PRINCIPALES POLITICAS Y PRÁCTICAS CONTABLES

defined by us in this text). We ignore these ugly possibilities and use a definition that will do for all practical purposes:

D _{IV.29. Let f : M → R be a real-valued function with open} domain M ⊆ Rℓ_{. f is called differentiable if all the partial derivatives}

∂f ∂xi

(i = 1, ..., ℓ)

exist and are continuous. In that case, the column vector

f′(x) :=          f1(x) f2(x) fℓ(x)          is called f ’s derivative at x.

T _{IV.3 (adding rule). Let f :R}ℓ _{→ R be a differentiable function}

and let g1, ..., gℓ be differentiable functionsR → R. Let F : R → R be defined

by F (x) = f (g1(x) , ..., gℓ(x)) . Then we have dF dx = ℓ i=1 ∂f ∂gi dgi dx.

6.2. Economics: the marginal rate of substitution. Consider two goods 1 and 2 (if other goods are present, hold them constant). A bundle y = (y1, y2) defines an indifference curve

Iy = (x1, x2)∈ R2+: (x1, x2)∼ (y1, y2) .

We now consider an amount x1 of good 1 and look for the amount x2 of

good 2 such that (x1, x2) is contained in Iy (see fig. 21 for an illustration).

In this manner, we can sometimes define a function Iy : x1 → x2.

Note that we have used the symbol Iyin two different ways, as a subset of

the goods space and as a function. We do this to economize on symbols and also to make clear that the function Iy is closely related to the indifference

curve.

D _{IV.30 (marginal rate of substitution). If the function Iy is} differentiable and if preferences are monotonic, we call

MRS = dIy(x1) dx1

the marginal rate of substitution between good 1 and good 2 (or of good 2 for good 1).

(

)

We can readily interpret the marginal rate of substitution: if one ad- ditional unit of good 1 is consumed while good 2’s consumption is reduced by MRS units, the consumer stays indifferent. We could also say: MRS measures the willingness to pay for one additional unit of good 1 in terms of good 2.

R +_{IV.2. Note that the above definition “does not work” if one of} the goods is a bad. In that case, consuming more of good 1 (nice music) leaves the consumer indifferent if he endures more of good 2 (filthy smoky air). However, since we deal with goods (in the sense of monotonic preferences) most of the time, there is no harm in that definition. Of course, if we are interested in the slope of the indifference curve, we can simply calculate

dIy(x1)

dx1 . In order to avoid tedious repetitions, we will not always point to the

fact that we have monotonic preferences.

R + _{IV.3. Marginal this and marginal that is standard staple for} economists. It is a somewhat peculiar way of saying that we consider the derivative of a function. Apart from the marginal rate of substitution, we will encounter “marginal utility”, “marginal cost”, “marginal revenue” etc.

As an example, consider the utility function given by U (x1, x2) = ax1+

bx2, a > 0 and b > 0, i.e., the case of perfect substitutes. Along an indif-

ference curve, the utility is constant at some level k so that we focus on all good bundles (x1, x2) fulfilling ax1+ bx2 = k. We find the slope of that

indifference curve by

• solving for x2 — x2(x1) = k_b −a_bx1 — and

• forming the derivative with respect to x1 — dx_dx2₁ =−a_b.

6. MARGINAL RATE OF SUBSTITUTION 77

So far, we did not make use of a utility function (possibly) representing the preferences. If such a function is available, calculating the marginal rate of substitution is an easy exercise:

L _{IV.10. Let be a preference relation on R}ℓ

+ and let U be the

corresponding utility function. If U is differentiable, the marginal rate of substitution between good 1 and good 2 can be obtained by

MRS (x1) = dIy(x1) dx1 = ∂U ∂x1 ∂U ∂x2 .

Here, we make use of the partial derivatives of the utility function, _∂x∂U₁ and ∂U

∂x2, for goods 1 and 2, respectively. They are called ... marginal utility.

P . _{Along an indifference curve, the utility is constant, i.e., we have} constant = U (x1, Iy(x1)) .

By the adding rule IV.3, differentiating with respect to x1 yields

0 = ∂U ∂x1 + ∂U ∂x2 dIy(x1) dx1 .

Thus, we can find the slope of the function Iy even if Iy is not given

explicitly. This is an application of the so-called implicit-function theorem. Let us return to the case of perfect substitutes considered above. The marginal rate of substitution is found easily:

MRS (x1) = ∂(ax1+bx2) ∂x1 ∂(ax1+bx2) ∂x2 = a b We note without proof:

L _{IV.11. Let U be a differentiable utility function and Iy an in-} difference curve of U . This indifference curve is concave if and only if the marginal rate of substitution is a decreasing function in x1.

This lemma is depicted in fig. 22. In that figure, we have x1 < y1 and

MRS (x1) > MRS (y1).

For example, the MRS of Cobb-Douglas utility functions (which is given by U (x1, x2) = xa1x12−a, 0 < a < 1) is MRS = ∂U ∂x1 ∂U ∂x2 = ax a−1 1 x12−a (1− a) xa 1x−a2 = a 1− a x2 x1.

If we increase x1, we need to decrease x2 > 0 along any indifference curve

(Cobb-Douglas preferences are monotonic) — x2

x1 is therefore a decreasing

function of x1. Thus, the MRS decreases in x1 and Cobb-Douglas in-

difference curves are concave (Cobb-Douglas preferences convex or Cobb- Douglas utility functions quasi-concave). Similarly, the utility function U (x1, x2) = x1x2 with MRS = x_x2₁ is quasi-concave.

1 y 1 x 2 x 1 x

F 22. _{Concave indifference curve, increasing MRS}

7. Topics The main topics in this chapter are

• preference relation • indifference

• strict preference • better set • worse set

• indifference set, indifference curve • lexicographic preferences

• Cobb-Douglas preferences • perfect substitutes

• perfect complements • utility function • the vector space Rℓ

• the first quadrant of Rℓ_,

Rℓ +

• ≥, >, and ≫ for vectors

• the distance between points x and y in Rℓ_{, x − y}

• the Euclidian norm, x − y 2=

'

ℓ

g=1(xg− yg)2

• the infinity norm, x − y _∞= maxg=1,...,ℓ|xg− yg|

• ε-ball with center x∗

• bounded set • interior point • boundary point • open set • closed set • sequence • convergence

8. SOLUTIONS 79