w_h0_value} + +ϕ.subs(symbol_subs) ``` Now let's compute $\frac{\partial \phi}{\partial D}$ and then evaluate it at the default values ```{code-cell} ipython3 -ϕ_D = ϕ(D, γ_h, γ_c, R, T, w_h0).diff(D) +ϕ_D = ϕ.diff(D) ϕ_D ``` ```{code-cell} ipython3 # Numerical value at default parameters -ϕ_D_func = Lambda((D, γ_h, γ_c, R, T, w_h0), ϕ_D) -ϕ_D_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value) +ϕ_D.subs(symbol_subs) ``` Thus, as with our earlier graph, we find that raising $R$ increases the initial college wage premium $\phi$. @@ -350,14 +372,13 @@ Thus, as with our earlier graph, we find that raising $R$ increases the initial Compute $\frac{\partial \phi}{\partial T}$ and evaluate it at default parameters ```{code-cell} ipython3 -ϕ_T = ϕ(D, γ_h, γ_c, R, T, w_h0).diff(T) +ϕ_T = ϕ.diff(T) ϕ_T ``` ```{code-cell} ipython3 # Numerical value at default parameters -ϕ_T_func = Lambda((D, γ_h, γ_c, R, T, w_h0), ϕ_T) -ϕ_T_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value) +ϕ_T.subs(symbol_subs) ``` We find that raising $T$ decreases the initial college wage premium $\phi$. @@ -367,14 +388,13 @@ This is because college graduates now have longer career lengths to "pay off" th Let's compute $\frac{\partial \phi}{\partial γ_h}$ and evaluate it at default parameters. ```{code-cell} ipython3 -ϕ_γ_h = ϕ(D, γ_h, γ_c, R, T, w_h0).diff(γ_h) +ϕ_γ_h = ϕ.diff(γ_h) ϕ_γ_h ``` ```{code-cell} ipython3 # Numerical value at default parameters -ϕ_γ_h_func = Lambda((D, γ_h, γ_c, R, T, w_h0), ϕ_γ_h) -ϕ_γ_h_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value) +ϕ_γ_h.subs(symbol_subs) ``` We find that raising $\gamma_h$ increases the initial college wage premium $\phi$, in line with our earlier graphical analysis. @@ -382,14 +402,13 @@ We find that raising $\gamma_h$ increases the initial college wage premium $\phi Compute $\frac{\partial \phi}{\partial γ_c}$ and evaluate it numerically at default parameter values ```{code-cell} ipython3 -ϕ_γ_c = ϕ(D, γ_h, γ_c, R, T, w_h0).diff(γ_c) +ϕ_γ_c = ϕ.diff(γ_c) ϕ_γ_c ``` ```{code-cell} ipython3 # Numerical value at default parameters -ϕ_γ_c_func = Lambda((D, γ_h, γ_c, R, T, w_h0), ϕ_γ_c) -ϕ_γ_c_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value) +ϕ_γ_c.subs(symbol_subs) ``` We find that raising $\gamma_c$ decreases the initial college wage premium $\phi$, in line with our earlier graphical analysis. @@ -397,27 +416,27 @@ We find that raising $\gamma_c$ decreases the initial college wage premium $\phi Let's compute $\frac{\partial \phi}{\partial R}$ and evaluate it numerically at default parameter values ```{code-cell} ipython3 -ϕ_R = ϕ(D, γ_h, γ_c, R, T, w_h0).diff(R) +ϕ_R = ϕ.diff(R) ϕ_R ``` ```{code-cell} ipython3 # Numerical value at default parameters -ϕ_R_func = Lambda((D, γ_h, γ_c, R, T, w_h0), ϕ_R) -ϕ_R_func(D_value, γ_h_value, γ_c_value, R_value, T_value, w_h0_value) +ϕ_R.subs(symbol_subs) ``` We find that raising the gross interest rate $R$ increases the initial college wage premium $\phi$, in line with our earlier graphical analysis. - ## Exercises +In the following exercises, we extend our previous model to a model of entrepreneurs versus workers. + ```{exercise-start} :label: edm_ex1 ``` -In this exercise, we add a parameter and reinterpret variables to get a model of entrepreneurs versus workers and do some computations like before. +We add a parameter $\pi \in (0,1)$ representing the probability that an entrepreneur's "project" succeeds. -We now let $h$ be the present value of a "worker". +We now let $h$ be the present value of a "worker". We define the present value of an entrepreneur to be @@ -425,21 +444,14 @@ $$ c_0 = \pi \sum_{t=4}^T R^{-t} w_t^c $$ -where $\pi \in (0,1) $ is the probability that an entrepreneur's "project" succeeds. - -For our model of workers and firms, we'll interpret $D$ as the cost of becoming an entrepreneur. +We interpret $D$ in the previous model as the cost of becoming an entrepreneur. This cost might include costs of hiring workers, office space, and lawyers. What we used to call the college, high school wage gap $\phi$ now becomes the ratio of a successful entrepreneur's earnings to a worker's earnings. -We'll find that as $\pi$ decreases, $\phi$ increases, indicating that the riskier it is to -be an entrepreneur, the higher must be the reward for a successful project. - -Now define `create_edm_π` and `compute_gap` following the previous *Computations* section, adopting the entrepreneur-worker interpretation of our model. - -Given: +In this exercise, update `create_edm_π` and `compute_gap` to formulate our entrepreneur-worker model with parameters below ```{code-cell} ipython3 R=1.05, # gross rate of return @@ -448,9 +460,13 @@ T=40, # time horizon γ_c=1.01, # college wage growth w_h0=1, # initial wage (high school) D=10, # cost for college -π=0 # chance of business success +π=1 # chance of business success ``` +and verify that when $\pi = 1$, the result is the same as our old model. + ++++ + ```{exercise-end} ``` @@ -458,6 +474,8 @@ D=10, # cost for college :class: dropdown ``` +Here is one solution + ```{code-cell} ipython3 # Define a model of entrepreneur-worker interpretation EqDiffModel = namedtuple('EqDiffModel', 'R T γ_h γ_c w_h0 D π') @@ -468,7 +486,7 @@ def create_edm_π(R=1.05, # gross rate of return γ_c=1.01, # college wage growth w_h0=1, # initial wage (high school) D=10, # cost for college - π=0 # chance of business success + π=1 # chance of business success ): return EqDiffModel(R, T, γ_h, γ_c, w_h0, D, π) @@ -486,6 +504,16 @@ def compute_gap(model): ϕ = A_h / A_c + D / (w_h0 * A_c) return ϕ ``` + +We validate our result by checking the result with $\pi = 1$ + +```{code-cell} ipython3 +ex_π = create_edm_π() +gap_π = compute_gap(ex_π) + +gap_π +``` + ```{solution-end} ``` @@ -505,9 +533,6 @@ If the probability that a new business succeeds is $0.2$, what is the initial wa ex3 = create_edm_π(π=0.2) gap3 = compute_gap(ex3) -gap3ex3 = create_edm_π(π=0.2) -gap3 = compute_gap(ex3) - gap3 ``` @@ -519,13 +544,13 @@ gap3 ``` Now let's study how the initial wage premium for successful entrepreneurs depend on the success probability. -Given +With $\pi \in [0.2, 1]$, ```{code-cell} ipython3 π_arr = np.linspace(0.2, 1, 50) ``` -Plot the relationship between the wage gap and the values of $\pi$. +plot the relationship between the wage gap and the values of $\pi$. ```{exercise-end} ``` @@ -533,6 +558,7 @@ Plot the relationship between the wage gap and the values of $\pi$. ```{solution-start} edm_ex3 :class: dropdown ``` + ```{code-cell} ipython3 π_arr = np.linspace(0.2, 1, 50) models = [create_edm_π(π=π) for π in π_arr] @@ -544,7 +570,8 @@ plt.xlabel(r'$\pi$') plt.show() ``` -Does the graph make sense to you? +We find that as $\pi$ decreases, $\phi$ increases, indicating that the riskier it is to +be an entrepreneur, the higher must be the reward for a successful project. ```{solution-end} ``` From 24fea06e0902755c3aa316cafb9a997c2d0dd058 Mon Sep 17 00:00:00 2001 From: Humphrey Yang Date: Wed, 24 Jul 2024 11:58:17 +1000 Subject: [PATCH 6/8] update the bound to include 1 --- lectures/equalizing_difference.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lectures/equalizing_difference.md b/lectures/equalizing_difference.md index 4396d072..8715bc84 100644 --- a/lectures/equalizing_difference.md +++ b/lectures/equalizing_difference.md @@ -434,7 +434,7 @@ In the following exercises, we extend our previous model to a model of entrepren ```{exercise-start} :label: edm_ex1 ``` -We add a parameter $\pi \in (0,1)$ representing the probability that an entrepreneur's "project" succeeds. +We add a parameter $\pi \in (0,1]$ representing the probability that an entrepreneur's "project" succeeds. We now let $h$ be the present value of a "worker". From 70a7e9d84e1db6cd5149cc67b7aed5d953ee8856 Mon Sep 17 00:00:00 2001 From: Humphrey Yang Date: Wed, 24 Jul 2024 12:07:41 +1000 Subject: [PATCH 7/8] update a small typo --- lectures/equalizing_difference.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lectures/equalizing_difference.md b/lectures/equalizing_difference.md index 8715bc84..e67abe25 100644 --- a/lectures/equalizing_difference.md +++ b/lectures/equalizing_difference.md @@ -451,7 +451,7 @@ This cost might include costs of hiring workers, office space, and lawyers. What we used to call the college, high school wage gap $\phi$ now becomes the ratio of a successful entrepreneur's earnings to a worker's earnings. -In this exercise, update `create_edm_π` and `compute_gap` to formulate our entrepreneur-worker model with parameters below +In this exercise, update `create_edm` and `compute_gap` to formulate our entrepreneur-worker model with parameters below ```{code-cell} ipython3 R=1.05, # gross rate of return From 4add8f4341f60f48784c9d8e38da2ebcf43a0c97 Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Fri, 26 Jul 2024 16:54:10 +1000 Subject: [PATCH 8/8] Modify context --- lectures/equalizing_difference.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/lectures/equalizing_difference.md b/lectures/equalizing_difference.md index e67abe25..80677cd6 100644 --- a/lectures/equalizing_difference.md +++ b/lectures/equalizing_difference.md @@ -429,7 +429,7 @@ We find that raising the gross interest rate $R$ increases the initial college w ## Exercises -In the following exercises, we extend our previous model to a model of entrepreneurs versus workers. +In the following exercises, we extend our previous model to include a choice between being an entrepreneur or a worker. ```{exercise-start} :label: edm_ex1