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BEEM103J

2021

Optimization Techniques for Economists

Exam

1. (10 points) Find the values of x that satisfy the inequality

where S3 and S4 are the third and fourth digits of your candidate number, re- spectively.

2. (14 points) Consider the function

Does this function have a global minimum or maximum? Does this function have a local minimum or maximum?

3. (14 points) The number 0 plays a central role in mathematics. Raising any nonzero number to the power of 0 gives 1 as a result suggesting that 00 may be equal to 1. However, raising 0 to any positive powers gives 0 as a result suggesting that 00 may be equal to 0.

(a) (4 points) Construct a sequence of positive numbers, xn where n = 1, 2, . . ., such that every number in the sequence gets closer and closer to zero and the sequence is converging to 0, i.e. limn二o xn = 0. Compute for a few elements in your sequence. Where is your sequence converging as n → o? (Note: There is no need to show formally where your sequence is converging. It is enough to report your numbers so you can get an insight into where your sequence is converging and what should 00 be equal to.)

(b) (10 points) Consider now the function f (x) = xx deﬁned over the open interval (0, o). Determine the limit limx二0 f (x). (Hint: You may want to consider working with a transformation of the function.) According to your result what should 00 be equal to?

4. (14 points) (a) (5 points) Linearize the equation

around the steady state values of the variables given in the following table:

(b) (9 points) Linearize the equation

around the steady state values of the variables given in the following table:

5. (14 points) A dealership sells t trucks at the price of £pt and s sedans at the price of £ps . The demand functions for trucks and sedans are given by

where S3 , S4 , S5 , and S6 are the third, fourth, ﬁfth, and sixth digits of your candidate number, respectively. These equations reﬂect a simple observation that when the price of trucks increases then customers buy fewer trucks and more sedans, and when the price on sedans increases they buy more trucks and fewer sedans. Find values of pt and ps such that the dealership maximizes its revenue. Are the second-order conditions for a maximum satisﬁed?

6. (14 points) A bus manufacturer produces buses in two diﬀerent colours: white and red. The company’s total cost function is given by

where BW and BR denote the quantities of white and red buses produced, re- spectively. The ﬁrm has managed to secure a contract from the government to produce 42 buses. The contract does not specify the colour of the buses supplied, it only speciﬁes that 42 units should be supplied in total. Use the method of La- grange multipliers to determine how many white and red buses will the company deliver to the government. Compute and interpret the Lagrange multiplier: how will total costs be aﬀected approximately if the government decides to order 43 buses instead?

7. (20 points) Consider the following optimization problem

where S4 and S5 are the fourth ﬁfth digits of your candidate number, respectively. Determine the solution to this problem using the Kuhn-Tucker method.