by Ajay Agrawal
these new technologies.
Appendix
Derivations for the Basic Model
Suppose that assumption (A1) holds. We fi rst derive the demand for
factors:
• Denote by p( x) the price of task x. Assumption (A1) implies
R
if x
N
1, I
( x)
(8A.1)
p( x) =
M
W
if x
I , N
(
.
( x)
L
• In addition, the demand for task x is given by
y( x) = Y .
p( x)
• Thus, the demand for smart machines in task x is
Y
if x
N
1, I
k( x) =
R
,
0
if x
I , N
(
and the demand for labor in task x is
0
if x
N
1, I
( x) =
Y
if x
I , N
(
.
W
• Aggregating the demand for machines from this expression and set-
ting it equal to the supply of capital, K, we have the following market-
clearing condition for capital:
K = Y ( I
N + 1).
R
Similarly, aggregating the demand for labor and setting it equal to its inelas-
tic supply, L, we obtain the market- clearing condition for labor as
230 Daron Acemoglu and Pascual Restrepo
L = Y ( N
I ).
W
• Rearranging these two equations, the equilibrium rental rate and wage
can be obtained as
(8A.2)
R = Y ( I
N + 1) and W = Y ( N
I ),
K
L
which are the expressions used in the text.
We next turn to deriving the expression for aggregate output.
• Because we normalized the price of the fi nal good to 1 as numeraire,
we have
N
ln p( x) dx = 0.
N 1
• Plugging in the expressions for p( x) from equation (8A.1) yields
I
N
ln R
ln
( x) dx +
ln W
ln
( x) dx = 0.
M
L
N 1
I
• Substituting the expressions for R and W from (8A.2), we obtain
I
ln Y
ln K /( I
N + 1)
(
) ln ( x) dx
M
N 1
N
+ ln Y ln L/( N I)
(
) ln ( x) dx = 0.
L
I
• This equation can be rearranged as
I
K
N
L
ln Y =
ln
+ ln
( x) dx +
ln
+ ln ( x) dx
I
N + 1
M
N
1
L
N 1
I
I
N
=
ln
( x) dx + ln
( x) dx
M
L
N 1
I
K
L
+ ( I N + 1)ln
+ ( N I)ln
,
I
N + 1
N
I
which, after taking exponentials on both sides of the equation, yields the
expression for aggregate output in equation (1) in the text.
Assumption (A1)
We now show that assumption (A1) is equivalent to the capital- labor ratio
of the economy taking an intermediate value. In particular, there exist two
positive thresholds < such that assumption (A1) holds whenever
Artifi cial Intelligence, Automation, and Work 231
K
(A2)
( , ).
L
Equation (8A.2) shows that
W
N
I
= K
.
R
L I
N + 1
Defi ne
( I )
( N )
= I N + 1 L
, and
= I N + 1
L
.
N
I
( I )
N
I
( N
I )
M
M
Then equation (A2) is equivalent to assumption (A1).
Derivations in the Presence of Technology- Skill Mismatch
• Denote by p( x) the price of task x. Assumption (A1) together with the fact that W > W (see footnote 12) implies
H
L
R
if x
N
1, I
( x)
M
W
p( x) =
L
if x
( I , S )
.
( x)
L
WH
if x
S, N ]
( x)
L
• Following the same steps as in our baseline model, we obtain the
market- clearing condition for capital,
K = Y ( I
N + 1).
R
• The demand for low- skill labor in task x is given by
0
if x
N
1, I
Y
( x) =
if x
( I , S )
.
WL
0
if x
S, N ].
• Aggregating the demand for low- skill labor and setting it equal to its
inelastic supply, L, we obtain the market- clearing condition for low-
skill labor as
L = Y ( S
I ),
WL
which implies the expression for W given in the main text.
L
232 Daron Acemoglu and Pascual Restrepo
• The demand for high- skill labor in task x is given by
0
if x
N
1, I
0
if x
( I , S )
h( x) =
.
Y
if x
S, N ].
WH
• Aggregating the demand for high- skill labor and setting it equal to
its supply, H , we obtain the market- clearing condition for high- skill
labor as
H = Y ( N
S ),
WH
which implies the expression for W given in the main text.
H
Derivations for the Model with Distortions
• Denote by p( x) the price of task x. The variant of assumption (A1) introduced in section 8.5 implies
R(1
)
if x
N
1, I
( x)
M
W (1 + )
p( x) =
if x
( I , J )
( x)
L
W
if x
J , N ].
( x)
L
• Following the same steps as in the model with no distortions, we obtain
&n
bsp; the market- clearing condition for capital,
K =
Y
( I
N + 1).
R(1
)
• The demand for labor in task x is
0
if x
N
1, I
Y
if x
( I , J )
( x) =
W (1 + )
.
Y
if x
J , N ]
W
• The expression for ℓ( x) implies that the total amount of labor employed
in tasks where labor gets rents is
L =
Y
( J
I ).
A
W (1 + )
Artifi cial Intelligence, Automation, and Work 233
The total amount of labor employed in tasks where labor does not get rents is
L
L = Y ( N
J ).
A
W
To derive the expression for (gross) output we proceed as follows:
• Again from our choice of numeraire, we have
N
ln p( x) dx = 0.
N 1
• Plugging in the expressions for p( x) we obtain
I
J
ln R
ln
( x) dx +
ln W + ln(1 + ) ln
( x) dx
M
L
N 1
I
N
+
ln W
ln
( x) dx = 0.
L
J
• Substituting for factor prices using the expressions for K, L , and A
L – L , we obtain
A
I
ln Y
ln K / ( I
N + 1)
(
) ln ( x) dx
M
N 1
J
+
ln Y
ln L / ( J
I )
(
) ln ( x) dx
A
L
I
J
+
ln Y
ln ( L
L ) / ( N
J )
(
) ln ( x) dx = 0.
A
L
I
• This equation can be rearranged as
I
K
J
L
ln Y =
ln
+ ln
( x) dx +
ln
A
+ ln ( x) dx
I
N + 1
M
J
I
L
N 1
I
N
L
+
ln
+ ln ( x) dx
N
J
L
J
I
N
K
=
ln
( x) dx + ln
( x) dx + ( I
N + 1)ln
M
L
I
N + 1
N 1
I
L
L
L
+ ( J I)ln
A
+ ( N J )ln
A
,
J
I
N
J
which yields equation (12) in the text.
234 Daron Acemoglu and Pascual Restrepo
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