ORIGINAL ARTICLE: David E. Bloom, Klaus Prettner, Jamel Saadaoui, Mario Veruete (2023), Artificial intelligence and the skill premium, arXiv:2311.09255. DOI: https://doi.org/10.48550/arXiv.2311.09255
What will likely be the effect of the emergence of ChatGPT and other forms of artificial intelligence (AI) on the skill premium?
​
To understand the consequences of ChatGPT and other forms of artificial intelligence, we developed a nested CES production function that distinguishes between industrial robots and AI. Industrial robots predominantly substitute for low-skill workers, whereas AI mainly helps to perform the tasks of high-skill workers. We show that AI reduces the skill premium as long as it is more substitutable for high-skill workers than low-skill workers are for high-skill workers.
​
Section 1 provides the notations and the assumptions that will be required for the calculations. Section 2 introduces the production function and a more readable notation. Then, Section 3 computes the wage rates for low- and high-skilled workers thanks to partial derivatives. Section 4 provides the wage-skill premium. Sections 5 and 6 calculate the partial derivatives of the different wage rates relative to industrial robots and AI. Section 7 provides interactive 2-D and 3-D plots to visualize the evolutions of the wage skill premium for different values of the parameters and the variables.
This notebook aims to replicate all the computations in the paper “Artificial intelligence and the skill premium” ( https://arxiv.org/abs/2311.09255 )
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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


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-α)
    α
    
    1-α
    γ
    -1
    M
    (1,0,0,0)
    M
    (
    L
    u
    ,
    L
    s
    ,P,)
    γ
    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​​]
    Out[]//TraditionalForm=
    (1-α)
    α
    
    1-α
    γ
    -1
    M
    (0,1,0,0)
    M
    (
    L
    u
    ,
    L
    s
    ,P,)
    γ
    In[]:=
    wageRateS=D[productionFunctionYt,
    L
    s
    [t]];​​TraditionalForm[wageRateS/.Trule]
    Out[]//TraditionalForm=
    (1-α)
    β
    2
    (1-
    β
    3
    )
    α
    
    φ-1
    L
    s
    γ
    φ
    -1
    (1-
    β
    2
    )
    φ
    
    +
    β
    2
    φ
    L
    s
    
    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
    
    

    4 Computation of the skill premium
    W
    s
    w
    u
    .

    In[]:=
    skillPremiumShort=
    wageRateSShort
    wageRateUShort
    ;​​TraditionalForm[​​skillPremiumShort/.Trule​​]
    Out[]//TraditionalForm=
    (0,1,0,0)
    M
    (
    L
    u
    ,
    L
    s
    ,P,)
    (1,0,0,0)
    M
    (
    L
    u
    ,
    L
    s
    ,P,)
    In[]:=
    skillPremium=
    wageRateS
    wageRateU
    ;​​TraditionalForm[skillPremium/.Trule]
    Out[]//TraditionalForm=
    β
    2
    (1-
    β
    3
    )
    φ-1
    L
    s
    1-θ
    L
    u
    1-
    γ
    θ
    
    β
    1
    θ
    L
    u
    +(1-
    β
    1
    )
    θ
    P
    
    γ
    φ
    -1
    (1-
    β
    2
    )
    φ
    
    +
    β
    2
    φ
    L
    s
    
    β
    1
    β
    3

    5 Computation of
    ∂
    w
    s
    ∂
    G
    t
    ,
    ∂
    w
    s
    ∂
    P
    t
    .

    ∂
    w
    s
    ∂
    G
    t

    In[]:=
    TraditionalForm[D[wageRateSShort,[t]]/.Trule]

    7 Plots

    If you are in the cloud, in order to see the following plots you should open the file as a copy on your cloud account OR download a copy of the file and open it with Mathematica or with WolframPlayer (for those who does not have access to Mathematica).

    CITE THIS NOTEBOOK

    Artificial intelligence and the skill premium​
    by David Bloom, Klaus Prettner, Jamel Saadaoui, Mario Veruete​
    Wolfram Community, STAFF PICKS, November 21, 2023
    ​https://community.wolfram.com/groups/-/m/t/3068735