Steady state law of demand in dynamic consumer problems∗
Davide Dragone†
Paolo Vanin‡
February 13, 2015
PRELIMINARY AND INCOMPLETE
Abstract
We study a broad class of dynamic consumer problems in continuous time to characterize
the response of the steady state demand for a good to a permanent increase in its market
price. We show that violations of the steady state law of demand can arise even in absence of
income effects. The results are robust to a variety of settings that are commonly used in the
economic literature on intertemporal consumer behavior. Possible applications include models
of habit formation, addiction, labor supply, learning, health capital and renewable resources.
JEL-Classification:
Keywords:
1
Introduction
One of the most common empirical regularities in economics is the law of demand: other things
equal, the quantity demanded of a good falls when its price rises. From a theoretical point of
view, the law of demand may not hold. In a static consumer problem, it is violated in the case of
Giffen goods, due to income effects. Less is known on the conditions for its validity in dynamic
models. In this paper we consider an intertemporal consumer problem and we show that the
interplay between impatience and dynamics can lead to violations of the law of demand even in
the absence of income effects.
For a broad and abstract class of dynamic consumer problems, we characterize the conditions
under which a permanent increase in the market price of a good leads to an increase in its steady
state demand. The class of problems we consider allows for the effect of past actions on current
choices, for either static or dynamic budget constraints, and for resource allocation to consumption
goods as well as for time allocation to labor and leisure.
∗
We thank Vincenzo Denicolò.
Department of Economics, University of Bologna
‡
Department of Economics, University of Bologna
†
1
We show that steady state price effects have two components, which are the dynamic counterpart of income and substitution effects in a static problem. As is true in a static framework,
income effects depend on interactions among the arguments of the objective function, and can
lead to violations of the law of demand. The difference is that such interactions are potentially
more complex, as they may include the interplay of state and control variables. Differently from
a static framework, however, the dynamic counterpart of the substitution effect can lead by itself
to violations of the law of demand, which can arise even in the absence of any interactions in the
objective function, and thus of income effects.
Despite the complexity of dynamic models, the conditions under which this possibility arises
can be surprisingly simple. For example, under a set of assumptions which are common in the
economic literature, we show that the sign of the steady state price effect only depends on the
relation between impatience and state dynamics: in such cases, the law of demand is violated if
and only if, in steady state, the marginal effect of the state variable on its own speed of change is
positive but lower than the discount rate. Using the formalization introduced below, this condition
can be written as fS ∈ (0, ρ). This condition implies that the law of demand is satisfied whenever
in steady state fS < 0. When this is not the case, the law of demand is violated when impatience
is sufficiently high.
To present the role of impatience in the most transparent way, Section 2 develops an illustrative
model that is simple, rules out income effects by construction, and allows for a closed form solution.
As it assumes fS = 1, the validity of the steady state law of demand only depends on impatience:
for patient consumers (ρ < 1) the law of demand holds, and for impatient consumers (ρ > 1) it
does not. Interestingly, starting from a steady state, a permanent increase in the price of a good
leads to a non monotonic response in the quantity demanded. For patient consumers demand
initially rises and then falls, eventually reaching a level that is lower in the new steady state than
in the old one. For impatient consumers, the reverse is true.
For later reference, in Section 3 we briefly review price effects in a static consumer problem.
In the following Sections we extend the static setup to different dynamic settings. Sections 4 and
5 consider the cases of static and dynamic budget constraint, respectively, to characterize steady
state price effects, highlight how violations of the law of demand can emerge in the absence of
income effects, and qualifies the results for the case of Frisch demand functions. Section 6 considers
an intertemporal model of labor supply and discusses the conditions under which, again absent
income effects, the steady state Frisch labor supply curve is backward bending.
Sections 3 to 6 are framed in abstract setups that can be applied to intertemporal models
of habit formation, human capital accumulation, health, addiction, and learning, to name a few.
Section 7 presents two specific applications highlighting how violations of the steady state law of
demand can arise in the standard Becker and Murphy (1988)’s model of rational addiction and
in the Grossman (1972) model of health capital accumulation. Section 8 concludes.
2
2
An illustrative example
Consider the following utility function, defined over two goods x ≥ 0 and y ≥ 0 and the state
variable S :
x2
S2
U (x, y, S) = x −
+y+S−
(1)
2
2
In static maximization problem, the consumer solves the following, taking S, the market price p
and income M as given:
x2
S2
+y+S−
max U (x, y, S) = max x −
x,y
x,y
2
2
s.t. M = px + y
(2)
(3)
Let p ∈ (0, 1) and M > p(1 − p), so as to grant an internal solution. Replacing the budget
constraint, the problem can be re-written in terms of V (x) = U (x, M − px):
x2
S2
max V (x) = max x −
+ M − px + S −
,
x
x
2
2
(4)
with first order condition (FOC) Vx = 1 − x − p = 0. The corresponding optimal consumption
choices are x∗ = 1 − p and y ∗ = M − p(1 − p). The quasi-linear specification of the utility function
ensures that all income effects are captured by y, so that changes in p have only substitution
∗
effects on the demand for x, and ∂x
∂p = −1.
Now consider a similar dynamic problem where the control variables x and y and the state
variable S are functions of time, the initial state is S(0) = S0 , and the future is discounted
exponentially at rate ρ > 0:
Z
max
x,y
∞
−ρt
e
Z
U (x, y, S)dt = max
x,y
0
s.t. M
∞
−ρt
e
0
x2
S2
x−
+y+S−
dt
2
2
(5)
= px + y
(6)
Ṡ = S − x.
(7)
Replacing the budget constraint, the current value Hamiltonian function associated to this problem is
x2
S2
H(x, S, µ; p, M ) = x −
+S−
+ M − px + µ(S − x)
2
2
where µ represents the shadow price of the state variable. The necessary and sufficient conditions
3
for an internal solution are
Hx = 1 − x − p − µ = 0
(8)
µ̇ = µ(ρ − 1) + S − 1
(9)
Ṡ = S − x
(10)
with the transversality condition limt→∞ e−ρt µS = 0. Note that in the dynamic problem the FOC
Hx = 0 equates marginal utility to the sum of the market price and the shadow value price of the
state variable (Ux = p + µ). In other words, the FOC equates the marginal utility of consumption
with its full price Π = p + µ. In the static problem the FOC Vx = 0 equates the marginal utility
of x to its market price only (Ux = p).
Differentiating Hx = 0 with respect to time and replacing µ = 1 − x − p from the FOC (8)
one can express the necessary and sufficient conditions in terms of the control variable x and the
state variable S:
ẋ = 1 − S + (x + p − 1) (ρ − 1)
(11)
Ṡ = S − x
(12)
By setting ẋ = Ṡ = 0, one obtains the steady state values
xss = S ss = 1 +
1−ρ
p
|J|
(13)
where |J| = ρ − 2 is the determinant of the Jacobian matrix. Saddle point stability (|J| < 0)
requires ρ ∈ (0, 2). The following Remark holds.
Remark 1 (Dynamic price effect: illustrative example) Consider a steady state with saddle point stability of problem (5)–(7). After a permanent increase in the market price of good x,
its steady state consumption decreases when the consumer is sufficiently patient (ρ < 1) and it
increases when she is sufficiently impatient (ρ > 1).
In Figure ??, for an impatient consumer (ρ > 1), we show the phase diagram in the (S, x)
space (upper panel) and in the (S, Π) space (lower panel), to illustrate how the optimal path and
the stable steady state react to a permanent (and unexpected) increase in p. The policy function
shifts downward. This implies that the demand for x initially jumps downward and subsequently
increases, to eventually reach a higher level in the new steady state than the initial one.1
This result is not due to income effects, which are nil by construction, but to the dynamic
adjustment of the state and costate variable. From FOC (8), at each point in time Ux = Π, where
Π = p + µ is the full price of x. Since the problem is additively separable, an increase in the full
1
When the agent is patient (ρ < 1), the pattern is reversed (see the Appendix for details).
4
Figure 1: Effect of an increase in the price of x for an for an impatient consumer (ρ > 1).
Upper panel: The policy function x(S) shifts downward, hence consumption of x first jumps from A
to B, and then follows the new policy function x(S)new until the new steady state C is reached. The
lower panel shows that the pattern for the full price of x is reversed.
∂x
= U1xx < 0. Yet, the full price needs not increase monotonically
price leads to a decrease in x: ∂Π
ρ−1
in the market price p. Given S, its response to a rise in p depends on impatience: ∂Π
∂p = |J| . Over
1
time, irrespective of the degree of impatience, the full price of x decreases in S: ∂Π
∂S = |J| < 0.
Thus, when the agent is impatient (ρ > 1), after a permanent increase in p, Π first rises and then
progressively falls. In the new steady state, despite the fact that the market price p is higher, the
full price Π is lower than in the initial steady state, which explains why xss is higher.
The economic intuition is the following. In a dynamic environment, a forward looking consumer must balance short run and long run considerations when choosing the optimal consumption
5
path. The balance depends on the intertemporal discount rate ρ, among other factors. If the agent
is impatient enough (ρ > 1), short run considerations are relatively more important than long run
ones. Essentially, on impact the agent behaves as in a static environment where, absent income
effects, the substitution effect induces a reduction in x after a rise in p. A lower x determines an
increase in the state variable S, which progressively lowers the full price of x and thus raises its
consumption, eventually resulting in a higher steady state level of both x and S. An impatient
consumer heavily discounts these long run consequences and reacts on impact as she would in a
static model.
If instead the agent is patient enough (ρ < 1), long run considerations are relatively more
important than short run ones. In steady state, as the full price of x increases when its market
price increases, a patient consumer aims at reducing x. To achieve this goal, she reacts on impact
in a way that is opposite to what she would do in a static environment, by initially consuming
more x in order to reduce S over time and thus allow herself in the future a lower consumption
of x.
3
A static consumer problem
In this section we briefly remind well known results from consumer theory, to emphasize that in
a static environment violations of the law of demand require a strong and negative interaction
between consumption goods in the utility function, and to have a benchmark for violations in a
dynamic environment.
Consider a utility function U (x, y, S), where x ≥ 0 and y ≥ 0 are two consumption goods and S
is a state variable (which can be simply considered as a parameter in a static problem). The utility
function is assumed to be concave, continuously differentiable and non satiated. Given income
M + w (S) > 0 (which may entail the state-dependent component w(S) for future comparability)
and market price p > 0, the problem is the following
max
x,y
s.t.
U (x, y, S)
(14)
M + w (S) = px + y
(15)
The Lagrangian function corresponding to this problem is
L = U (x, y, S) + λ [M + w (S) − px − y]
where λ is the Lagrangian multiplier associated to the static budget constraint.
6
(16)
In an internal solution the following first order conditions (FOCS) hold2 :
Lx = Ux − pλ = 0
(17)
Ly = Uy − λ = 0
(18)
Lλ = M + w (S) − px − y = 0
(19)
and implicitly determine x̂, ŷ and λ̂ as functions of p and M. As it is well known, applying the
implicit function theorem, the static price effect is
∂ x̂
∂ x̂
λ̂
= − − x̂
∂p
∂M
Ω̂
(20)
where Ω̂ = −Uxx +2pUxy −p2 Uyy > 0 by concavity. The first term represents the static substitution
effect, which is positive under non satiation (see Lx and Ly ), while the second term represents
the static income effect, with
Uxy − pUyy
∂ x̂
=
.
(21)
∂M
Ω̂
Hence Uxy , the interaction between x and y in the utility function, crucially determines the sign
of the static income effect. If such interaction is sufficiently negative (Uxy < pUyy ), good x is
inferior; if it is below an even lower threshold (Uxy < − λ̂x̂ + pUyy ), good x is a Giffen good and the
x
law of demand is violated. The reason is that, at the optimum the FOCs imply M RSxy = U
Uy = p.
When p increases, for ordinary goods the M RS is raised by consuming less x and more y, thus
reducing Ux and increasing Uy . For Giffen goods, instead, increase in the M RS is indeed obtained
by consuming more x in response to an increase in its price, because increasing the consumption
of x reduces Ux , but it also reduces Uy more than proportionally.
4
A dynamic consumer problem without saving
In the following problem we consider a generic consumer’s maximization problem in an dynamic
setting, in which we allow both utility and income to be state dependent. In this section we ignore
the possibility of saving, borrowing and lending, which are considered in next section.
Let the instantaneous utility function be U (x (t) , y (t) , S (t)) and the budget constraint be
M + w(S (t)) ≥ px (t) + y (t) , where x (t) ≥ 0 and y (t) ≥ 0 are consumption goods, S (t) is a
state variable and w (S) is the state-dependent component of income, assumed to be continuously
differentiable and quasiconcave. Given an initial condition S (0) = S0 , the state variable evolves
according to Ṡ = f (x (t) , y (t) , S (t)) . Omitting the time arguments for brevity, we assume that
U (x, y, S) and f (x, y, S) are continuously differentiable and concave functions in (x, y, S) , with
Uxx < 0. We will focus on the case in which the budget constraint is binding and the non-negativity
2
Throughout the paper subscripts denote partial derivatives.
7
constraints are not.
Given an intertemporal discount rate ρ > 0 and initial state S(0) = S0 , the consumer’s
problem is
Z
∞
max
{x,y}
e−ρt U (x, y, S) dt
(22)
0
s.t. M + w(S) = px + y
(23)
Ṡ = f (x, y, S) .
(24)
The current-value Hamiltonian function is
H(x, y, S, λ, µ; p, M ) = U (x, y, S) + λ[M + w(S) − px − y] + µf (x, y, S)
(25)
where µ is the costate variable associated to the state S. If the Hamiltonian function is concave
in state and control variables, the following conditions are necessary and sufficient for an internal
solution (Mangasarian, 1966; Seierstad and Sydsaeter, 1977):3
Hx = Ux (x, y, S) − λp + µfx (x, y, S) = 0
(26)
Hy = Uy (x, y, S) − λ + µfy (x, y, S) = 0
(27)
Hλ = M + w(S) − px − y = 0
(28)
µ̇ = ρµ − HS (x, y, S)
(29)
Ṡ = f (x, y, S)
(30)
with the transversality condition limt→∞ e−ρt µ (t) S (t) = 0.4
The FOCS (26)-(28) determine the optimal value of x, y and λ as functions of the state and
costate variable, of the market price and income:
x∗ = x∗ (S, µ; p, M )
(31)
y ∗ = y ∗ (S, µ; p, M )
(32)
λ
∗
∗
= λ (S, µ; p, M )
(33)
Replacing (x∗ , y ∗ , λ∗ ) in (29)-(30) yields the optimal state and costate dynamics
µ̇ = ρµ − HS (x∗ , y ∗ , λ∗ , S, µ; p, M )
(34)
Ṡ = f (x∗ , y ∗ , S)
(35)
3
If f (x, y, S) is non linear, the above conditions are necessary and sufficient for a maximum provided along the
optimal solution µt ≥ 0 for all t.
4
The budget constraint is binding if, along the optimal path, λ is positive, which occurs when Ux (x, y, S) +
µfx (x, y, S) > 0 and Uy (x, y, S) + µfy (x, y, S) > 0 at all t. We focus on this case.
8
Given the initial state and the terminal conditions, the solution {S (t) , µ (t)} of the above system
of differential equations represents the optimal trajectories of the state and costate variables,
which depend on p and M . Plugging them into (31) to (33), the optimal path is:
4.1
x (t) = x∗ (S (t) , µ (t) ; p, M )
(36)
y (t) = y ∗ (S (t) , µ (t) ; p, M )
(37)
λ (t) = λ∗ (S (t) , µ (t) ; p, M )
(38)
Steady state price effect without saving
We now focus on the optimal path of x. An increase in market price changes x(t) in (36) as
follows:
∂x∗ ∂x∗ ∂S (t) ∂x∗ ∂µ (t)
∂x (t)
=
+
+
.
(39)
∂p
∂p
∂S ∂p
∂µ ∂p
where the first term is the direct effect and the last two terms represent the indirect effects through
changes in the state and costate variables S and µ.
We will focus on stable steady states, for which conclusions can be reached without assuming
further structure, and denote steady state variables with ss superscript:
xss = x∗ (S ss , µss ; p, M )
(40)
With respect to the illustrative example of Section 2, this general model allows for considerably
richer interactions among x, y and S, both in the utility function and in the law of motion. Yet,
steady state price effects admit a surprisingly compact representation, as shown below.
Let Ω = −Hxx + 2pHxy − p2 Hyy and |J| be the determinant of the Jacobian matrix associated
to (34)-(35) (see equation 72 in the Appendix). Let the exogenous income effect in steady state
be −xss ∂xss /∂M , where ∂xss /∂M is given in equation (81) in the Appendix. The following holds
(all proofs are in the Appendix):
Proposition 1 (Steady state price effect: no saving) Suppose problem (22)–(24) admits a
steady state with saddle point stability. The response of the steady state consumption of x to a
permanent increase in its market price p is described by
(fS + fy wS ) (fS + fy wS − ρ)
∂xss
∂xss
= λss
− xss
.
∂p
|J| Ω
∂M
(41)
ss
The steady state law of demand is violated when ∂x
∂p > 0. Equation (41) highlights two
5
possible sources of violation. One is the exogenous income effect, described by the second term.
5
Although our interest is in steady states that can be reached, notice that equation (41) holds for both stable
and unstable steady states.
9
In a static environment income effects are the only source of violation of the law of demand. The
second one, captured by the first term of equation (41), is a dynamic analogue of the substitution effect. Yet, contrary to the static substitution effect, it can be positive due to the specific
intertemporal features of the model: impatience and the dynamics of S. To focus on this latter
aspect, the following Remark provides sufficient conditions under which there are no exogenous
income effects.
Remark 2 (No income effects: sufficient conditions) In a stable steady state of problem
ss
(22)–(24), there are no exogenous income effects ( ∂x
∂M = 0) if the utility function is quasi linear
in y, and y does not affect the evolution of S, i.e. fy = HyS = Hxy = Hyy = 0.
The conditions presented in Remark 2 are commonly assumed in the economic literature, as
shown in the applications discussed in Section 7. They imply that λ∗ > 0 along the optimal path
(and hence in steady state, see equation (27) above). Since Ω > 0 by concavity and |J| < 0 in a
steady state with saddle point stability, the following holds:
Corollary 1 (Steady state price effect absent income effects) Under the conditions of Remark 2, which rule out income effects, steady state consumption of x increases after a permanent
ss
increase in its market price p ( ∂x
∂p > 0) if and only if fS ∈ (0, ρ) .
Absent income effects, if in steady state fS is negative, the steady state law of demand cannot
be violated. By construction, this holds in models with quasi-linear utility functions and with
a law of motion of the form Ṡ = g(x) − δS. By contrast, models in which in steady state fS is
positive allow for possible violations of the law of demand even in the absence of income effects. A
particularly simple application is presented in the illustrative example of Section 2, where fS = 1.
By Corollary 1, the steady state law of demand is violated if and only if ρ > 1, in which case a
rise in the market price p reduces the full price Π in steady state, which in turn translates into a
higher steady state consumption of x.
The possibility of violations of the law of demand in the absence of income effects is a specific
feature of intertemporal consumer problems, and it cannot arise in a static environment. It is due
to intertemporal substitution effects, and it is driven by the interplay between time preferences ρ
and the technology of state accumulation f .
Clearly, income effects remain a possible source of violation of law of demand in dynamic
models. Proposition 1 shows that market prices may affect both the value of the exogenous and
the endogenous component of income, M and w(S). As is true for Giffen goods, such income
effects are related to interactions among the arguments of the utility function, but these now
also include interdependences between state and control variables, as shown in more detail in the
Appendix.
10
5
A dynamic consumer problem with saving
The possibility of violating the law of demand holds even when the agent can save or borrow.
In such a case the static budget constraint considered in the previous section is replaced by a
dynamic budget constraint. Returns on assets r(A) are assumed to be increasing and concave.
Given initial wealth A (0) = A0 , the problem becomes
Z
∞
e−ρt U (x, y, S) dt
(42)
s.t. Ȧ = r(A) + M + w(S) − px − y
(43)
max
{x,y}
0
Ṡ = f (x, y, S)
(44)
with S (0) = S0 and A (0) = A0 . The corresponding current-value Hamiltonian function is:
H(x, y, S, A, µ, λ; p, M ) = U (x, y, S) + µf (x, y, S) + λ [r(A) + M + w (S) − px − y]
(45)
where µ and λ are the costate variables associated to the states S and A, respectively.
The following conditions are necessary for an internal solution:
Hx = Ux (x, y, S) − λp + µfx (x, y, S) = 0
(46)
Hy = Uy (x, y, S) − λ + µfy (x, y, S) = 0
(47)
λ̇ = λ (ρ − rA )
(48)
µ̇ = ρµ − HS (x, y, S, A, µ, λ; p, M )
(49)
Ȧ = r(A) + M + w(S) − px − y
(50)
Ṡ = f (x, y, S)
(51)
together with the transversality conditions limt→∞ e−ρt µ (t) S (t) = 0 and limt→∞ e−ρt λ (t) A (t) =
0. The above conditions are also sufficient for a maximum if H(x, y, S, A, µ, λ; p, M ) is concave in
the state and control variables (Mangasarian, 1966; Seierstad and Sydsaeter, 1977). We additionally assume Hxx and Hyy to be strictly negative.
Proceding as in the previous section allows to obtain the steady state consumption of x:
xss = x∗ (S ss , Ass , µss , λss ; p, M )
(52)
˜
where x∗ represents the value of x that satisfies the first order conditions (46) and (47). Let |J|
be the determinant of the Jacobian matrix associated to (48)-(51) once the first order conditions
are satisfied (see equation (88) in the Appendix). The following holds:
Proposition 2 (Steady state price effect with saving) Suppose problem (42)–(44) admits
˜ > 0. The response of the steady state consumption
a steady state with saddle point stability and |J|
11
of x to a permanent increase in its market price p is described by (41).
Proposition 2 shows that the steady state price effect in the consumer’s problem with saving
is formally identical to that in the problem without saving. Note, however, that the terms Ω, |J|
and ∂xss /∂M , are now derived from a different Hamiltonian function.6
5.1
Steady state price effects with a given interest rate
Proposition 2 implicitly requires strict concavity of the asset return function, because this is
˜ > 0, as shown by equation (88) in the Appendix. Such strict concavity is a
necessary to have |J|
conceivable assumption, but in many economic models the consumer takes the market interest rate
r > 0 as given, asset income is just r(A) = rA, and it is assumed that r = ρ (see, e.g., Heckman,
1974; Becker and Murphy, 1988). In such a case, the shadow price of wealth is constant over time,
λ(t) = λ0 for all t. In principle λ0 depends on the parameters of the model, but the economic
literature often studies Frisch demand functions, in which price changes are compensated so that
the marginal utility of wealth λ0 remains constant.
With constant λ, the dynamic system reduces to equations (49)–(44), where the optimal
values of controls satisfy the first order conditions. In a steady state with saddle point stability,
ˆ < 0 (see
the determinant of the associated 3-dimensional Jacobian matrix must be negative, |J|
equation (110) in the Appendix). The following holds.
Proposition 3 (Steady state price effect for Frisch demand functions) Suppose problem
ˆ < 0).
(42)–(44), with r (A) = rA and r = ρ, admits a steady state with saddle point stability (|J|
The response of the steady state consumption of x to a permanent increase in its market price p
that is compensated to maintain the marginal utility of assets constant is
fS (fS − ρ) Hyy + fy [fy HSS + (ρ − 2fS ) HyS ]
∂xss
= −ρλ0
ˆ
∂p
Ω|J|
(53)
It can be checked that in steady state exogenous income effects are nil, ∂xss /∂M = 0, so the
above price effect is only due to intertemporal substitution.
In the special case in which the dynamics of state S is only influenced by control x, one obtains
Corollary 2 (Steady state price effects for Frisch demand functions when fy = 0)
If fy = 0, then the Frisch steady state consumption of x increases after a permanent increase in
ss
its market price p ( ∂x
∂p > 0) if and only if fS ∈ (0, ρ) .
We thus have that in a common class of consumer problems with dynamic budget constraint,
violations of the steady state Frisch law of demand are characterized by the same simple condition
obtained in Corollary 1 for a problem without saving and without income effects: fS ∈ (0, ρ) .
6
In the special case of additive separability, with fy = Hxy = HyS = HxS = 0, it holds that
fS ) f
λss −xss pHyy
(Hxx +p2 Hyy )−fx2 HSS
S (ρ−fS )
, whose sign depends on fS (ρ − fS ) as in Remark 1.
12
∂xss
∂p
= fS (ρ −
More generally, if the dynamics of state S is also affected by control y, another possible
source of violations comes from the term fy (ρ − 2fS ) HyS (notice that the term fy2 HSS cannot
be positive). As we know from the static theory of Giffen goods, and as already discussed above,
interactions in the objective function can be a source of violation of the law of demand. What is
interesting to notice here is that the relevant interaction is now between state S and control y.
6
A dynamic labor supply problem
An important class of intertemporal consumer problems concern the choice between consumption
and leisure. A slight modification of the problem discussed in the previous section allows studying
this case. The natural counterpart of price effects is now a wage effect. In particular, we will
characterize the steady state slope of the (Frisch) labor supply curve. Consider the following
individual problem:
Z
max
x,y
∞
e−ρt U (x, y, S) dt
(54)
0
s.t. Ṡ = f (x, y, S)
Ȧ = rA + M + wg(x, S) − y,
(55)
(56)
where x is leisure, y is consumption. Given a time endowment normalized to one, ` = 1 − x is
labor supply. The term g(x, S) represents effective labor and w is its exogenous return. The main
formal difference with respect to the previous section is that now state S and control x do not
enter the dynamic budget constraint in an additively separable way, but can potentially interact.
We assume g (x, S) to be continuously differentiable and decreasing in leisure, gx (x, S) < 0, but
we make no specific assumption on the the effect of state S on effective labor, thus accommodating
for a variety of models, including human capital, health and addiction.
Consider the current-value Hamiltonian
H(x, y, S, A, µ, λ; w, M ) = U (x, y, S) + µf (x, y, S) + λ [rA + M + wg (x, S) − y] .
(57)
ˇ denote the relUnder the usual concavity requirements and transversality conditions, letting |J|
evant determinant of the Jacobian matrix, the following holds.
Proposition 4 (Steady state slope of Frisch labor supply) Suppose problem (54)–(56), with
ˇ < 0). The response of steady state
r = ρ, admits a steady state with saddle point stability (|J|
13
Frisch labor supply ` to a permanent increase in the wage w paid per unit of effective labor is
fS (fS − ρ) Hyy + fy [fy HSS + (ρ − 2fS ) HyS ]
∂`ss
gx
= − ρλ0
ˇ
∂w
Ω|J|
fS fx Hyy + fy [fy HxS − fS Hxy − fx HyS ]
+ ρλ0
gS
ˇ
Ω|J|
(58)
As in the previous section, in steady state exogenous income effects are nil, ∂`ss /∂M =
0. If effective labor does not depend on state S, gS = 0, and gx = −1, then expression 58
essentially reduces to 53, with the obvious difference between the Hamiltonian and Jacobian
functions between the two cases, and with the opposite sign, since we are now looking at labor
supply rather than at leisure demand. Moreover, in the special case in which gS = fy = 0, one
obtains
Corollary 3 (Steady state Frisch labor supply when gS = fy = 0)
If gS = fy = 0, then the steady state Frisch labor supply decreases after a permanent increase in
ss
the wage w ( ∂`
∂w < 0) if and only if fS ∈ (0, ρ) .
Again, this is the same simple condition obtained in the previous sections. If gS 6= 0 and
fy 6= 0, then interactions give rise to several additional channels that may generate a backward
bending labor supply curve.
7
Applications
7.1
Example 1: Addiction and habit formation
Becker and Murphy (1988) develop a rational addiction model in which the consumer chooses
between an addictive good x and a non addictive good y. Consumption of the addictive good
contributes to the accumulation of an addiction capital S, which in turn makes x more desirable
(UxS > 0). Their model is a special case of model (42) to (44), in which f (x, y, S) = x − δS, the
utility function is quadratic, M = 0, and r = ρ. Since fy = 0 and fS = −δ < 0, according to
Corollary 2, steady state consumption of the addictive good satisfies the law of demand.
Becker and Murphy mention, but do not formally develop, the possibility of an investment
in addiction reduction (for instance, participation in a rehabilitation program). Suppose y is
interpreted as such an investment and for simplicity assume f (x, y, S) = x + y − δS. One can
imagine that participating to the rehab program is physically or mentally more costly the higher
individual addiction, that is UyS < 0. According to Proposition 3, if such an effect is large, then
an increase in the price of the addictive good can lead to a long run increase in its consumption.7
7
Formally, ∂xss /∂p > 0 if and only if UyS <
UyS < kUxS , where k > 1 if Uxy = 0 .
δ(δ+ρ)Uyy +USS +λ̄wSS
2δ+ρ
14
< 0, which in a stable steady state implies
Yet, surprisingly, it can be shown that it also leads to a lower stock of addiction thanks to the
countervailing effect of higher rehab expenditure.
Consider now a variation of their model in which y is also addictive (i.e. UyS > 0) and
contributes to the addiction stock S (e.g. x is beer and y is vodka). According to Proposition 3,
in a stable steady state the consumption of x increases in its price when good y is sufficiently more
addictive than good x.8 In this case, responding to an increase in the price of x by substituting y
for x would excessively increase addiction. To avoid it, the consumer reduces y and increases x
(as the latter is less addictive) and ultimately the stock of addiction decreases. A higher tax on
beer may thus reduce alcoholism but, surprisingly, through a reduction in vodka consumption and
an increase in beer consumption.
7.2
Example 2: Health capital
Grossman (1972) proposes a health accumulation model in discrete-time. Its essence is captured
by the continuous-time model (42) to (44), where S is health capital, x is a health investment (say,
medical care) and y is a consumption good, with U (x, y, S) = V (y, S) and f (x, y, S) = g (x)−δS,
M = 0, where g (·) is an increasing and concave function and r = ρ. Since fy = 0 and fS = −δ < 0,
according to Corollary 2, the steady state investment in health satisfies the law of demand.
Consider now a variation of the model in which the consumer can increase health through
medical care x and physical activity y. For simplicity, consider f (x, y, S) = g (x) + y − δS. It is
natural to assume that physical activity is more enjoyable (or less costly) for healthier people,
that is UyS > 0. This effect has the potential to lead to violations of the law of demand: when it
is sufficiently strong, according to Proposition 3 , in a stable steady state an increase in the price
of medical care can lead to an increase in the consumption of medical care, a decrease in physical
activity and a decrease in health 9
Hence, responding to an increase in the price of medical care by reducing medical care would
decreases health, which in turns reduces the marginal utility of physical exercise. If such an effect
is large enough, the consumer may choose to reduce physical exercise, which would result in a
further decrease in health. To counter this effect, the agent increases medical care, although not
enough to avoid a reduction in health and in physical exercise. A higher price of medical care
may thus reduce health but, surprisingly, through a reduction in physical exercise and an increase
in medical care.
8
δ(δ+ρ)U
+U
+λ̄w
yy
SS
SS
Formally, ∂xss /∂p > 0 if and only if UyS > −
> 0, which in a stable steady state implies
2δ+ρ
UyS > kUxS , where k > 1.
δ(δ+ρ)Uyy +USS +λ̄wSS
9
Formally, ∂xss /∂p > 0 if and only if UyS > −
> 0.
2δ+ρ
15
A
A.1
Appendix
Illustrative example
Solving the system of linear ordinary differential equations Ṡ and ẋ, given S (0) = S0 , S (∞) = S ss
and x (∞) = xss , yields
x (t) = xss + (S0 − S ss ) (1 − e1 ) ee1 t
(59)
S (t) = S ss + (S0 − S ss ) ee1 t
(60)
p
where e1 = ρ − 8 − 4ρ + ρ2 /2 is the smaller eigenvalue of the Jacobian matrix
J=
1
1
1 ρ−1
!
(61)
whose determinant is |J| = ρ − 2. Note that neither the determinant nor e1 depend on the market
price, and that stability requires ρ < 2, which is assumed to hold hereafter. The steady state for
t → ∞ is:
xss = S ss =
µss =
p
,
|J|
(ρ − 1)(1 − p) − 1
|J|
(62)
y ss = M − pxss .
(63)
Rearrange (59) and (60) to obtain the following policy function:
x (S) = xss + (e1 − 1) (S ss − S) .
(64)
For a given value of S, after an increase in p the whole policy function x (S) shifts downward
when ρ > 1, and upward when ρ < 1, as
1−ρ
∂x (S)
= e1
∂p
|J|
(65)
The steady state response of x to an increase in p is opposite in sign to the response on impact:
∂xss
1−ρ
=
∂p
|J|
(66)
To understand the effect of a change in the market price p along the transition to the new steady
state, remember that the FOC requires
Ux = Π
(67)
16
where Π := p + µ denotes the full price of x (see, e.g. Becker and Murphy, 1988). We will show
that, after an increase in the market price p, for ρ > 1 the full price first jumps upward and then
it decreases over time until the new steady state is reached. Since x is negatively related to its full
price, x first jumps downward; then it increases over time until the new steady state is reached.
The crucial point is that ρ > 1 implies that in such a new steady state the full price has decreased
despite the fact that the market price has increased, i.e. ∂Πss /∂p < 0. Hence the new steady
state xss is higher then it was before the price change. If, instead, ρ < 1, the full price first jumps
upward, then it decreases along the transition until a new steady state is reached where the full
price has increased.
After the jump in the value of x, the transition to the new steady state depends on the change
of S and µ over time. Consider the optimal costate as a function of S
µ (S) = µss + (e1 − 1) (S − S ss )
(68)
and focus on the case where ρ > 1. After x has jumped downward, S starts increasing and µ
starts decreasing because
∂µ (S)
= e1 − 1 < 0
(69)
∂S
This is accompanied by a progressive increase in x.
The reduction in the shadow value µ continues until the new steady state is reached. Note
that
1
∂µss
=
< 0.
(70)
∂p
|J|
Hence
∂Πss
∂µss
ρ−1
=1+
=
,
∂p
∂p
|J|
(71)
and we can conclude that, in the new state state, the full price decreases, even if the market price
p has increased, when ρ > 1. The reason is that ∂µss /∂p < −1 for ρ > 1, so the reduction in µ
more than compensates the increase in p. When this is the case, the new steady state value of
x is higher then the initial one. When, instead, ρ < 1, the opposite changes occur and the new
steady state value of x is lower.
A.2
Consumer problem without saving
The determinant of the Jacobian matrix associated to (34)-(35) is
(2fS − ρ) (pHyS − HxS ) + (fx − pfy ) HSS
Ω
2(fx − pfy )(fx HyS − fy HxS ) + [fx (pHyy − Hxy ) + fy (Hxx − pHxy )](2fS − ρ)
+wS
Ω
2
2
fy Hxx − 2fx fy Hxy + fx Hyy
+wS2
,
(72)
Ω
|J| = fS (ρ − fS ) + (fx − pfy )
17
where Ω = −Hxx + 2pHxy − p2 Hyy > 0 by concavity.10 We are interested in assessing
ss
∂xss
∂x∗ ∂x∗ ∂S
∂x∗ ∂µss
=
+
+
.
∂p
∂p
∂S ∂p
∂µ ∂p
From the FOCS (26)-(28) one obtains:
where
∂x∗
λ∗
∂x∗
= − − x∗
.
∂p
Ω
∂M
(73)
Hxy − pHyy
∂x∗
=
.
∂M
Ω
(74)
In general expression (73) is different from the static price effect (20) and from the overall price
effect of a dynamic consumer’s maximization problem, which we will compute below.11 In analogy
to the Frischian demand function, which is obtained maintaining the shadow value of income
constant, (73) allows to identify the slope of the demand function keeping the state and the
shadow price of the state (i.e. the costate) fixed.12
Moreover:
∂x∗
∂S
∂x∗
∂µ
=
=
HxS − pHyS
Hxy − pHyy
+ wS
;
Ω
Ω
fx − pfy
;
Ω
(75)
(76)
and the terms ∂S ss /∂p and ∂µss /∂p can be computed by applying the implicit function theorem
to (34)-(35), which yields:
∂S ss
∂p
ss
∂µ
∂p
ss
+ fy wS − ρ ∂x∗
∂S
= −λ
− xss
|J|
∂µ
∂M
ss
∗
∗
λ
∂x
∂x
∂µss
=
HSS
− fS
− xss
|J|
∂µ
∂S
∂M
fy (HxS + pHyS ) − 2fx HyS + (fy Hxy − fx Hyy ) wS
−wS λss
Ω|J|
ss fS
10
(77)
(78)
When the evolution of the state is linear and additively separable in the controls (fxx = fyy = fxy = 0), the
income effect obtained in a static setting coincides with the income effect obtained in the dynamic setting when S
and µ are fixed: ∂x∗ /∂M = ∂ x̂/∂M .
11
Since the direct effect ∂x∗ /∂p is computed keeping state and costate fixed, it is the analogue of expression (20),
with which it coincides for µ = 0.
12
As for the static and the dynamic price effects, expression (73) can be interpreted as the sum of substitution and
income effects. Note that (73) takes into account the role of state and costate variables (which makes it different
from the static case), but it takes them as given (which makes it different from the dynamic case).
18
where
SS
∂S
∂M
∂µss
∂M
fy HxS − fx HyS
fy (pHxy − Hxx ) + fx (Hxy − pHyy )
+ (fS − ρ)
Ω |J|
Ω|J|
2
2
fy Hxx − 2fx fy Hxy + fx Hyy
,
(79)
−wS
Ω|J|
2 − p (H H
Hxx HSS − HxS
SS xy − HxS HyS )
= fy
Ω|J|
2
HxS HyS − Hxy HSS + p HSS Hyy − HyS
+fx
Ω|J|
HxS (Hxy − pHyy ) + HyS (pHxy − Hxx )
+fS
Ω|J|
2
fy (Hxx HyS − HxS Hxy ) + fx (HxS Hyy − Hxy HyS ) − fS Hxx Hyy − Hxy
(80)
+wS
Ω|J|
= (fx − pfy )
Rearranging yields ∂xss /∂p in Proposition 1, where the income effect results from
ss
fy (HxS − pHyS ) − fS (Hxy − pHyy )
∂xss ∂x∗
∂x∗ ∂S
∂x∗ ∂µss
=
+
+
= (fS − ρ)
∂M
∂M
∂S ∂M
∂µ ∂M
Ω |J|
2
fy HxS − fy (fS Hxy + fx HyS ) + fS fx Hyy
fS HyS − fy HSS
+ wS
.
+ (fx − pfy )
Ω |J|
Ω |J|
(81)
To appreciate the role of interactions in the utility function in generating income effects and
violations of the steady state law of demand, let us focus on the first order condition with respect
to x, given by (26). Define the full price as Π ≡ Ux − Hx . As in the illustrative example, FOC
(26) requires Ux = Π along the optimal path, although now Π = λp − µfx . Differentiate (26) to
get Uxx dx + Uxy dy + UxS dS = dΠ. If Uxy = UxS = 0, as assumed in the illustrative model, then
along the optimal path x is inversely related to Π. Thus, in that model violations of the steady
state law of demand can only be due to the fact that Πss is decreasing in p. In general, however,
the steady state law of demand can be violated even if Πss is not decreasing in p, due to the fact
that Uxy and UxS can be different from zero. The role of Uxy closely resembles the mechanism
behind Giffen goods: in the static consumer problem (14)–(15), violations of the law of demand
are due to a sufficiently negative Uxy : in response to a rise in p, Ux increases through a rise in x
and a reduction in y when Uxy < − λ̂x̂ + pUyy . The role of UxS is instead specific to a dynamic
environment.
19
A.3
Consumer problem with saving
The FOCS (46)-(47) determine the optimal value of x, y as functions of the state and costate
variables, of the market price and income:
x∗ = x∗ (S, A, µ, λ; p, M )
(82)
y ∗ = y ∗ (S, A, µ, λ; p, M ).
(83)
2 > 0 by concavity, then application of Cramer’s rule yields:
Denoting Ω̃ = Hxx Hyy − Hxy
∂x∗
∂p
∂x∗
∂λ
∂x∗
∂µ
=
=
=
Hxy HyS − HxS Hyy
λ
∂x∗
Hyy ;
=
;
∂S
Ω̃
Ω̃
pHyy − Hxy ∂x∗
;
= 0;
∂A
Ω̃
fy Hxy − fx Hyy
.
Ω̃
Replacing (x∗ , y ∗ ) in (29)-(30) yields the optimal state and costate dynamics
Ṡ = f (x∗ , y ∗ , S)
(84)
Ȧ = r (A) + M + w(S) − px∗ − y ∗
(85)
µ̇ = ρµ − HS (x∗ , y ∗ , S, A, µ, λ; p, M )
(86)
λ̇ = λ (ρ − rA )
(87)
In steady state, the determinant of the Jacobian associated (84)-(87), is
˜ = λss Ω rAA |J|
|J|
Ω̃
(88)
where Ω and |J| have been defined in Section (A.2) in relation to the problem without saving.
Proceeding as in the previous section, steady state consumption of x is
xss = x∗ (S ss , Ass , µss , λss ; p, M )
(89)
The change of steady state consumption of x after a price increase in p is computed as follows:
ss
∂x∗ ∂x∗ ∂S
∂x∗ ∂Ass ∂x∗ ∂µss ∂x∗ ∂λss
∂xss
=
+
+
+
+
.
∂p
∂p
∂S ∂p
∂A ∂p
∂µ ∂p
∂λ ∂p
20
where
∂S ss
∂p
∂Ass
∂p
∂λss
∂p
∂µss
∂p
=
λss
rA (ρ − rA ) fy2 HxS − (fx Hyy − fy Hxy ) (ρ − fS ) − fy fx HyS
Ω̃|J|
)
ss
(λss )2
∂S
+
rAA (pfy − fx ) (fS − ρ + fy wS ) − xss
∂M
Ω̃|J|
λss
(ρ − rA ) {pfy HyS (2fS − ρ) − fS fy HxS − fy (pfy HSS − fx HSS )
Ω̃|J|
+ [fS (Hxy − Hyy p) − fx HyS + (fy Hxy − fx Hyy ) wS ] (fS − ρ)
ss
ss ∂A
+fy (fx HyS − fy HxS ) wS − x
∂M
2
ss
(λ )
=
rAA fS2 (Hxy − pHyy ) − pfy2 HSS − pfy HyS ρ
Ω̃|J|
−fS [fy (HxS − 2pHyS ) + (Hxy − pHyy )] − fx [HyS (fS − ρ) − fy HSS ]
∂λss
− fy2 HxS + fx Hyy (fS − ρ) − fy (fS Hxy + fx HyS − Hxy ρ) wS − xss
∂M
ss
λ
2
= −
rA (ρ − rA ) fy (HSS Hxy − HxS HyS ) + fx HyS
− HSS Hyy
Ω̃|J|
λss
−
rA (ρ − rA ) fS (HxS Hyy − HyS Hxy )
Ω̃|J|
(90)
=
(91)
(92)
(93)
(94)
(λss )2
rAA {HSS (pfy − f x) + fS (HxS − pHyS )
(95)
Ω̃|J|
+ [fS (Hxy − pHyy ) − 2fx HyS + fy (HxS + pHyS ) + (fx Hxy − fx Hyy )] wS } (96)
∂µss
−xss
(97)
∂M
−
21
and
SS
∂S
∂M
SS
∂A
∂M
SS
∂λ
∂M
λss rAA 2
fx HyS + pfy2 HxS − fx fy (HxS + pHyS )
˜
Ω̃|J|
+ [fy (Hxx − pHxy ) − fx (Hxy − pHyy )] (fS − ρ)}
(ρ − rA ) n 2
2
2
=
fy HSS Hxx − HxS
+ fx2 HSS Hyy − HyS
+ fS Ω̃ (fS − ρ)
˜
Ω̃|J|
+fy [2fx (HxS HyS − HSS Hxy ) + (HxS Hxy − Hxx HyS )(2fS − ρ)]
= −
+fx (HxS Hyy − Hxy HyS ) ρ + 2fx fS (Hxy HyS − HxS Hyy )}
λss rAA n 2
2
2
=
fy HSS Hxx − HxS
+ fx2 HSS Hyy − HyS
+ fS (fS − ρ) Ω̃
˜
Ω̃|J|
+fy [2fx (HxS HyS −HSS Hxy ) + (HxS Hxy − Hxx HyS )(2fS − ρ)]
+fx (HxS Hyy − Hxy HyS ) ρ + 2fx fS (Hxy HyS − HxS Hyy )}
SS
∂µ
∂M
(98)
(99)
(100)
2
λss rAA
(fy HxS
− HSS Hxx + (HSS Hxy − HxS HyS ) p
˜
Ω̃|J|
+fS [Hxx HyS − HxS Hxy + (Hxy HyS + HxS Hyy ) p]
= −
+fx [HyS (HyS p − HxS ) + HSS (Hxy − Hyy p)]
i
h
+wS fy (HxS Hxy − Hxx HyS ) + fx (Hxy HyS − HxS Hyy ) + fS Ω̃
(101)
(102)
We now focus on the case in which the Jacobian admits two negative eigenvalues, which ensures
˜ is strictly positive. Under
saddle point stability to the steady state. When this is the case, |J|
this stability requirement the following obtains
2
λSS
∂xss
∂xss
rAA (fS + fy wS ) (fS + fy wS − ρ) − xss
=
∂p
∂M
Ω̃ J˜
(103)
in Proposition 1, where ∂xss /∂M has been defined in Section (A.2) in relation to the problem
without saving,
A.3.1
Frisch demand function
Optimality conditions (46)-(51) become
Hx = Ux (x, y, S) − λ0 p + µfx (x, y, S) = 0
(104)
Hy = Uy (x, y, S) − λ0 + µfy (x, y, S) = 0
(105)
µ̇ = ρµ − HS (x, y, S, A, µ, λ0 ; p, M )
(106)
Ȧ = rA + M + w(S) − px − y
(107)
Ṡ = f (x, y, S)
(108)
22
Proceding as in the previous section allows obtaining the steady state consumption of x:
xss = x∗ (S ss , Ass , µss , λ0 ; p, M )
(109)
where function x∗ yields the value of x that satisfies the first order conditions (104) and (105).
The change of steady state consumption of x after a price increase in p is computed as follows:
ss
∂xss
∂x∗ ∂x∗ ∂S
∂x∗ ∂Ass ∂x∗ ∂µss ∂x∗ ∂λ0
=
+
+
+
+
.
∂p
∂p
∂S ∂p
∂A ∂p
∂µ ∂p
∂λ ∂p
When a price change is compensated to keep the marginal utility of wealth constant, the last term
is zero. In steady state, the determinant of the Jacobian associated to (106)-(108) is
ˆ =ρfS (ρ − fS ) +
|J|
−
ρ
ρ 2
(fy HxS − fx HyS )2 −
fy Hxx − 2fx fy Hxy + fx2 Hyy HSS
Ω
Ω
ρ (2fS − ρ)
[(fy Hxy − fx Hyy ) HxS + (fx Hxy − fy Hxx ) HyS ]
Ω
(110)
where Ω, |J| and ∂xss /∂M have been defined in the Appendix in relation to the problem without
saving.
A.4
Labor supply
When the Mangasarian conditions hold, the following conditions are necessary and sufficient:
Hx = Ux (x, y, S) + µfx (x, y, S) + λwgx (x, S) = 0
(111)
Hy = Uy (x, y, S) + µfy (x, y, S) − λ = 0
(112)
λ̇ = λ (ρ − r)
(113)
µ̇ = ρµ − HS (x, y, S, A, µ, λ; w, M )
(114)
Ȧ = rA + M + wg(x, S) − y
(115)
Ṡ = f (x, y, S) .
(116)
with the transversality conditions limt→∞ e−ρt µ (t) S (t) = 0 and limt→∞ e−ρt λ (t) A (t) = 0.
References
Becker, G. S. and K. M. Murphy (1988). A theory of rational addiction. Journal of Political
Economy 96 (4), 675–700.
Grossman, M. (1972). On the concept of health capital and the demand for health. Journal of
Political Economy 80 (2), 223–55.
23
Heckman, J. (1974). Life cycle consumption and labor supply: An explanation of the relationship
between income and consumption over the life cycle. The American Economic Review , 188–194.
Mangasarian, O. L. (1966). Sufficient conditions for the optimal control of nonlinear systems.
SIAM Journal on Control 4 (1), 139–152.
Seierstad, A. and K. Sydsaeter (1977). Sufficient conditions in optimal control theory. International Economic Review , 367–391.
24