Extra notes for Question 2(b) – Implicit differentiation
Boromeus Wanengkirtyo
February 10, 2013
We see from today’s seminar that we can get the elasticity of labour market tightness θ, wrt y by
log-linearising. We can also do this by implicit differentiation. You can decide which method you
prefer.
From slide 59, combine together the social value of a job S, with the job-creation condition:
S=
y−b
r + λ + βp(θ)
(1)
k = (1 − β)q(θ)S
and substitute the job-finding and vacancy filling rates:
k(r + λ + βθ1−µ ) = (1 − β)θ−µ (y − b)
(2)
Take logs on both sides, and then rearrange to make some function F :
F ≡ ln(k) + ln(r + λ + βθ1−µ ) − ln(1 − β) + µ ln(θ) − ln(y − b) = 0
(3)
Differentiate bit by bit wrt ln y and use chain rule (remember k, β, b are constants)
d ln(r + λ + βθ1−µ )
d ln(r + λ + βθ1−µ )
dθ
d ln θ
=
·
·
d ln y
dθ
d ln θ d ln y
β(1 − µ)θ−µ
d ln θ
=
·θ·
r + λ + βθ1−µ
d ln y
(4)
d ln(y − b)
d ln(y − b)
dy
=
·
d ln y
dy
d ln y
1
=
·y
y−b
(5)
Combine together:
dF
β(1 − µ)θ1−µ d ln θ
d ln θ
y
=
·
+µ
−
=0
1−µ
d ln y
r + λ + βθ
d ln y
d ln y y − b
1
(6)
And rearrange for the elasticity that we want:
) 1−µ
+ µ(r + λ + 1−µ
µβθ
βθ
y
d ln θ βθ1−µ − =
1−µ
d ln y
r + λ + βθ
y−b
⇒
d ln θ
r + λ + βθ1−µ
y
=
·
1−µ
d ln y
µ(r + λ) + βθ
y−b
(7)
(8)
2