(i) Verbally explain the difference and relationship between a maximum and a maximizer.
(ii) Relate the arg max to the maximum using a mathematical statement (refer to a function ).
(iii) Let such that and where is a constraint set. Can you have ? Explain why (not).
Hint: Less formally, this question asks whether you can have a strictly larger value in a constrained optimization problem than in an unconstrained problem with the same objective.
(iv) Can be empty? Can it have more than one argument if there is a strict global maximizer?
A first step to solving an optimization problem is forming an expectation of whether one will find a solution. Our common tool to approach this issue is the Weierstrass Extreme Value Theorem. A number of practice problems for this theorem can be found on Problem Set 4. Here, we again focus on the role of continuity in the requirements of the theorem.
(i) Give an example of a discontinuous function on a compact domain that does not have a global maximum.
Hint: Think about the intuition that Chapter 4 has discussed. You may want to define a “split function”, i.e. , either explicitly or using an indicator term.
(ii) Show that the Weierstrass Extreme Value Theorem is sufficient but not necessary by giving an example of a discontinuous function that attains both a global maximum and minimum.
(iii) If there are no “border solutions” that we need to consider, meaning that we use only the Lagrangian method to identify potential solutions, do you need to worry about continuity?
In practical applications, a common issue with the Weierstrass Extreme Value Theorem is that the support is not compact. For example, this is the case whenever we optimize over the whole or in unconstrained optimization or non-compact constraint sets, such as open intervals/balls. Fortunately, in many cases, we can “compactify” the domain and avoid issues with solution existence in an elegant way. In this exercise, you will establish a corollary of Weierstrass that is a concrete example of this method.
Corollary: Optimizing a Univariate Function with Non-vanishing Limits.
Consider a function , i.e. a function that is twice continuously differentiable. Assume that
Then, assumes both a global maximum and minimum, and the global extremizers are critical points of .
(i) Assume that has exactly two critical points (i.e. points with ). Can you illustrate the intuition of this corollary graphically for Case 1?
(ii) When has exactly two critical points, can you say one is the global maximum/minimum of depending on which case (Case 1 or Case 2) you are in?
(iii) Why do we need in Case 1 and in Case 2 to ensure existence of the global extrema?
(iv) Give a formal argument why the corollary holds, i.e. put the graphical intuition in a mathematical argument. You may restrict attention to Case 1.
Comment: An analogous argument can be made for Case 2. Because this case does not add an interesting particularity, we do not investigate the argument establishing it here.
Hint 1: Recall the definition of the limit : If , then
Use this definition to restrict the investigation to a compact domain and apply Weierstrass.
Hint 2: It may be easier to investigate existence of the global maximum and minimum in isolation.
and, if there are global maximizers, compute the maximum value.
Hint: Explicitly think about the limit behavior.
and, if there are global extremizers, compute the extreme values.
Hint: The function may have a particular shape that we investigated in an earlier exercise; an argument exploiting this shape may be used to elegantly circumvent the computation-intensive second order condition.
Determine the global extrema of .
Determine the global extrema of .
Hint: Recall that for unconstrained multivariate optimization, a different approach to limit behavior analysis is required.
In the exercises of Chapter 3, we investigated definiteness of the second derivative of for . Recall that the second derivative was and that for ,
(i) Can you use an optimization problem approach to find the values for and where is not positive semi-definite, i.e. where there exist for which ?
To help you with the solution, note that if we write , we have
Hence, the magnitude does not matter for the property of being strictly negative, and you can reduce the search for with to a direction vector. Because the magnitude of the direction vector does not matter and as if , then the expression is weakly positive, there is no loss in setting and restricting the search to the value that yields .
(ii) Can you find the range of where is positive definite? (Solve (i) first.)
Consider the function .
(i) For a given , which direction of the yields the largest and smallest value of , when the set of directions is defined to have normalized length, i.e. ? You may consider the Euclidean norm.
Hint 1: For , is equivalent to , which may be an easier constraint to work with.
Hint 2: is closed and bounded and thus compact, so you need not worry about solution existence.
Hint 3: By hint 2, it is not necessary to consult the second order condition, you can simply compare the values for each candidate.
Hint 4: There will be 6 candidates in total, resulting from two cases. To determine which is the largest and smallest, you can exploit that once you solved for in either case, you should obtain a function that is is strictly monotonic in .
As you may guess from the fact that there are 4 hints, the problem is not particularly easy. Don’t feel bad peaking into the solution if you get stuck.
(ii) Can you use the result of (i) to say something about the unconstrained optimization problems
Suppose we have two individuals, Martin and Anna. Both like to spend their free time performing only two activities: relaxing () and going climbing (). Otherwise, they don’t derive utility from any other source. Suppose that an hour of relaxation costs (e.g. for a Netflix account, food and drinks, or whatever you like to consume in your free time), and an hour of climbing costs (equipment, gym subscription, etc.). Suppose that both Martin and Anna are employed at the same job, and can work for a net hourly wage of to generate income; they both have no initial wealth. Their preferences differ: we have
for Martin and
for Anna. This means that Martin puts more weight on climbing whereas both activities are equally weighted for Anna.
(i) Formulate the problem that Anna faces when maximizing utility within a given day that has 24 hours. Simplify the problem as much as possible.
(ii) How can you interpret the budget constraint that you obtain after simplification?
(iii) Solve Anna’s utility maximization problem (finding the optimal distribution of time across activities is enough; the value of utility does not matter).
(iv) Assume now that Anna has some savings and does not need to work on the day we consider in our optimization problem here. Given her utility function, what is the minimum amount of money that she needs to spend to receive at least the same as before? (Even though she does not need to work here, she can still not spend more than 24 hours on both activities combined.)
Hint 1: Once you have simplified the problem to have only one choice variable, it may be instructive to investigate whether or not the time budget constraint binds by looking at the first derivative of the objective.
Hint 2: Don’t worry if your results don’t give nice numbers anymore, you will need a calculator for this exercise. You may round all intermediate results to two digits.
(v) How do you explain the difference in results of (iv) and (iii)?
(vi) How much would Martin need to earn per hour to afford Anna’s level of utility if he has no savings? You may not be able to solve for the wage to the cent; thus, assume that the wage is an integer value. You can use without proof that utility is strictly increasing in the wage and check in steps of 1 whether a given wage yields at least the desired level of utility.
Hint: Solve Martin’s utility maximization problem in analogy to Anna’s with variable wage to derive the maximal utility as a function of the wage in a first step.
(vii) What level of utility can Martin maximally attain as his wage increases? Why is utility not unbounded above? What can you say about Martin’s utility and time allocation when he does not have to work from this investigation?
(viii) Consider the problem where now, and are tradeable goods, e.g. cookies and rice (in kg). Suppose that Martin and Anna live on a deserted island, on which there are 10 boxes of cookies and 12 kg of rice. Formulate the welfare-maximizing resource allocation problem when equal weight is given to both individuals and simplify it as much as possible. How do you proceed to find the solution (verbally)?
(ix) What is the allocation that solves the problem of (viii)?
Hint: Beware of border solutions.
(xi) Finally, consider again the island scenario but now assume that Anna initially owns all the cookies and Martin owns all the rice. Assume that they can freely discuss exchanging the goods, have perfect information about all aspects of the trade and there is no power asymmetry between the two. At what ratio do they trade the goods, and what is the final allocation? How does aggregate welfare compare to the previous exercise?
Hint: You can express both agents’ utility to trading a given quantity of goods at a fixed ratio as a function of the ratio and the goods quantity, and then solve for concrete values that sustain an equilibrium by thinking about the condition that ensures that both agents do not want do deviate.