dx1
If the household consumes one additional unit of good 1,
how many units of good 2 movement on the does he have to forgo so as to remain budget line within his budget.
F 8. Willingness to pay and opportunity cost inequality
MRS = dx2
dx1
absolute value of the slope of the indifference curve
> dx2
dx1
absolute value of the slope of the budget line
= MOC.
If, now, the household increases his consumption of good 1 by one unit, he can decrease his consumption of good 2 by MRS units and still stay on the same indifference curve. Compare fig. 9. However, the increase of good 1 necessitates a decrease of only MOC < MRS units of good 2. Therefore, the household needs to give up less than he would be prepared to. In case of strict monotonicity, increasing the consumption of good 1 leads to a higher indifference curve.
Thus, we cannot have MRS > MOC at the optimal bundle unless it is impossible to further increase the consumption of good 1. This is the situation depicted in fig. 10.
Thus, if the household consumes both goods in positive quantities, we can derive the optimality condition
MRS = MOC!
x2
1 unit of good 1 indifference
curve budget line
MRS MOC
x1
F 9. Not optimal
indifference curves budget line
household
optimum x1
x2
F 10. The willingness to pay can be higher than the cost.
(if both terms are defined).
Alternatively, we can derive this first-order condition with the help of a utility function (if we have one). The household tries to maximize
U x1,m p2 −p1
p2x1 .
If the household increases the consumption of good 1 by one unit, we have two effects. First, his utility increases by ∂x∂U1. Second, an increase in x1leads to a reduction in x2 by MOC = dxdx21 = pp12 and this reduced consumption of good 2 decreases utility (chain rule). Therefore, the household increases
2. THE HOUSEHOLD OPTIMUM 129
x1 as long as
∂U
∂x1
marginal benefit of increasingx1
> ∂U
∂x2
dx2
dx1
marginal cost of increasingx1
holds. Dividing by ∂x∂U2, an increase in x1 leads to an increase in utility if MRS =
The MRS- versus-MOC rule can help to derive the household optimum in some cases:
and, together with the budget line, the household optimum x1(m, p) = am
p1, x2(m, p) = (1− a)m
p2.
• Goods 1 and 2 are perfect substitutes if the utility function is given by U (x1, x2) = ax1+ bx2 with a > 0 and b > 0. An increase of good 1 enhances utility if
a
b = MRS > MOC = p1
p2
holds so that we find the household optimum
x (m, p) = We have the marginal rate of substitution
M RS =
holds for sufficiently large x1 which calls for an increase of x1. In-versely,
x1
x2 = MRS < MOC = p1
p2
holds for sufficiently large x2 so that an increase of x2 seems a good idea. Therefore, we need to compare the extreme bundles mp1, 0 and 0,mp2 and obtain
2.3. Household optimum and monotonicity. We now turn to spe-cific implications that can be drawn from the fact that some x∗is a household optimum and that some sort of monotonicity holds.
L VI.4. Let x∗ be a household optimum of the decision situation
∆ = (B (p, m) , ). Then, we have the following implications:
• Walras’ law: Local nonsatiation implies p · x∗= m.
• Strict monotonicity implies p >> 0.
• Local nonsatiation and weak monotonicity imply p ≥ 0.
P . We use proofs by contradiction for each statement:
• Because of x∗ ∈ B, we can exclude p · x∗ > m. Assume p· x∗< m.
Then, the household can afford bundles sufficiently close to x∗. By local nonsatiation, within the set of those affordable bundles, a bundle y exists that the household strictly prefers to x∗ (see fig.
11). This is a contradiction to x∗ being a household optimum.
• Turning to the second implication, assume a household optimum and a price pg which is zero or negative. Then, the household can afford more of good g. By strict monotonicity, the household is better off implying the desired contradiction (existence of household optimum).
• Assume a negative price for some good g. By weak monotonic-ity the household can “buy” additional units of that good without being worse off. Since the price is negative, the household has ad-ditional funding for preferred bundles which exist by nonsatiation.
Again, a contradiction to the existence of a household optimum follows.
3. COMPARATIVE STATICS AND VOCABULARY 131
p1
m p2
m x2
x1
x*
y
F 11. Proving Walras’ law
3. Comparative statics and vocabulary
3.1. Vocabulary. In household theory, we carefully distinguish para-meters and variables. If we focus on the price pg of good xg, we treat the other prices and the income or endowment as parameters. That means, they are fix for the time being. If we plot xRg (pg, ...) as a function of pg, we obtain a demand curve. A change in pg results in a movement along the demand curve while a parameter change shifts the whole demand curve:
D VI.5. In household theory, we omit the R and often write x (p, m) or x (p, ω) instead of xR(p, m, ) or the like. Holding some of the parameters constant, we distinguish
• the (Marshallian) demand function for good g, denoted by xg(pg),
• the Engel function for good g, denoted by xg(m), and
• the cross demand function for good g with respect to the price pk of some other good k = g, denoted by xg(pk) .
In case of an endowment budget, the household is called a net supplier of good g in case of xg(p, ω) < ωgand a net demander in case of xg(p, ω) > ωg. Once we know the three functions defined in the above definition, we can form the derivatives:
D VI.6. We call a good g
• ordinary if
∂xg
∂pg ≤ 0
holds and non-ordinary otherwise (demand function),
• normal if
∂xg
∂m ≥ 0
holds and inferior otherwise (Engel function),
• a substitute of good k if
∂xg
∂pk ≥ 0 holds and
• a complement of good k if
∂xg
∂pk ≤ 0 holds.
E VI.7. Consider the demand functions x1 = apm1 and x2 = (1− a)pm2, 0 < a < 1, in case of a money budget (arising from a Cobb-Douglas utility function) and find out
• Is good 1 an ordinary good?
• Is good 1 normal?
• Is good 1 a substitute or a complement of good 2?
3.2. Price-consumption curve and demand curve. The demand function defined in the previous section is sometimes qualified as “Marshal-lian” in order to differentiate between Marshallian demand and Hicksian demand which is a central topic in the next chapter. In this section, we show how to derive demand curves.
Assume a money budget for two goods 1 and 2. For fixed values p2 and m, we vary the price p1 of good 1 in x1-x2-space (prices pB1, pC1 and pD1 with pB1 > pC1 > pD1) and obtain the price-consumption curve which is the geometric locus of household optima (see the upper part of fig. 12). We then associate all the different prices of good 1 with the demand for that good. The graph obtained (the lower part of our figure) is called the demand curve where — normally — the ordinate is the price axis.
E VI.8. Assuming that good 1 and good 2 are complements, sketch a price-consumption curve and the associated demand curve for good 1.
Assuming the utility function U (x1, x2) = x113 · x223, we now calculate the price-consumption curve and the demand curve for good 1. You remember that the household optimum (x∗1, x∗2) is given by
x∗1 = 1 3
m
p1, x∗2 = 2 3
m p2.
Thus, the demand curve for good 1 is x∗1= f (p1) = 13mp1.
In general, the price-consumption curve for variing p1 is determined by the following procedure:
3. COMPARATIVE STATICS AND VOCABULARY 133
F 12. The price-consumption curve and the demand curve
• We associate each p1with the household optimum (x∗1(p1) , x∗2(p1)).
• We look for the geometric locus of these optima and express it as a function x2 = h (x1) . It is important that h is not a function of p1.
In our example, x∗2 = h (x1) = 23pm2 is already the price-consumption curve
— the price of good 1 affects the demand of good 1, but not of good 2.
Therefore, the price-consumption curve is a horizontal line.
E VI.9. Can you also find the demand function for good 2?
Be careful and check for zero and negative prices; you can use the case distinction given by
Determine the price-consumption curve for the the case of perfect comple-ments, U (x1, x2) = min (x1, 2x2)!
Sometimes, demand curves hit the axes (see fig. 13):
xg
pg
sat
xg proh
pg
demand curve
F 13. Saturation quantity and prohibitive price
D VI.7. Let xg(pg) be the quantity demanded for any price pg≥ 0. We call
xsatg := xg(0) the saturation quantity and
pproh1 := min{pg ≥ 0 : xg(pg) = 0} the prohibitive price.
3.3. Income-consumption curve and Engel curve. Fig. 14 shows how to derive the Engel curve from the income-consumption curve. The latter one connects the household optima in our x1-x2 diagram.
E VI.10. Assuming that good 1 and good 2 are complements, sketch an income-consumption curve and the associated Engel curve for good 1!
Assuming the same Cobb-Douglas utility function as in the previous section, we determine the income-consumption curve and the Engel curve.
Algebraically, the Engel curve can be obtained from the household optimum and is given by
x∗1 = q (m) = 1 3
m p1.
In order to express the income-consumption curve algebraically, we have to write x2as a function of x1, but not of income m (which takes on all values).
We solve good 1’s demand for m and obtain m = 3p1x∗1. Subsituting in x∗2 yields
3. COMPARATIVE STATICS AND VOCABULARY 135
x1
x2
C B
D mB
mC
mD
x1
m
mB
mC
mD
income-consumption curve
x1B x1C x1D x1B
x1C
xD1
Engel curve
F 14. Deriving the Engel curve
x∗2 = 2 3
m p2
= 2
3 3p1x∗1
p2
= 2p1
p2x∗1
and hence the income-consumption curve x∗2= g (x∗1) = 2pp12x∗1.
E VI.11. Determine the income-consumption curve for the the case of perfect complements, U (x1, x2) = min (x1, 2x2)! Can you also find the Engel-curve function for good 2?
3.4. Defining substitutes and complements. The definitions of sub-stitutes and complements seem innocuous. However, they are highly prob-lematic as you will realize when you solve the following exercise:
E VI.12. Determine ∂x1∂p(p,m)2 and ∂x2∂p(p,m)1 for the quasi-linear utility function given by
U (x1, x2) = ln x1+ x2 (x1> 0)!
Assume positive prices and pm2 > 1, in order to avoid a corner solution!
Thus, good g can be the substitute of good k while k is not a (strict) substitute of g. We will see how to avoid this problem in the next chapter.
3.5. Price elasticities of demand. An important characteristic of demand is dxdpgg, i.e., the question how demand changes if the price of a good changes. However, this slope of the demand curve depends on the units of measurement — do we have Euros or dollars? This problem can be avoided by using relative quantities. By how many percent does demand change if price is changed by one percent?
D VI.8. Let xg(pg) be the demand at price pg(and other prices which are hold constant). The price elasticity (of demand) is denoted by εxg,pg and given by
E VI.13. Calculate the price elasticities of demand for the de-mand function (individual or aggregate) given by
xg(pg) = 100− pg and xk(pk) = 1 pk
.
If we know that we are dealing with ordinary goods, we can consider the absolute value of the price elasticity. Then |εx,p| < 1 and εx,p > −1 are equivalent. The price elasticity can help to assess the effect of a price change on expenditure, i.e., we are interested in
d (px (p)) dp .
If the absolute value of the price elasticity is smaller than 1, the expenditure increases if the price increases. You can see this from
d (px (p))
This result can be used as an argument for a liberal drug policy. It is plau-sible that demand for drugs is inelastic, |εx,p| < 1. Assume the government increases the price of drugs by taxing them or by criminalizing selling or buy-ing. Then the expenditure of drug users increases and so does drug-related crime (stealing money in order to finance the addiction).
3. COMPARATIVE STATICS AND VOCABULARY 137
inferior goods
necessity goods
normal goods
luxury goods
m x ,1
0 1 ε
F 15. Inferior and normal, luxury and necessity goods 3.6. Income elasticity of demand. In a similar fashion, we can de-fine the income elasticity. Income increases by one percent. By how many percent does demand increase?
D VI.9. Let xg(m) be the demand at income m. The income elasticity (of demand) is denoted by εxg,m and given by
εxg,m:=
dxg
xg
dm m
= dxg
dm m xg.
If a good is normal, its income elasticity is positive. We can subdivide normal goods in
• luxury goods such as Kaviar: your consumption increases stronger than your income and
• necessity goods such as oat groates: your consumption increases weaker than your income.
D VI.10. We call a good g
• a luxury good if
εxg,m≥ 1 holds
• a necessity good if
0≤ εxg,m≤ 1 holds.
E VI.14. Calculate the income elasticity of demand for the Cobb-Douglas utility function U (x1, x2) = x113 · x223! How do you classify (demand for) good 1?
In a sense, an income elasticity of 1 is very normal.
L VI.5. Assume local nonsatiation and the household optimum x∗. Then the average income elasticity is 1:
ℓ
g=1
sgεxg,m= 1
( )p
F 16. Aggregation of individual demand curves where the weights are the relative expenditures, sg := pgmxg.
According to Walras’ law (lemma VI.4, p. 130), the household chooses x∗ on the budget line, p· x∗(m) = m. Of course, if there is only one good,
To show the lemma in the general case, we form the derivative of the budget equation m = ℓ
3.7. Aggregation of individual demand curves. Household theory shows how to derive individual demand curves. We now aggregate several individual demand curves in order to arrive at an aggregate demand curve.
D VI.11. Let xi(p) be the demand functions of individuals i = 1, ..., n. Aggregate demand is then given by
x (p) :=
n
i=1
xi(p) .
Consider fig. 16 to see how "horizontal aggregation" works. For every price, the quantities demanded by households A and B are added.
4. SOLUTION THEORY 139
xg
pg
sat
xg
( )
g g p x ˆproh
pg
pˆg
demand curve inverse
demand curve
xˆg
( )
g g x p ˆF 17. Demand curve and inverse demand curve
E VI.15. Consider the individual demand functions for good g given by
x1g(pg) = max (0, 100− pg) , x2g(pg) = max (0, 50− 2pg) and x3g(pg) = max (0, 60− 3pg) .
Find the aggregate demand function. Hint: Find the prohibitive prices first!
Individual and aggregate demand functions can sometimes be inversed:
Have a look at fig. 17. It can be considered as a graphical representation of the demand curve. At price ˆpg (ordinate) we obtain the quantity demanded xg(ˆpg) . Inversely, for a given quantity ˆxg of good g, we can ask for the price pg(ˆxg) that is just sufficient to yield the quantity ˆxg. The resulting function is called the inverse demand function:
D VI.12. Let x1 : 0, pproh1 → 0, xsat1 be an injective (indi-vidual or market) demand function. The inverse of this function is called the inverse (individual or market) demand function and given by
p1 = x−11 : 0, xsat1 → 0, pproh1
x1 → p1(x1) where p1(x1) is the unique price resulting in x1.
4. Solution theory
4.1. The general setup of an optimization problem. We now want to take a broader, more general view of the problem facing a house-hold. (We present solution theory in this section and in a later chapter on pp. 228.) Usually, an optimization problem consists of a feasible set and a
preference relation on that set. If the preference relation is expressed by a function, this function is often called an objective function. For example, a firm’s profit function or a household’s utility functions are examples of objective functions.
Feasible sets can be strategy sets as in chapter II or budgets as in the present chapter. In chapter VIII, we introduce production sets which de-scribe the production possibilities open to a firm.
Summarizing these examples,
• in decision theory (and similarly in game theory), we have
— feasible set = strategy set and
— preference relation ω on the set of strategies S for a belief ω on W defined by s ωs′ :⇔ u (s, ω) ≥ u (s′, ω)
• in household theory, we find
— feasible set = budget and
— preference relation on the set of good bundles, possibly defin-able by a utility function
• in the theory of the firm, we encounter
— feasible set = production set and
— preference relation given by a profit function
D VI.13. Let f :Rℓ→ R be an objective function and M ⊆ Rℓ the feasible set. (f, M) is called an optimization problem if it is a maximiza-tion or a minimizamaximiza-tion problem, i.e., if we look for a maximal or a minimal element in f (M), respectively. x∗ ∈ M is called a solution to the
• maximization problem if f (x∗)≥ f (x) for all x ∈ M holds,
• minimization problem if f (x∗)≤ f (x) for all x ∈ M holds.
x∗ ∈ M is called a solution if it is a solution to the maximization or the minimization problem.
Mathematically, the above definition of the optimization problem (f, M) is equivalent to looking for a maximum or minimum of the function f|M.
Solution theory deals with two questions:
• Existence: Can we be sure that a solution to the optimization problem exists?
• Uniqueness: Are there several solution or is there just one solution?
In the best of all worlds, we have exactly one solution.
4.2. Existence. Not every maximization (or minimization) problem needs to have a solution. The two most relevant examples can be shown with the help of fig. 18. In both subfigures, the value of the objective function can be increased by increasing the variable x. In (a), the domain is R+ (and hence not bounded) and for any x from the domain, x + 1 yields a higher value. In (b), for every x ∈ [0, 1) (which is not closed), we have 1 > 1+x2 > x≥ 0. That is, we cannot find a greatest number smaller than 1.
4. SOLUTION THEORY 141
x x
domain: the real line
(a) (b)
( )x
f f( )x
excluded domain: the interval[ )0,1
1
F 18. No solutions
In household theory, these problems do not need to bother us a lot. The reason lies in the following theorem and its corollary:
T VI.1. Let f : M → R be a continuous function where M ⊆ Rℓ is nonempty, closed and bounded. Then, f adopts a maximum and a minimum on M.
We only hint at a proof. By the continuity of f, the image f (M) is also closed and bounded. Every closed and bounded subset of R takes on a maximum and a minimum.
C $$ VI.1. Let be a continuous preference relation and let p∈ Rℓ+ obey p >> 0. Then, the household’s decision problem has a solution, i.e., x (p, m) or x (p, ω) is nonempty for every m≥ 0 and ω ∈ Rℓ+.
The corollary follows from the above theorem together with theorem IV.1 (p. 69) and lemma VI.2 (p. 120).
4.3. Uniqueness. Optimization problems can have several solutions as you have seen in fig. 7 (c), p. 126. Subfigure (a) hints at the importance of convexity:
T VI.2. Let f : M → R be a strictly quasi-concave function on a convex domain M ⊆ Rℓ. Then, we cannot have two different solutions to a maximization problem.
Let f : M → R be a quasi-concave function that obeys strict monotonicity or local nonsatiation (see exercise IV.17, 69) and let M be a strictly con-vex domain M ⊆ Rℓ. Then, we cannot have two different solutions to a maximization problem.
P . We provide a proof (by contradiction) of the first assertion. Let x, y∈ M be two solutions with x = y. By the convexity of M, we have
kx + (1− k) y ∈ M.
Strict quasi-concavity implies
f (kx + (1− k) y) > min (f (x) , f (y)) = f (x) = f (y) .
Thus, we have found an element from our feasible set M that yields a higher value. Therefore, neither x nor y can be solutions which is the desired contradiction.
C $$ VI.2. Let U be a strictly quasi-concave utility function.
Then, the household’s decision problem has at most one solution.
Returning to fig. 7 (c) on p. 126, note that
• the utility function is quasi-concave but not strictly quasi-concave and
• the budget is convex but not strictly convex.
Therefore, this example does fit neither the first nor the second part of the theorem.
The corollary is derived from the first part of the theorem. With respect to the second part, just reconsider fig. 7 (d) and exchange the indifference and budget curves.
Uniqueness in the above theorem is possible from strict quasi-concav-ity of the objective function or strict convexquasi-concav-ity of the domain. If we have neither, we cannot hope for uniqueness, but we can still ensure that the solutions form a convex set:
T VI.3. Let f : M → R be a quasi-concave function on a convex domain M ⊆ Rℓ. Then the set of solutions is convex.
Therefore, if you have two solutions, every convex combination of these solutions is also a solution. A good example is provided by perfect sub-stitutes where x (p, m) is an interval if the budget line’s slope happens to coincide with the indifference curves’ slopes.
4.4. Local solutions and global solutions.
D VI.14. A solution x∗ ∈ M is also called a global solution. A solution x∗ ∈ M is called a local solution if we have an ε-ball K with center x∗ such that x∗ is a solution to (f, M∩ K) .
Of course, every global solution is a local one (consider fig. 19).
The inverse is more interesting:
T VI.4. Let f : M → R be a strictly quasi-concave function on a convex domain M ⊆ Rℓ. Then, a local maximum of f on M is a global one.
P . Assume a local maximum at x∗ ∈ M and hence an ε-ball K with center x∗ such that f (x∗)≥ f (x) for all x ∈ M ∩ K (see fig. 20).
Assume further that x∗ is no global solution. Then we have an x′ ∈ M obeying f (x∗) < f (x′) . By the convexity of M, we have
kx∗+ (1− k) x′ ∈ M
4. SOLUTION THEORY 143
x
F 19. Every global solution is a local one.
( 1 ) '
*
k x
kx + −
x
*x '
F 20. A local solution is global.
for every k∈ (0, 1). Strict quasi-concavity yields f kx∗+ (1− k) x′ > f (x∗) .
Now, we can have k so large that kx∗ + (1− k) x′ is included in the ε-ball K with center x∗. This is the desired contradiction to the local maximum at x∗.
C $$ VI.3. Let U be a strictly quasi-concave utility function.
Then, a local solution to the household’s decision problem is already a global one.
4.5. Interior solutions, boundary solutions, and corner solu-tions. Assume an optimization problem (f, M) . We distinguish between different kinds of solutions:
D VI.15. A solution x∗ ∈ M is called an interior solution if it
D VI.15. A solution x∗ ∈ M is called an interior solution if it