Chp 3 - Classical demand theory Flashcards
What is monotone and strongly monotone?
Monotone
x=(3,2,1)
x’=(4,3,2)
Strongly monotone:
x=(1,2,3)
x’=(2,2,3)
CD is strict, Leontif is monotone.
Define the indifferent set, upper level set (contour set), lower level set.
Indifference set:
x in X = {y: y~x}
Upper level set
x in X = {y: y≥x}
Lower level set
x in X = {y: y≤x}
Det ska vara preferens linjer
What feature will monotonicity give our indifference curves?
They will be downward sloping.
Which assumptions are necessary for preferences?
- Rationality
- Local non-satiation
- Convexity
What is convexity in preferences and what feature will it yield?
A preference relation is convex on x if the upper level contour set in convex.
This makes the consumer prefer a diverse bundle of goods (de vill alltid ha en blandning av varor). No corner solutions tror jag?
Which preferences can not be represented by utility functions?
Lexo-graphical preferences.
To avoid this, preferences needs to be continuous.
What assumption will yield that Walras law is satisfied så we have xp=w?
Local non-satiation or monotonic preferences.
Note, we will inly have this when we have utility functions where the exponent parameters adds to 1. Other wise it could be that we derive negative utility from one good etc….
Suppose that u(.) is a continuous function representing a locally non satiated preference relationship. Then the Walrasian demand correspondence x(p,w) possesses the following properties
H.M of degree zero in (p,w): x(ap, aw) = x(p,w) for any p,w and scalar a>0.
Walras law holds. That is, px=w with equality.
Convexity/uniqueness. If the preference relationship is convex then u(.) is quasiconcave, then x(p,w) is a convex set. If the preference relation ship is strictly convex, then u(.) is strictly convex and x(p,w) consists of a single element, i.e., we have a single solution.
What are the features of the Slutsky demand vs the Hicksian demand?
Slutsky Demand: The previous bundle needs to be affordable
Hicksian demand: Previous utility needs to be affordable.
When can we say that X_1 ≥ 0 and X_2 ≥ 0 holds with equality?
When we have a function where the terms put together, like in the CD case.
X_1 X_2.
Because if one of then then is zero, then the whole expression is zero and we have a solution at origio.
Write the budget set for L goods and the budget set for two goods
L goods:
B_{p,w} = {x ⊂ R^L_+: px ≤ w}
Two goods
B_{p,w} = {x ⊂ R^2_+: p_1 x_1 + p_2 + x_2 ≤ w}
Derive the Slutsky Equation
See notes or notation.
Prove that the expenditure function is homogeneous of degree one in prices
See notion
e(ap,u^) = min_x apx s.t.u(x) ≥ u^
take out a
ae(p,u^) = a min_x px s.t.u(x) ≥ u^
Boken skriver
e(ap,u) = apx* = ae(p,u)
Show that the expenditure function is non decreasing in p.
Lets say we have
p < p’
multiplying with h(p,u)
ph(p’,u) < p’h(p’,u)
Where RHS is the expenditure function when price is p’, we thus have:
ph(p’,u) < e(p’u)
Then one can claim that when p < p’
ph(p,u) ≤ ph(p’,u) < e(p’u)
Or using ph(p,u) = using e(p,u)
e(p,u) ≤ ph(p’,u) < e(p’u)
Thus the expenditure function is non decreasing in prices.
u should be \bar u…. See notion for neater expressions.
Show that the expenditure function is concave in price.
Show a diagram with e(p,u) on the y-axis and p_i on the x-axis. Show how his total expenditures increases linearly with p_i as a 45 degree line. Then show a second concave line that lies under the 45 degree line representing what happens i the consumer minimizes. See notes.
Also see the formal proof in notion where one uses p’’ as a linear combination of p and p’’.
e(p’‘,u) = aph(p,u) + (1-a)p’h(p’,u)
etc until
e(ap + (1-a)p’,u) ≥ ae(p,u) + (1-a)e(p’,u).
