Gedächtnisprotokolle Flashcards
State the utility maximization problem.
max x≥0 u(x) s.t. px ≤ w
Want to maximise the utility of our consumption bundle given our budget constraint that the price of our consumption cannot exceed wealth.
We know that the UMP has a solution when the prices are strictly positive (p»0) and our utility function is continuous.
State and explain the FOC to the UMP.
Since the UMP is a constrained problemm we use the Langragian to derive the FOC
L= u(x) – λ [p * x-w]
1) Partial derivative wrt xi
δL/δxi = δ u(x) / δ xi – λ pi = 0
rearranged to: δ u(x)/ δxi = λpi
- gives us the marginal utility of consumption
- price of the good scaled by the multiplier, representing the marginal cost of spending one more unit of currency on the good
- shadow price of langragian
2) Partial derivative wrt λ
λ = -(px-w) = 0
- ensures that the budget constraint is binding: it implies that p * x=w and confirms our assumption.
How do we get from the Marshallian demand to the indirect utility?
- Indirect utility is the utility of the optimal consumption bundle given our constraints by prices and wealth.
- We can derive the indirect utility by plugging in the marhsallian demand into our utility function: u(x(p,w))
How can we recover the Marshallian demand from the indirect utility?
Roy’s Identity:
x(p,w)= - ∇p v(p,w)/ ∇w v(p,w)
It tells us how demand of good l responds to changes in prices of good l AFTER adjusting for the consumers wealth - holding u constant.
State the cost minimization problem.
min wz s.t. f(z) ≥ q
Minimize the cost of production - subject to productive feasibility.
Result of cost minimization problem: conditional factor demand correspondence z(w,q)
State FOC and SOC to Cost Minimization Problem.
CMP: min wz st F(z) ≥ q
L= wi zi - f(z)-q
FOC wi- λδf(z)/δzi
Rearrange to wi = λδf(z)/δzi
= cost wi of using an additional unit of input = the value of the output produced by that additional unit of input (the marginal product of input i times the shadow price λ).
SOC = second partial derivative must be larger or equal to 0
then objective function must be convex
for multiple variables the determinant of the bordered hessian must be positive semidefinite
Interpreting a hessian further
For a more accurate interpretation, one would need to examine the signs of the determinants of all the leading principal minors of the bordered Hessian matrix, and not just the determinant of the entire matrix.
SOC Max Problem
- ≤0
- concave in x
- H: negative semidefinite
SOC Min Problem
- ≥ 0
- convex in x
- H: positive semidefinite
What do we have to assume for the cost min problem?
Problem should be well defined
1. Convex cost function with respect to input choices. This means that the cost increases at an increasing rate as you use more inputs, reflecting the principle of diminishing returns.
2. Well-Behaved Production Function: The production function f(z) is typically assumed to be quasi-concave and displays non-decreasing returns to scale up to a point
3. No Externalities?
Properties of production function
- non-decreasingin all inputs
- If k=1, the function has constant returns to scale.
- If k>1, it exhibits increasing returns to scale.
- If k<1, it shows decreasing returns to scale.
- concave in inputs = diminishing marginal returns
State and explain Shephard’s Lemma. What does it give us?
Case of deriving h(p,u) from e(p,u)
1. we know that e(p,u) = p h(p,u), we take the derivative of this wrt p and use the product rule for doing that
2. the result includes the FOC from the EMP problem which are per definiton = 0
3. we know that utility is held constant so the change in utility from a change in price = 0, ergo we dont need to worry about utility further
4. we can take the FOC from the EMP and rearrange it for the p
5. plugging the result from EMP for p into our partial derivative of e(p,u) wrt p then shepherds lemma is the result
6. here we also see the so-called indirect and direct effect- the direct effect is the second term which remains in the shepherds lemma but the first term falls away since the result is zero already by definition from the FOC
State and explain Hotelling’s Lemma. What does it give us?
Not the envelope theorem version:
- connects profit maximization + supply function
- follows from duality theorem
- it says that the change in profits from a change in price is proportional to the quantity produced
- If the supply function is single valued, then the gradient of the profit function wrt p gives us the supply at these prices
∇ π(p) = δπ(p) / δp = y * (p)
Proof:
- function g(p) is maximised by y * at p
g(p) = π(p) - p y *
- How would max value change if p changes? Just take partial derivative of the function above and rearrange so that ∇ π(p) = δπ(p) / δp = y * (p)
State and explain the Envelope Theorem. When is which term equal to zero? Why? Tell us about direct and indirect effects.
1. Optimization Problem Set U
You have an objective function, f(x,α) that you want to maximize by choosing x subject to constraint g(x,α)=0
2. Langrangrian and FOC
L= f(x,α)- λg(x,α)-0 and FOC: δα/δx= δf(x,α)/ δx - λ δg(x,α)/ δx = 0 AND δα/δx= -g(x,α)=0 —> x(α)
3. Optimal Value Function v(α) is max value of objective function f(x,α) and we found best x(α)
4. Differentiate the value function- here envelope theorem comes in!
How does v(α) change when α changes? So we differentiate v(α) at α!
5. Applying Envelope Theorem
When differentiating v(α), the term involving δx/δα drops out bc of FOC - only the partial derivative of objective
function wrt δ remains = the way that v(α) changes as α changes does not depend on how x adjusts to α
State and explain a Walrasian equilibrium. What is a Walrasian equilibrium? When does it hold?
In a private ownership economy, an allocaton and a price vector consitute a WE when
1) firms maximize profits
2) consumer maximize their utility subject to their budget constraint that the value of their consumption cannot exceed their earnings from their endowments (p ω) and the sum of profits from firm shares they get which depend on firms profit
3) market clears that not more can be consumed as endowed and produced
Tell us about desirability. When does it hold?
Laws of desirability
1) monotonicity: if you have strictly more of x than y than x is preferred over y
2) strict monotonicity: if you have at least as much of x than y than x is preferred over y
3) LNS: in every distance epsilon to x there will be another x’ which is more preferred
Describe a quasiconcave function. What is the definition of quasiconcavity?
For every two bundles x,y e X, the utility of consuming the convex combination of these two bundles u(αx + 1-α)y), is weakly higher the the minimal utility from consuming each bundle seperately, min {u(x); u(y)} or more compactly:
u(αx+(1-α)y) ≥ min {u(x), u(y)}
Graphically:
* Starting from bundles x and y, with associated utility levels u(x) and u(y), respectively, we can construct a convex combination of the two bundles
* if the utility that the consumer derives from conming this convex combination is larger than consuming the bundle associated with the lowest utility level (in this case y), we say this individual utility function satisfies quasi-concavity
Strict quasi-concavity
u(.) satsifies strict quasi-concavity if for every two bundles x,y e X the utility of consuming its convex combination u(αx+(1-α)y), where 0 < α < 1 is STRICTLY higher than the minimal utility from consuming wach bundle seperately.
Hence the utility function that yields the indifference curve in the fig satisfies strict quasi-concavity.
What is an example of a utility representing preferences of the Gorman form?
It allows for straightforward aggregation of individual preferences into a representative consumer’s utility function.
Example would be:
