maths

1. FEP-UP Doctoral Program in Economics
Mathematical Economics (2010-11)
Some economic applications
of differential equations
João Correia da Silva
This version: November 12th
, 2010.

2. 1 A Model of Stock Prices
Consider a simple financial market structure consisting of only two assets:
(1) government bonds which pay a fixed interest rate, r;
(2) shares of stock, which pay a constant stream of dividends, d.
Taking r and d as given, we want to construct a model to predict the evolution of
share prices, p.1
1.1 Bonds
The return on bonds depends on the annual interest rate, r, and on the period of
capitalization T (in years).
If capitalization is annual, then, the value at t of an investment, I, made at t0 is:
VB(t) = I(1 + r)t−t0
.
With the same interest rate, but monthly capitalization (the monthly interest rate is
r/12), the value at t of the same investment made at t0 is: VB(t) = I(1 + r
12
)12(t−t0)
.
If capitalization occurs n times per year, the value at t of the investment made at t0
is: VB(t) = I(1 + r
n
)n(t−t0)
.
Finally, with instantaneous capitalization, the value at t of the investment made at
t0 is:
VB(t) = lim
n→+∞
I
1 +
r
n
n(t−t0)
= I
lim
n→+∞
1 +
r
n
n
t−t0
= Ier(t−t0)
.
In the continuous time version of the model, we assume that capitalization is instan-
taneous (takes place in continuous time).
1
This model is borrowed from De la Fuente (2000, pp. 503-513).
2

3. The growth at t of the value of the investment made at t0 is:
V 0
B(t) = Irer(t−t0)
.
Growth at t0 is simply:
V 0
B(t0) = Ir.
1.2 Stocks
The return on an investment in stocks has two components:
i) a fixed annual dividend, d, per unit of stock;
ii) a capital gain, due to variations in the stock price.
For simplicity, we assume a continuous flow of dividends.
An investment, I, made at t0 consists in buying I
p(t0)
units of stock, at the prevailing
market price, p(t0).
The value at t of an investment in stock, I, made at t0 is:
VS(t) = I
p(t0)
[d(t − t0) + p(t)].
The growth at t of the value of the investment made at t0 is:
V 0
S(t) = I
p(t0)
[d + ṗ(t)].
Growth at t0 is:
V 0
S(t0) = I
p(t0)
[d + ṗ(t0)].
The present value of the dividend stream is designated as the fundamental value of
the stock. Notice that it is constant over time.
V FS =
Z ∞
0
d e−rt
dt =
d
r
(1)
3

4. 1.3 Arbitrage and Equilibrium
Arbitrage is the practice of taking advantage of price differentials between different
markets. It consists of a trade that involves no possibility of loss and a positive
probability of gain.
No-Arbitrage Principle: In equilibrium, there are no arbitrage opportunities that
can be explored.
The absence of arbitrage opportunities implies that the two assets (bonds and stocks)
have the same expected return, in each moment in time. The only uncertainty con-
cerns the variation in the price of the stock.
No-arbitrage at implies the following dynamic condition:
Ir =
I
p(t0)
[d + ṗe
(t0)] . (2)
It must hold for all t0, therefore, we can write it as:
ṗe
(t0) = rp(t0) − d. (3)
1.4 Adaptive Expectations
We still haven’t defined the way in which agents form their expectations. In a first
approximation, we assume that agents have adaptive expectations. If the actual price
of the stock exceeds the expected price, agents revise their forecasts in the direction
of the observed price.
ṗe
(t) = α[p(t) − pe
(t)] (4)
Some manipulation yields a differential equation governing expected stock prices.
4

5. ṗe
(t) = α
d
r
+
ṗe
(t)
r
− pe
(t)
⇔ ṗe
(t) = −
rα
r − α
pe
(t) +
αd
r − α
(5)
To make sure that the system is stable, assume that α r (expectations are not too
volatile).
The steady-state value of pe
, denoted pe∗
, can be determined by setting ṗe
= 0. It
coincides with the fundamental value of the stock.
0 = −
rα
r − α
pe∗
+
αd
r − α
⇔ pe∗
=
d
r
(6)
Solving the differential equation, we find that the evolution of pe
is described by the
following expression.
pe
(t) = pe∗
+ [pe
(t0) − pe∗
]e− rα
r−α
(t−t0)
⇒
⇒ pe
(t) =
d
r
+
pe
(t0) −
d
r
e− rα
r−α
(t−t0)
(7)
The fact that the system is stable means that a discrepancy between the fundamental
value of the stock and the expected share price tends to diminish and disappear
asymptotically. In the long run, the expected value of a stock is equal to the present
value of the dividends.
And how is the evolution of the market price of the stock? With recourse to previous
conditions, and some manipulation, we obtain the law of motion of the stock price.
(
ṗe
(t) = rp(t) − d
ṗe
(t) = α[p(t) − pe
(t)]
⇒ rp(t) − d = α[p(t) − pe
(t)] ⇔
⇔ (r − α)p(t) = d − αpe
(t) ⇔
5

6. p(t) =
d
r − α
−
α
r − α
d
r
−
α
r − α
pe
(t0) −
d
r
e− rα
r−α
(t−t0)
⇔
⇔ p(t) =
d
r
−
α
r − α
pe
(t0) −
d
r
e− rα
r−α
(t−t0)
.
The expression above shows that (with α r) the stock price also converges asymp-
totically to the fundamental value.
1.5 Rational Expectations
A disturbing feature of this model is that the forecast error is predictable by an agent
that is able to make the calculations we made above. This presents an opportunity
for quick profits. A way to avoid this criticism is to consider the more sophisticated
formulation of rational expectations (Muth, 1961).
Since this model is purely deterministic, we use the much simpler notion of perfect-
foresight.
pe
(t) = p(t) (8)
The dynamics of the system change drastically.
rp(t) = d + ṗe
(t) = d + ṗ(t) ⇔ ṗ(t) = rp(t) − d (9)
There is still a single steady-state solution.
0 = rp∗
− d ⇔ p∗
=
d
r
(10)
But now the system is unstable, because r 0.
6

7. p(t) = p∗
+ [p(0) − p∗
]ert
(11)
The second term, that may be designated as the bubble, grows to plus or minus
infinity unless the initial price equals the fundamental value of the stock. If we rule
out these ever-growing bubbles, the only solution corresponds to share prices jumping
instantaneously to the fundamental value of the stock.
Let’s now introduce a tax rate, τ, on dividends and see how share prices reacts to
changes in this tax rate. The equations must be modified to account for the fact that
net dividends diminish to (1 − τ)d.
ṗ(t) = rp(t) − (1 − τ)d (12)
p(t) = p∗
+ [p(0) − p∗
]ert
(13)
p∗
=
(1 − τ)d
r
(14)
An unexpected change in the tax rate, to a new value that is expected to remain
forever, makes stock prices jump instantaneously to the new fundamental value.
A more interesting problem is to predict the movements of the stock prices in response
to a credible announcement, at t0 of a future change in the dividend tax rate, from
τ0 to τ1, that will take place at t1.
At t1, the stock price will be at the new fundamental value, p∗
(τ1). For t t1, the
law of motion derived for τ0 applies. Now it is not reasonable to exclude the diverging
solutions, because the divergence is only temporary.
p(t) = p∗
(τ0) + [p(0) − p∗
(τ0)]ert
This expression must assume the value p∗
(τ1) at t1. Otherwise, there would be a pre-
7

8. dictable jump in stock prices at t1, allowing agents to make profits through arbitrage.
p∗
(τ0) + [p(0) − p∗
(τ0)]ert1
= p∗
(τ1)
The stock price jump caused by the announcement is given below.
p(0) − p∗
(τ0) = [p∗
(τ1) − p∗
(τ0)] e−rt1
(15)
In sum, stock prices jump after the announcement, then diverge from the old funda-
mental value in a precise way in order to coincide with the new fundamental value
when the tax change enters into effect. During the transition, stock prices are given
by the following expression.
p(t) = p∗
(τ0) + [p∗
(τ0) − p∗
(τ1)]er(t−t1)
8

9. 2 The Solow Model of Economic Growth
The standard model in Growth Theory was introduced by Solow (1956). It considers
only one sector and two factors of production: labor (L) and capital (K). The avail-
able technology determines what quantity of the homogeneous good can be produced
as a function of the quantities of capital and labor that are employed.2
Y = F(K, L) (16)
We make several convenient assumptions on the shape of this aggregate production:
(1) constant returns to scale, (2) positive and decreasing marginal products, (3)
capital and labor are essential, (4) Inada conditions.
F(aK, aL) = aF(K, L) (17)
FK, FL 0 ; FKK, FLL 0 (18)
F(0, L) = 0 ; F(K, 0) = 0 (19)











FK → ∞ as K → 0
FL → ∞ as L → 0
FK → 0 as K → ∞
FK → 0 as K → ∞
(20)
The assumption of constant returns to scale allow us to define it in the so-called
intensive form, relating output per worker, y = Y/L, with capital stock per worker,
K/L.
2
Similar dynamic analysis of this model can be found in De la Fuente (2000, pp. 518-527) and
Gandolfo (1997, pp. 175-184).
9

10. F(K/L, L/L) = F(K/L, 1) = f(k) (21)
The Cobb-Douglas specification will frequently be assumed. Note that the sum of
the exponents must be equal to unity, because we are assuming constant returns to
scale.
Y = AKα
L1−α
⇔ y = Akα
(22)
It can be verified that the derivative of the intensive form production function is
positive and decreasing, as it coincides with the marginal productivity of capital.
Output per worker is an increasing and concave function of capital intensity.
F(K, L) = LF(K/L, 1) ⇒ FK(K, L) = Lf0
(k)
1
L
= f0
(k) (23)
Assume that a constant fraction of output is invested (capital formation), and the
remaining is consumed. It is equivalent to assume that households save a constant
fraction of their income, and the financial sector channels it to firms for investment.
K̇ = sY − δK = sLf(k) − δK (24)
It is convenient to write the dynamic system in terms of the capital intensity, k. The
expression is simplified if we assume that the labor force grows at the constant rate,
n.
k̇ = ˙
K/L =
K̇L − KL̇
L2
= K̇/L − K/L
L̇
L
⇒
⇒ k̇ = sf(k) − (δ + n)k (25)
The second term, (δ + n)k, represents the volume of investment per worker that is
needed to maintain capital intensity constant. Part of it is to compensate for capital
10

11. depreciation, δk, and the remaining is to be combined with the workers entering
the labor force, δn. In steady-state, saving and investment per worker, sf(k), must
coincide with these investment needs.
f(k∗
)
k∗
=
δ + n
s
(26)
As a consequence of the Inada conditions, f(k)
k
, varies from infinity to zero as k varies
from zero to infinity (use L’Hôpital’s rule to find the first limit).
lim
k→0
f(k)
k
= lim
k→0
f0
(k)
1
= lim
k→0
FK(kL, L) = FK(0, L) = +∞ (27)
lim
k→+∞
f(k)
k
= lim
k→+∞
F(k, 1)
k
= lim
k→+∞
F(1, 1/k) = F(1, 0) = 0 (28)
Given continuity of f(k)
k
, the theorem of Bolzano guarantees existence of the steady-
state solution, k∗
.
11

12. 3 A Dynamic IS-LM Model
In the static IS-LM model, income and interest rate are determined by the intersection
between the IS curve and the LM curve. The IS curve represents the combinations
of output and interest rate that are such that the market for goods and services is in
equilibrium. The supply of output must be equal to the demand for output, which
increases with income and decreases with the interest rate. Denoting by y the natural
logarithm of real output (or income), a simple formulation is the following:
y = β0 + βy − γr (IS) (29)
The LM curve is composed by the combinations of output and interest rate that are
such that the money market is in equilibrium. The real money supply must be equal
to the demand for real balances, which is increasing in income and decreasing in the
nominal interest rate. Denoting by m the natural logarithm of real money supply
and by πe
the expected rate of inflation, a simple version of the LM curve is written
as:
m = ky − α(r + πe
) (LM) (30)
To draw the curves, it is convenient to write the system as:
(
r = β0
γ
− 1−β
γ
y
r = k
α
y − m
α
− πe
(31)
With some manipulation, the static equilibrium solution can be obtained.



y∗
= β0α+γm+αγπe
γk+(1−β)α
r∗
= β0
γ
γk
1−β
α+ γk
1−β
− m+απe
α+ γk
1−β
(32)
The following change of parameters is convenient to simplify the writing of the system.
Let y0 = β0α
γk+(1−β)α
, ay = γ
γk+(1−β)α
, r0 = β0
γ
γk
1−β
α+ γk
1−β
, and ar = 1−β
γk+(1−β)α
.
12

13. (
y∗
= y0 + ay(m + απe
)
r∗
= r0 − ar(m + απe
)
(33)
To introduce dynamics in the model, assumptions are now introduced regarding (1)
the evolution of the money supply, (2) the price movements and (3) the expectations
of the agents about the rate of inflation.
In this simple economy, the money supply grows at a constant rate:
Ṁ
M
= µ (34)
The rate of inflation is a function of the expected rate of inflation and of the output
gap. The relation below combines a Philips curve with Okun’s law.
Ṗ
P
= π = πe
+ θ(y − ȳ) (35)
Agents are assumed to have adaptive expectations, updating their forecasts of infla-
tion by a fraction of the current forecast error.
˙
πe = δ(π − πe
) (36)
To analyze the dynamics, the model is reduced below to a system of differential
equations in m and πe
. Using the expression for the actual rate of inflation we find
the relation between expected inflation and real output, which is in turn related to the
real money supply and the expected rate of inflation. This yields the first differential
equation.
˙
πe = δθ(y − ȳ) = δθ[y0 + ay(m + απe
) − ȳ] = δθay(m + απe
) − δθ(ȳ − y0) (37)
The growth rate of the real money supply is equal to the difference between the
13

14. growth of the nominal money supply and the actual inflation rate.
m = ln(M/P) = ln(M) − ln(P) ⇒ ṁ =
Ṁ
M
−
Ṗ
P
= µ − π (38)
Using again the expression for the actual rate of inflation, and the expression for
equilibrium output, we arrive at the second differential equation.
ṁ = µ − πe
− θ(y − ȳ) = µ − πe
+ θ(ȳ − y0) − θay(m + απe
) (39)
The formulation of the dynamical system consists of two linear and autonomous
differential equations in m and πe
.
(
˙
πe = δθαayπe
+ δθaym − δθ(ȳ − y0)
ṁ = −(1 + θαay)πe
− θaym + µ + θ(ȳ − y0)
(40)
14

15. 4 Dornbusch’s Overshooting Model
Now we study an open economy IS-LM model with perfect foresight and sticky prices.
We seek to find out how price rigidity in markets for goods and services affects the
short-run responses of exchange rates to policy shifts and other exogenous distur-
bances.
The concern is with short-run dynamics, so we will make the simplification of as-
suming that the output supply, y, is fixed. The finding will be that the sticky price
assumption makes exchange rates more volatile than the underlying fundamentals,
as observed empirically.
The nominal exchange rate, s, is defined as the price of a foreign unit of currency in
terms of domestic currency. An increase in s corresponds to a loss of value of the
domestic currency.
Demand for domestic output depends positively on the real exchange rate (the ratio
between foreign and domestic prices, expressed in a common currency unit), s+pf
−
p. Aggregate demand is also positively related to government expenditures, g, and
negatively related to the real interest rate, i−π. Here and everywhere below, i stands
for the nominal interest rate, and π is the (actual and expected) rate of inflation.
yd
= δ(s + pf
− p) − σ(i − π) + g (41)
If demand exceeds supply, inventories decrease and prices increase. We model this
through the following Phillips curve.
π = α(yd
− y) (42)
Equilibrium in the money market occurs in the LM curve: the real money supply,
m−p, must be equal to demand, which is positively related to income y and negatively
related to the nominal interest rate.
15

16. m − p = φy − λi (43)
The final condition is an interest parity condition. The return of domestic bonds must
be equal to the return of foreign bonds, when expressed in the same currency unit.
In other words, the interest rate differential must compensate for the depreciation of
the domestic currency.
i = if
+ ṡ (44)
To study a small economy, we can take the foreign interest rate, if
, as an exogenous
constant. For the sake of simplicity, assume also that the nominal money supply, m,
and the foreign price level, pf
, remain constant. The time-dependent state variables
are, then, i, p and s.
Is is not difficult to solve for the steady-state, i̇ = ṗ = ṡ = 0. For the exchange rate
to stabilize, the domestic and foreign interest rates must be equal.
i∗
= if
(45)
With recourse to the money market equilibrium condition, the steady-state price level
can be written in terms of the fixed parameters.
m − p∗
= φy − λi∗
⇔ p∗
= m − φy + λif
(46)
Finally, the steady-state exchange rate is that which induces equilibrium in the market
for goods and services. The real exchange rate is calculated below.
y = δ(s∗
+ pf
− p∗
) − σi∗
+ g ⇔ s∗
+ pf
− p∗
=
1
δ
(y + σif
− g) (47)
To obtain the nominal exchange rate, substitute the expression for the steady-state
price level.
16

17. s∗
= m +
1
δ
− φ
y +
σ
δ
+ λ
if
− pf
+
g
δ
(48)
Observe that δ stands for the real exchange rate elasticity of the demand for domestic
output. If domestic and foreign goods were perfect substitutes, then only the cheapest
would be demanded, and thus prices would necessarily be such that s∗
= p∗
− pf
.
This relation is known as the absolute purchasing power parity relation: one unit of
domestic currency buys the same output in both countries.
To analyze the dynamics of this model, we reduce it to a system of differential equa-
tions in p and s. First, we write the interest rate in terms of the fixed parameters,
and substitute it in the law of motion of the exchange rate.
i =
φy − m + p
λ
(49)
ṡ = i − if
=
φy − m + p
λ
− if
(50)
To find the law of motion of the price level, we substitute the expression for aggregate
demand in the Phillips curve.
π = α(yd
− y) = α
δ(s + pf
− p) − σ(i − π) + g − y
It is straightforward to solve for π.
π =
α
1 − ασ
h
δ(s + pf
− p) −
σ
λ
(φy − m + p) + g − y
i
⇔
The dynamics of the model are described by the following system of differential
equations.
(
ṡ = 1
λ
p + φy−m
λ
− if
π = α
1−ασ
δs − (δ + σ
λ
)p + δpf
− λ+σφ
λ
y − σ
λ
m + g
(51)
17

18. 5 Diamond’s Overlapping Generations Model of
Growth
A limitation of the model of Solow is that is takes the saving rate as an exogenous
constant. Cass and Koopmans considered instead a representative agent that lived
forever, and decided how much to consume and to save with the objective of maxi-
mizing utility.
The model of Diamond came to introduce a demographic structure. The economy
is supposed to be populated by successive generations of finitely lived agents. In
the simple version of the model, each generation lives for two periods only. In every
period, there are two generations in the economy: the young and the old.
An agent born at time t, receives incomes w1
t and w2
t at time t and time t + 1,
respectively. The problem of this representative agent is to maximize utility, U(c1
t , c2
t ).
This objective function satisfies the following assumptions: each period’s consumption
has positive and decreasing marginal utility, cross derivatives are non-negative, and
Inada conditions hold.
U1, U2 0 ; U11, U22 0 ; U12 = U21 ≥ 0 (52)
(
U1 → ∞ as c1
t → 0 ; U1 → 0 as c1
t → ∞
U2 → ∞ as c2
t → 0 ; U2 → 0 as c2
t → ∞
(53)
The agent may lend or borrow money from the first period to the second, subject to
the interest rate rt+1.
max
c1
t ,c2
t
U(c1
t , c2
t ) s.t.
(
c1
t + st = w1
t
c2
t = w2
t + st(1 + rt+1)
(54)
A convenient way to write the agent’s problem is with recourse to the notion of
permanent income, wt = w1
t +
w2
t
1+rt+1
. The model becomes equivalent to one in which
the agent receives all the income in the first period and only has to decide how much
18

19. to save.
max
st
U(wt − st, st(1 + rt+1)) (55)
Differentiating U with respect to s yields the first order condition. Since U is a
concave function of s, the second order condition is satisfied.
−U1(wt − st, st(1 + rt+1)) + (1 + rt+1)U2(wt − st, st(1 + rt+1)) = 0 ⇔
⇔
U1(wt − st, st(1 + rt+1))
U2(wt − st, st(1 + rt+1))
= 1 + rt+1 (56)
19

20. 6 The Basic Model of Job Search
Suppose that you expect to receive wage offers drawn from a given probability dis-
tribution, arriving at fixed or random time dates. Every time you receive an offer,
you should decide whether to accept it and become employed, or to reject it and wait
for the next offer. What is the minimum offer that you should be willing to accept?
In other words, what is your reservation wage? To answer this question, we will use
dynamic programming.
The objective of the agent is to maximize the discounted value of lifetime income.
E
( ∞
X
t=0
βt
yt
)
(57)
At time t, the income of the agent, yt
, may be equal to the wage rate, x, or to the
unemployment benefit, b. Unemployed workers receive one offer in each period. All
jobs are permanent and pay the same wage in every period. However, wages differ
across jobs: x is a random variable drawn from a time-invariant distribution.
F(w) = pr(x ≤ w) (58)
If a worker accepts a job offer, then the income is determined for life (it would be
nonsensical to accept the offer, and later quit because the wage distribution is time-
invariant). The value of accepting an offer of x is given below.
Wa(x) =
∞
X
t=0
βt
x =
x
1 − β
(59)
The value of rejecting an offer, Wr, is not a function of x. The consequences of
rejecting an offer with low or high x are the same. A rational worker chooses the
option with the highest value, therefore, the lifetime income of a worker who received
an offer x is:
20

21. v(x) = max[Wa(x), Wr] = max[
x
1 − β
, Wr] (60)
The job is accepted if x is greater than a critical value, x∗
, that we call the reservation
wage.
x∗
= Wr(1 − β) (61)
If an agent declines an offer, then he receives the unemployment benefit and a new
offer, later in time. The value of this new offer is corrected by the discount factor.
The value of rejecting an offer is, then:
Wr = b + βE {max[Wa(x), Wr]} (62)
Substituting and working the expression:
x∗
= Wr − βWr = b + βE {max[Wa(x), Wr]} ⇔
⇔ x∗
= b + βE {max[Wa(x) − Wr, 0]} (63)
This equation can be solved.
x∗
= b + β
Z ∞
0
max[Wa(x) − Wr, 0]dF(x) =
= b + β
Z x∗
0
max[Wa(x) − Wr, 0]dF(x) + β
Z ∞
x∗
max[Wa(x) − Wr, 0]dF(x) =
= b + β0 + β
Z ∞
x∗
[Wa(x) − Wr]dF(x)
21

22. Recalling the expressions for Wa and Wr, we can arrive at the fundamental reser-
vation wage equation, implicitly defining the reservation wage as a function of the
unemployment benefit, the time-discount factor and the probability distribution of
the wage offers.
x∗
= b + β
Z ∞
x∗
[
x
1 − β
−
x∗
1 − β
]dF(x) ⇔
⇔ x∗
= b +
β
1 − β
Z ∞
x∗
(x − x∗
)dF(x) (64)
22

23. Continuous-Time and Stochastic Offer Arrivals
Let’s introduce the time-dimension, by setting the duration of each period to h, and
the probability of receiving an offer in the period to λh. In the limit when h converges
to zero, this stochastic process is a Poisson with parameter λ.
In this setting, it is natural to assume that the wage and unemployment benefit are
wh and bh, and that the time discount is e−ρh
.
With t being an index for the periods, the value of accepting an offer becomes:
Wa(x) =
∞
X
t=0
e−ρht
xh =
xh
1 − e−ρh
(65)
The reservation wage must be such that Wa = Wr.
x∗
=
1 − e−ρh
h
Wr (66)
Adapting the expression for Wr, we obtain:
Wr = bh + e−ρh
E {λh max[Wa(x), Wr] + (1 − λh)Wr} (67)
(1 − e−ρh
)Wr = bh + e−ρh
E {λh max[Wa(x), Wr] − λhWr} (68)
(1 − e−ρh
)Wr = bh + e−ρh
E {λh max[Wa(x) − Wr, 0]}
1 − e−ρh
h
Wr = b + e−ρh
E {λ max[Wa(x) − Wr, 0]}
x∗
= b + λe−ρh
Z ∞
x∗
[Wa(x) − Wr]dF(x)
23

24. x∗
= b + λe−ρh
Z ∞
x∗
xh
1 − e−ρh
−
x∗
h
1 − e−ρh
dF(x)
x∗
= b +
λhe−ρh
1 − e−ρh
Z ∞
x∗
(x − x∗
)dF(x) (69)
The model of job search can also be defined in continuous time. Consider the limit
when h goes to zero and using L’Hôpital’s rule:
lim
h→0
1 − e−ρh
h
= ρ (70)
The expression for the reservation wage becomes:
x∗
= ρWr = b +
λ
ρ
Z ∞
x∗
(x − x∗
)dF(x) (71)
Rearranging, we compare the opportunity cost of rejecting an offer with the gain
associated with searching. These are equal when the job offer equals the reservation
wage.
x∗
− b =
λ
ρ
Z ∞
x∗
(x − x∗
)dF(x) (72)
To do comparative statics using this expression, write:
H(x∗
; b, λ, ρ) = x∗
− b −
λ
ρ
Z ∞
x∗
(x − x∗
)F0
(x)dx = 0 (73)
To calculate the partial derivatives, Leibniz’s rule is useful:
φ(x) =
Z b(x)
a(x)
G(x, s)ds ⇒
⇒ φ0
(x) =
Z b(x)
a(x)
∂G(x, s)
∂x
ds + b0
(x)G(x, b(x)) − a0
(x)G(x, a(x)) (74)
24

25. Hx∗ =
∂H(x∗
; b, λ, ρ)
∂x∗
= 1 −
λ
ρ
Z ∞
x∗
(−1)F0
(x)dx − (x∗
− x∗
)F0
(x∗
)
⇒ (75)
Hx∗ = 1 +
λ
ρ
[1 − F(x∗
)] 0 (76)
Hb = −1 0 (77)
Hλ = −
1
ρ
Z ∞
x∗
(x − x∗
)F0
(x)dx 0 (78)
Hρ = −
λ
ρ2
Z ∞
x∗
(x − x∗
)F0
(x)dx 0 (79)
By the implicit-function theorem:
∂x∗
∂b
= −
Hb
H∗
x
0 (80)
∂x∗
∂λ
= −
Hλ
H∗
x
0 (81)
∂x∗
∂ρ
= −
Hρ
H∗
x
0 (82)
Unsurprisingly, the reservation wage increases with the unemployment benefit and
with the frequency of offers, and decreases with the time discount factor.
25

26. References
De la Fuente, A. (2000), “Mathematical Methods and Models for Economists”, Cambridge
University Press.
Gandolfo, G. (1997), “Economic Dynamics”, Springer.
26