106920 REPARACION ESCUELA MARCO AURELIO SOTO
F082031 TECHO CANALETA 4", LAMINA ALUMINIZADA (EEB/M) M2
6.6.1
Definition of the expenditure function
The consumer’s expenditure function minimisation problem involves finding a minimum rather than a maximum value function. This minimum value function is called the expenditure function. It is the minimum expenditure needed to obtain utility ¯u or more at prices p1 and p2. The compensated demand functions h1(p1, p2, ¯u) and h2(p1, p2, ¯u)
are the solutions to the problem of minimising expenditure p1x1+ p2x2 subject to the
constraint u(x1, x2) ≥ ¯u. Thus the expenditure function e(p1, p2, ¯u) is the value of the
objective p1x1+ p2x2 when x1 = h1(p1, p2, ¯u) and x2 = h2(p1, p2, ¯u), so:
e(p1, p2, ¯u) = p1h1(p1, p2, ¯u) + p2h2(p1, p2, ¯u).
Because compensated demand is a function of (p1, p2, ¯u) the expenditure function is also
a function of p1, p2 and ¯u.
6.6.2
Properties of the expenditure function
Monotonicity in utility
It follows from the general result that a maximum value function is non-decreasing in the level of the constraint k. Here the level of the constraint is ¯u. If (x1, x2) solves the
consumer’s expenditure minimisation problem for ¯u and (x01, x02) solves the problem for ¯
u0 > ¯u, then (x01, x02) satisfies the constraint u(x10, x02 ≥ ¯u). Thus (x01, x02) satisfies the constraint for the problem of minimising the cost of getting utility ¯u which implies that e(p1, p2, ¯u) ≤ p − 1x01+ p − 2x
0
2 = e(p1, p2, ¯u0).
The stronger result that the expenditure function is increasing in ¯u, so if ¯u < ¯u0 then e(p1, p2u) < e(p¯ 1, p2, ¯u) can be established provided compensated demand (x01, x
0 2) at
utility ¯u0 has one or both of x01 > 0 or x02 > 0 and utility is a continuous function of (x1, x2). In this case u(x01, x02) ≥ ¯u0 > ¯u, so starting with consumption (x01, x02) it is
possible to reduce expenditure whilst still having utility above ¯u. This implies that the cheapest way of getting utility ¯u must be less than the cheapest way of getting utility ¯
u0, so e(p1, p2, ¯u)<e(p1, p2, ¯u).
Monotonicity in prices
Intuitively, when the price of a good increases it costs more to get the same level of utility. However a consumer who does not buy the good can maintain the same level of
6.6. The expenditure function
utility at the same expense so the result is that the expenditure function is non-decreasing in prices.
To see why this is so think about a fall in the price of good 1 from p1 to p01 whilst the
price of good 2 does not change. The expenditure function is the minimum cost of getting utility ¯u at prices (p1, p2). At prices (p1, p2) this is done by buying h1(p1, p2, ¯u)
and h2(p1, p2, ¯u).
When prices change to (p01, p2) it is still possible to obtain utility ¯u by buying
h1(p1, p2, ¯u) and h2(p1, p2, ¯u), at a cost of:
p01h1(p1, p2, ¯u) + p2h2(p1, p2, ¯u)
which implies that e(p01, p2, ¯u) the minimum cost of getting utility ¯u at these prices
cannot be more than p01h1(p1, p2, ¯u) + p2h2(p1, p2, ¯u) so:
e(p01, p2, ¯u) ≤ p01h1(p1, p2, ¯u) + p2h2(p1, p2, ¯u).
As we require that h1(p1, p2, ¯u) ≥ 0 and p01 < p1 it follows that
p01h1(p1, p2, ¯u) ≤ p1h1(p1, p2, ¯u) so:
p01h1(p1, p2, ¯u) + p2h2(p1, p2, ¯u)
≤ p1h1(p1, p2, ¯u) + p2h2(p1, p2, ¯u) = e(p1, p2, ¯u).
Putting these inequalities together gives:
e(p01, p2, ¯u) ≤ e(p1, p2, ¯u)
so when the price of good 1 falls the expenditure function cannot rise, or put another way the expenditure function is non-decreasing in p1. A similar argument establishes
that the expenditure function is non-decreasing in p2.
Homogeneity
We have already seen that the compensated demand function is homogeneous of degree zero in prices meaning that if s > 0:
h1(sp1, sp2, ¯u) = h1(p1, p2, ¯u)
h2(sp1, sp2, ¯u) = h2(p1, p2, ¯u).
This implies that the expenditure function is homogeneous of degree one in prices, meaning that if s > 0, e(sp1, sp2, ¯u) = se(p1, p2, ¯u). This means that if all prices are
multiplied by s, for example s = 2, so prices double then the cost of maintaining a particular level of utility also doubles. To see why this is, note that as compensated demand is homogeneous of degree zero in prices:
e(sp1, sp2, ¯u) = sp1h1(sp1, sp2, ¯u) + sp2h2(sp1, sp2, ¯u)
= sp1h1(p1, p2, ¯u) + sp2h2(p1, p2, ¯u)
= s[p1h1(p1, p2, ¯u) + p2h2(p1, p2, ¯u)]
= se(p1, p2, ¯u)
Figure 6.3: Concavity of the expenditure function.
Concavity
The expenditure function is concave in prices. This means that for any (p01, p02) and (p001, p002) if:
(p∗1, p∗2) = t(p01, p02) + (1 − t)(p001, p002) where 0 ≤ t ≤ 1, then:
te(p01, p02, ¯u) + (1 − t)e(p001, p002, ¯u) ≤ e(p∗1, p∗2, ¯u).
Theorem 20
This is illustrated in Figure 6.3 which shows the expenditure function as a function of p1 with p2 being held constant at p∗2.
Proof. Suppose that (x01, x02), (x001, x002) and (x∗1, x∗2) are the cheapest ways of getting utility ¯u at prices (p01, p02), (p001, p002) and (p∗1, p∗2). Thus:
p∗1x∗1+ p∗2x∗2 = e(p∗1, p∗2, ¯u).
As (x01, x02) is the cheapest way of getting utility ¯u at prices (p01, p02) and (x∗1, x∗2) is a way of getting utility ¯u:
p01x01+ p02x02 = e(p01, p02, ¯u) ≤ p01x∗1+ p02x∗2.
Similarly:
p001x001 + p002x002 = e(p001, p002, ¯u) ≤ p001x∗1+ p002x∗2.
6.6. The expenditure function
and 1 − t cannot be negative because 0 ≤ t ≤ 1, and then adding gives:
te(p01, p02, ¯u) + (1 − t)e(p100, p002, ¯uj) ≤ t(p01x1∗+ p02x∗2) + (1 − t)(p001x∗1+ p002x∗2) = (tp01+ (1 − t)p001)x∗1+ (tp02+ (1 − t)p002)x∗2 = p∗1x∗1+ p∗2x∗2 = e(p∗1, p∗2, ¯u) because (p∗1, p∗2) = t(p01, p02) + (1 − t)(p001, p002) = (tp01+ (1 − t)p001, tp02+ (1 − t)p002). Shephard’s Lemma
Another important property of the expenditure function, called Shephard’s Lemma, is that if the expenditure function has partial derivatives with respect to prices they are equal to compensated demand, that is:
∂e(p1, p2, ¯u)
∂p1
= h1(p1, p2u).¯ (6.1)
Proof. Suppose that the price of good 2 is fixed at p∗2 and that the price of good 1 changes from p∗1 to p1. Suppose that (x1, x2) is the cheapest way of getting utility ¯u at
prices (p1, p∗2). Because (x∗1, x∗2) is the cheapest way of getting utility ¯u at prices (p∗1, p∗2)
it is also a way of getting utility ¯u at prices (p1, p∗2) so:
p1x1+ p∗2x2 = e(p1, p∗2, ¯u) ≤ p1x∗1+ p ∗ 2x ∗ 2 = p∗1x∗1+ p∗2x∗2+ (p1− p∗1)x ∗ 1 = e(p∗1, p∗2, ¯u) + (p1− p∗1)x ∗ 1. Thus: e(p1, p∗2, ¯u) − e(p ∗ 1, p ∗ 2, ¯u) ≤ (p1− p∗1)x ∗ 1.
If p1 > p∗1 this implies that:
e(p1, p∗2, ¯u) − e(p − 1∗, p∗2, ¯u)
p1 − p∗1
≤ x∗1.
If the expenditure function has a derivative with respect to p1 at the point (p∗1, p∗2, ¯u) the
left-hand sides of these inequalities tend to the derivative as p1 tends to p∗1. The only
way this can happen is if the derivative is x∗1.
Downward sloping compensated demand again
We have already seen in Section 6.1.3 of this chapter that the compensated demand curve cannot gradient upwards, that is compensated demand is a non-increasing function of price. We established this with a simple direct argument. There is another way of showing the same thing with the result we have just got. As:
∂e(p1, p2, ¯u)
∂p1
differentiating again with respect to p1 gives: ∂2e(p 1, p2, ¯u) ∂p2 1 = ∂h1(p1, p2, ¯u) ∂p1 .
As the expenditure function is concave in prices:
∂2e(p1, p2, ¯u)
∂p2 1
≤ 0 so from the above equation:
∂h1(p1, p2, ¯u)
∂p1
≤ 0
that is compensated demand is a non-increasing function of price.
Activity 6.3 The expenditure function with a Cobb-Douglas utility function
Find the expenditure function for the Cobb-Douglas utility function u(x1, x2) = xa1xb2
where a > 0, b > 0 and a + b = 1. Make sure that the function you write down depends only on the parameters of the utility function a and b, the level of utility ¯u and prices p1 and p2. Confirm that the expenditure function has the properties listed
in the previous section.
Activity 6.4 The expenditure function with a linear utility function Find the expenditure function for the linear utility function u(x1, x2) = 2x1+ x2.
Make sure that the function you write down depends only on the level of utility ¯u and prices p1 and p2. Confirm that the expenditure function has the properties listed
in the previous section.