Old Exams

Question 1

State differences between the profit maximization and the cost minimization problems regarding the SOCs. Tell us about conditions for existence and uniqueness.

Answer 1

Profit maximization and cost minimization problems are dual to each other but focus on different objectives and constraints.

Profit Maximization (PM):

• Objective: To maximize profits by choosing the optimal level of outputs (and inputs) given the prices.
• SOCs: The Hessian matrix of the profit function wrt output levels must be negative semidefinite at the optimum to ensure a maximum.
• Existence: Profit maximization assumes the existence of a well-behaved production function, price-taker behavior in output and input markets, and sufficient regularity conditions like continuity and differentiability of the profit function.
• Uniqueness: Production set is strictly convex, as it implies strictly diminishing marginal rate of substitution between inputs and outputs. This translates to a strictly concave profit function, ensuring a unique maximum.

Cost Minimization (CM):

• Objective: To minimize costs for a given level of output by choosing the optimal combination of inputs.
• SOCs: The Hessian matrix of the cost function with respect to input quantities must be positive semidefinite at the optimum to ensure a minimum.
• Existence: Cost minimization requires a convex production set, ensuring that the combination of inputs to produce a certain level of output is feasible. Moreover, input prices must be positive, and technology must exhibit non-increasing returns to scale.
• Uniqueness: If the production technology exhibits strictly convex isoquants, then the input combination for producing a certain level of output is unique. In terms of the cost function, this translates to a convex cost function in input space, ensuring a unique cost-minimizing point.

In both problems, the existence of an interior solution says that the marginal products of inputs should go to infinity as the input goes to zero and should decrease to zero as the input goes to infinity. This ensures that the firm will use a positive amount of each input. The curvature of the objective function (profit or cost) and the shape of the constraint set (production or input requirement set) play critical roles in determining the uniqueness of the solutions.

Question 2

State properties of the expenditure function.

Answer 2

The expenditure function, denoted by E(p,u), represents the minimum expenditure required to achieve a certain level of utility u given a vector of prices p. Here are the key properties of the expenditure function:

Homogeneity of Degree One in Prices

Quasi-Concavity in Prices:
The expenditure function is quasi-concave in prices, which means that the set of price vectors p for which the expenditure function is less than or equal to a certain level of expenditure forms a convex set.

Quasi-Convexity in Utility:
The expenditure function is quasi-convex in utility, which means that the set of utility levels u for which the expenditure function is less than or equal to a certain level of expenditure forms a convex set.

Non-Decreasing in Prices:
The expenditure function is non-decreasing in all prices. This implies that if the price of any good increases, the expenditure required to achieve a given level of utility will also increase.

Non-Increasing in Utility:
The expenditure function is non-increasing in utility. This means that if utility increases, the expenditure required to achieve that utility level will decrease or remain the same.

Walras’ Law:
The total expenditure across all goods equals total income when markets clear.

Question 3

Explain the Envelope Theorem + proof.

Answer 3

The Envelope Theorem states that the derivative of the value function with respect to a parameter is equal to the derivative of the objective function with respect to that parameter, evaluated at the optimal solution. In other words, if you have optimized a function with respect to some parameter, then the derivative of the optimized value with respect to that parameter is the same as the derivative of the original function with respect to that parameter, holding the optimizing variables constant at their optimal values.

Question 4

Write down and explain a Walrasian equilibrium.

Question 5

State the utility maximization problem.

Answer 5

• In the utility maximisation problem, we want to maximise the utility given our budget constraint that the price of our consumption cannot exceed wealth.
• Mathmatically, it says: max x ≥0 u(x) subject to p x ≤ w
• We know that the UMP has a solution when the prices are strictly positive (p»0) and our utility function is continuous.

Question 6

UMP: State and explain the FOC

Answer 6

As the UMP is a constrained problem we use the langragrian to derive the FOC

L= u(x) – λ [px-w]
FOC:
δL/δx_i= δu(x)/ δx_i – λp_i = 0
* this can be rearranged to: δu(x)/ δx_i = λp_i
* 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

Second part is the partial δL/δλ= -(px-w)=0
* ensures that the budget constraint is binding
* It implies that px=w and confirms our assumption

Question 7

How do we get from the Marshallian demand to the indirect utility?

Answer 7

• We know that the 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))

Question 8

How can we recover the Marshallian demand from the indirect utility?

Answer 8

• This way around is called Roy’s Identity. When we take the negative gradient of the indirect utility function wrt p and divide this through the gradient of the indirect utility function wrt w, the result will be x(p,w).

This only holds under the following assumptions:
* U(.) is a continuous utility function representing a locally non-satiated preference relation defined on X=R and v(.) is differentiable at (p,w)»0

• Roy’s Identity tells us how the demand of good l responds to changes in prices after adjusting for consumer’s wealth- holding utility constant.
• Provides way to express demand functions directly in terms of observable variables (p & w) without having to specify the unobservable utility function.
• It is negative because when prices increase, demand decreases.
• It is an application of the envelope theorem