ECO 3145 Mathematical Economics I Module 1

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ECO 3145 Mathematical Economics I

Module 1 – Static Optimization I – unconstrained – local optimum

Practice Problems

I.    Univariate Functions

1.    Use the vocabulary of necessary and sufficient conditions to describe the relationship   between global and local optima. Use arrows (⇒, ⇐, or  ⇔) to indicate the relationship between global and local optima

2.    Consider the following two statements: (i.) f (x) = 0 , and (ii.) f(x)  achieves an optimum at x . Is statement (i.) a necessary or sufficient condition for statement (ii.), or both? Arrange   statements (i.) and (ii.) using the appropriate arrows to show the correct relationship, i.e.        ,  , or  .

3.    C&W pp.87-88, #1-2.

4.    C&W pp.226-227, #1 (do a couple), #2 (do a couple), #3*, #4* (* means more difficult)

5.    C&W p.241, #1 (do a couple), #2

 

II.   Multivariate Functions

1.   For the following two functions, derive (i) the total differential and (ii) the partial differential1 with respect to x1..

a.) f(x1 , x 2 ) = 5x1(2)  −10x1 x 2  − x2(2)

b.) f(x1 , x 2 , x 3 ) = x 1(2)  + 2x2(2)  + 3x3(2)  + 4x1 x 2  − 6x1 x 3  + 8x 2 x 3

2.   Find the stationary points of each of the following functions.

a.) f(x1 , x 2 ) = 4x1  − 2x2(2)  + x1(2)  + x 2

b.) f(x1 , x 2 ) = 8x1  − x1(2)  +14x 2  − 7x2(2)

c.) f(x1 , x 2 ) = 4x1  −   +  x 2  − 2

d.) f(x1 , x 2 ) = x 1  + 3ex 2    − ex1    − e3x2

e.) f(x1 , x 2 ) = 100 − 5x1  + 4x1(2)  − 9x 2  + 5x2(2)  + 8x1 x 2

f.) f(x1 , x 2 ) =  x 1(3)  + 3x1 x 2  + 2x1  −  x2(2)

3.   As a consultant to the Journal of Important Stuff, you must determine the effect on sales of the number of pages devoted to economics (E) and the number of pages devoted to other    subjects (A). After careful study, you have decided that the relationship between sales (S)  and the allocation of pages is given by the following function:

S = 100A + 310E − A 2  − 2E2  − AE

If the goal is to maximize sales, what will be your recommendation regarding the allocation of pages between economics and other subjects?

4.   The profits of two cigarette manufacturers, Eventual Death (E) and Painful Death (P), depend upon advertising expenditures according to the following two functions.

 E  = 1000AE  − AE(2)  − AP(2)

 P  = 1000AP  − AP A E  − AP(2)

In these equations,  represents profit and A represents advertising expenditures.

a.)  If each producer believes the advertising expenditure of the competitor is fixed, what will

be the optimal level of expenditure and the maximum profit of each?

b.) Now the two companies have decided to merge. Nonetheless, they keep the two distinct  brands. The managers would like to maximize the combined profit of the two brands, i.e.  =  E  +  P . What levels of advertising expenditure, AE and AP, will they choose? Are the combined profits greater before or after the merger? (Note: You can deduce the          answer to this last question by comparing AE and AP before and after the merger without calculating the new  . Explain how.)

c.)  The managers of the merged company must decide whether to keep the distinct brands or

merge them as well. In the case of a single, merged brand (Eventual Painful Death?) the profit function of the merged company is the same as in (b) except that AE  = AP  = A .  What is the optimal level of A? Which is the best choice, separate brands or a single      brand?

5.    Find all the points that satisfy the first-order conditions for the function (1,  2,  3) = ln 1  + ln 2  + 3 ( − 1  − 2 )
















2022-06-10