ChatGPT, AI, and wage inequality

Version of August, 23, 2023.
David Bloom (Harvard TH Chan School of Public Health, dbloom@hsph.harvard.edu)​
Klaus Prettner (Vienna University of Economics and Business, klaus.prettner@wu.ac.at)​
Jamel Saadaoui (University of Strasbourg, saadaoui@unistra.fr)​
Mario Veruete (Quantum DataLab, veruete.mario@quantum-datalab.com)
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1
.
Production function
Y
t
.
2
.
Computation of the wage rates
w
u,t
=
∂Y
∂
L
u
and
w
s,t
=
∂Y
∂
L
s
.
3
.
Computation of the skill premium =
W
s
w
u
.
4
.
Computation of
∂
w
s
∂
,
∂
w
s
∂P
.
5
.
Computation of
∂
w
u
∂
,
∂
w
u
∂P
.
6
.
Plots
1 Notation and Assumptions
In[]:=
β[i_]:=
β
i
In[]:=
$Assumptions=And[​​0<θ<=1(*θ==1meansfullsubsituabilityofworkersbyindustrialrobots*),​​0<γ<1,​​0<φ<1,​​0<α<1,​​0<β1<1,​​0<β2<1,​​0<β3<1,​​0<β[1]<1,​​0<β[2]<1,​​0<β[3]<1​​];
In[]:=
Clear[style];​​style[x_,c_]:=Framed[x,Background->c,RoundingRadius->20];
In[]:=
Clear[colorRules];​​colorRules={​​
L
u
[t]->style[
L
u
[t],RGBColor[0.64,0.87,0.32]],​​
L
s
[t]->style[
L
s
[t],RGBColor[0.84,0.68,0.3]],​​P[t]->style[P[t],RGBColor[0.21,0.79,0.74]],​​[t]->style[[t],RGBColor[1,0.29,0.35,0.53]],​​φ->Style[φ,Red,18,Bold],​​θ->Style[θ,Blue,18,Bold],​​γ->Style[γ,Orange,18,Bold],​​α->Style[α,Purple,18,Bold]​​};
In[]:=
stateVariables={
L
u
[t],
L
s
[t],P[t],[t]};
In[]:=
parameters={θ,γ,φ,α,β[1],β[2],β[3]};
In[]:=
Mrule={M[
L
u
,
L
s
,P,]->M};​​MTrule={M[
L
u
[t],
L
s
[t],P[t],[t]]->M};​​Trule={f_[t]:>f};
2
Production function
Y
t
◼
  • P[t] = industrial robots
    • ◼
  • L
    s
    [t]
    = skilled workers
    • ◼
  • L
    u
    [t]
    = unskilled workers
    • ◼
  • [t] = AI
    • ◼
  • [t] = capital stocks, machines, etc.
    • In[]:=
    Clear[productionFunctionYt,productionFunctionYtM];​​productionFunctionYtM=
    1-α
    γ
    (M[
    L
    u
    [t],
    L
    s
    [t],P[t],[t]])
    α
    [t]
    ;​​productionFunctionYt=
    1-α
    γ
    β[3]
    γ/θ
    β[1]
    θ
    (
    L
    u
    [t])
    +(1-β[1])
    θ
    (P[t])
    
    +(1-β[3])
    γ/φ
    (β[2]
    φ
    (
    L
    s
    [t])
    +(1-β[2])
    φ
    ([t])
    )
    
    α
    [t]
    ;
    In the following, for readability purposes, we introduce the notation:​​
    M=
    γ/θ
    β
    3
    
    θ
    β
    1
    (
    L
    u
    [t])
    +(1-
    β
    1
    )
    θ
    (P[t])
    
    +(1-
    β
    3
    )
    γ/φ
    (
    φ
    β
    2
    (
    L
    s
    [t])
    +(1-
    β
    2
    )
    φ
    ([t])
    )
    .
    In[]:=
    TraditionalForm[productionFunctionYtM/.MTrule/.Trule]
    Out[]//TraditionalForm=
    α
    
    1-α
    γ
    M
    In[]:=
    TraditionalForm[productionFunctionYt/.Trule]
    Out[]//TraditionalForm=
    α
    
    1-α
    γ
    
    β
    3
    γ/θ
    
    β
    1
    θ
    L
    u
    +(1-
    β
    1
    )
    θ
    P
    
    +(1-
    β
    3
    )
    γ/φ
    (1-
    β
    2
    )
    φ
    
    +
    β
    2
    φ
    L
    s
    
    
    3
    Computation of the wage rates
    w
    u,t
    =
    ∂Y
    ∂
    L
    u
    and
    w
    s,t
    =
    ∂Y
    ∂
    L
    s

    Wage unskilled
    w
    u

    In[]:=
    wageRateUShort=D[productionFunctionYtM,
    L
    u
    [t]];​​TraditionalForm[​​wageRateUShort/.MTrule/.Trule​​]
    Out[]//TraditionalForm=
    1
    γ
    (1-α)
    α
    
    1-α
    γ
    -1
    M
    (1,0,0,0)
    M
    (
    L
    u
    ,
    L
    s
    ,P,)
    NB. In Wolfram Language,
    Derivative
    [{
    n
    1
    ,
    n
    2
    ,…}][f]
    represents the derivative of
    f[{
    x
    1
    ,
    x
    2
    ,…}]
    taken
    n
    i
    times with respect to
    x
    i
    . In general, arguments given in lists in
    f
    can be handled by using a corresponding list structure in
    Derivative
    . ​ If
    f=f(
    x
    1
    ,
    x
    2
    ,...,
    x
    i,
    ...,
    x
    n
    ),then,thefirstorderpartialderivative
    ∂f
    x
    i
    isgivenby:
    ​ ​
    ∂f
    x
    i
    (
    x
    1
    ,
    x
    2
    ,...,
    x
    i,
    ...,
    x
    n
    )=
    Derivative[0,0,...,1,0,...,0][f][
    x
    1
    ,
    x
    2
    ,...,
    x
    i
    ,...,
    x
    n
    ]
    =
    (0,0,...,1,0...,0)
    f
    (
    x
    1
    ,
    x
    2
    ,...,
    x
    i,
    ...,
    x
    n
    )
    . In particular,
    (0,1,0,0)
    M
    (
    L
    u
    ,
    L
    s
    ,P,)
    represents
    ∂M
    ∂
    L
    s
    . Similarly,
    (0,1,0,1)
    M
    (
    L
    u
    ,
    L
    s
    ,P,)
    represents
    2
    ∂
    M
    ∂
    L
    s
    ∂
    .​For more details, the documentation page of the function
    Derivative
    .
    In[]:=
    wageRateU=D[productionFunctionYt,
    L
    u
    [t]];​​TraditionalForm[wageRateU/.Trule]
    Out[]//TraditionalForm=
    (1-α)
    β
    1
    β
    3
    α
    
    θ-1
    L
    u
    γ
    θ
    -1
    
    β
    1
    θ
    L
    u
    +(1-
    β
    1
    )
    θ
    P
    
    1-α
    γ
    -1
    
    β
    3
    γ/θ
    
    β
    1
    θ
    L
    u
    +(1-
    β
    1
    )
    θ
    P
    
    +(1-
    β
    3
    )
    γ/φ
    (1-
    β
    2
    )
    φ
    
    +
    β
    2
    φ
    L
    s
    
    

    Wage skilled
    w
    s

    In[]:=
    wageRateSShort=D[productionFunctionYtM,
    L
    s
    [t]];​​TraditionalForm[​​wageRateSShort/.MTrule/.Trule​​]
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