Definitions Flashcards
Completeness
For all goods bundles, you can decide if you prefer one or the other, or if you’re indifferent
Transitivity
If you prefer one goods bundle over another, and that other over a third one, then you must prefer the one goods bundle over the third one
Representation Theorem
If a preference relation is rational and continuous, then there exists a continuous utility function representing it
When is a preference relation continuous?
If the upper and lower contour sets for all goods bundles in this preference relation are closed
When is a preference relation preserved under limits?
For a sequence of a pair of bundles, where every element of the sequence of one bundle is preferred to the respective element of the sequence of the other bundle, the “limit bundle” of the preferred sequence is preferred to the “limit bundle” of the other sequence.
Representation Theorem with Monotonicity
If a preference relation is rational, continuous, and monotone, then there exists a continuous, strictly increasing utility function representing it
When is a preference relation monotone?
When more is always better, meaning that when there are more of both goods in one bundle compared to another, then the bundle with more goods is strictly preferred
When is a preference relation convex?
When the upper contour set is convex. Which is the case if when both y and z are preferred to x, this implies that a linear combination of y and z is also preferred to x.
WARP
If bundle x is ever chosen when bundle y is available, then there can be no budget sets containing both bundles where bundle y is chosen and bundle x is not.
When is a bundle revealed preferred to another?
When one bundle is chosen over the other when both are feasible.
When does a rational preference relation rationalize a choice rule?
When the optimal choices generated by it coincide with the choices produced by the choice rule for all budget sets
When does there exist a rational preference relation that is consistent with a choice structure satisfying WARP?
When lit B includes all subsets of X of up to three elements
When is a demand function homogenous (of degree zero)?
When the consumption choice does not change if both prices and wealth change, i.e. the consumer’s consumption choice depends only on the set of feasible points
What does Walras’ law say?
That a consumer fully expends his wealth
When is a good normal / inferior?
If the demand for it increases / decreases with wealth
When is a good a Giffen good?
When demand increases / decreases with an increase / decrease in the price
When does the Walrasian demand function satisfy WARP?
If when for one combination of prices and wealth, the consumer chooses one bundle over another that was also affordable, then when she chooses the other bundle, this must mean that the preferred bundle was not available
What is the Slutsky wealth compensation?
To isolate the effect of the change in relative cost of different commodities when prices changes from the effect of the wealth change, we adjust the consumer’s wealth such that her initial consumption bundle is just affordable at the new prices.
What is the s_l,k-th element of the Slutsky matrix?
derivative of the l-th demand function with respect to the k-th price + derivative of the l-th demand function with respect to wealth * k-th demand function
When is the Slutsky matrix negative semi-definite?
When v^TSv <= 0
When is a preference relation locally non-satiated?
If for any consumption bundle x there is another arbitrarily close bundle that is preferred to x
When is a utility function quasiconcave?
When it is a monotonic transformation of a concave function
What does Shephard’s lemma say?
h(p,u) = gradient with respect to p of e(p,u)
in other words: the hicksian demand function for good l is equal to the partial derivative of the expenditure function with respect to the price of good l
What does Roy’s identity say?
x(p,w) = gradient with respect to p of v(p,w) / partial derivative of v(p,w) with respect to w
in other words: the marshallian demand function for good l is equal to the partial derivative of the indirect utility function with respect to the price of good l divided by the partial derivative of the indirect utility function with respect to wealth
What is the compensating variation?
The change in wealth which makes the consumer just as well off after the price change as before
What is the equivalent variation?
How much wealth the consumer would accept in place of the price change
What does the law of supply say?
That quantities respond in the same direction as price changes. I.e., when an output price goes up, output goes up and when an input price goes up, output goes down
Why does the law of supply hold for any price change?
Because firms do not face a budget constraint, so that price changes do not induce wealth effects, i.e. only substitution effects affect the output decision
When is a production vector efficient?
When there is no other feasible production vector that produces more of some output using less of some input and that generates as much output using no additional inputs
What is the elasticity of substitution between factors of production?
1/(1-rho)
What is the difference between first- and second-order stochastic dominance?
If everyone with an increasing utility function (hence, regardless of their attitude to risk) prefers lottery 1 over lottery 2, then lottery 1 first-order stochastically dominates lottery 2.
If everyone with a concave utility function (hence, everyone who is risk-averse) prefers lottery 1 over lottery 2, then lottery 1 second-order stochastically dominates lottery 2.
What does the independence axiom say?
Preferring one lottery over another is equivalent to preferring a linear combination of the first lottery and a third lottery over a linear combination of the second lottery and the third lottery
