CHAPTER 2
THE MATHEMATIC OF OPTIMIZATIONS
TRUE-FALSE
1. The first order condition state that any activity that contributes to the agent’s goal should be expanded up to the point at which the marginal contribution of further expansion is highest.
2. The Lagrangian multiplier can be intrepreted as the implicit value or shadow price of the objective function.
3. The envelope theorem is useful for examining how these optimal choices change when the problem’s parameters change.
4. b = f(x), when b is constant, then db/dx < 0
5. d
/ dq = 0 is a sufficient condition for an optimization process.
6. The elasticity ey,x will have different sign with the partial derivative 
y / 
7. When the elasticity is calculated, there are unit-free -- the unit of measurement “cancel out” because they appear either in numerator or denominator.
8. Lagrangian multiplier method is used for solving unconstrained maximization problems.
9. When a function is homogeneous of degree 0 (zero), a doubling of all of its arguments, doubles the value of the function itself. But for function that are homogeneous of degree 1 (one), a doubling of all its arguments leaves the value of the function unchanged.
10. Young’s theorem show that, for homogeneous function, there is definite relationship between the values of the function and the values of its partial derivative.
ESSAY
1. Explain following theory briefly:
a. Envelope theorem
b. Young’s theorem
c. Implicit function
d. Euler’s theorem
2. What is the different between cardinal and ordinal properties?
3. Explain following statement with mathematical explanation and draw the proper graph!
“First order condition is a necessary condition for an optimal condition, but not a sufficient one.”
4. What is elasticity? How it’s measured? Can it be calculated through “logarithmic differentiation”? Give an example!
5. Suppose U (x,y) = 5 
+ 2
a. Calculate the partial derivative U /
b. Evaluate these partial deriative at x=2 , y = 3
c. Write the total differential for U
d. What is the implied trade off between x and y holding U constant?
6. Explain about the Lagrange multiplier. Why sometimes it’s used as shadow price?
7. Suppose a firm’s total revenue depend on the quantity of output produced (q) ccording to the function: R = 50 q - 
With total cost: C = 
+ 40q + 150
a. What level of output should the firm produce in order to maximize profits? What will profits be?
b. Show the second order condition for a maximum are satisfied at the output level found in part (a)
8. Suppose that f (x,y) = xy. Find the maximum value for f if x and y are constrained to sum to 1. Solve this problem in two ways : by substitution and by using the Lagrangian multiplier method.
9. Suppose y is a function of two variables 
dan 
.
y = 
- 
+ 24
Use envelope shorcut to measure the optimal value of y (y*) changes when a parameter of it’s function changes.
10. Rachmat’s utility function:
U = 
+
where his income is $1000. If the whole income assume to spend fully for consumption of X and Y, what is the maximum utility he can reach if price of X $5 and price of Y $2?
*Use Lagrangian multiplier method to solvethis problem. Differ clearly about both objective function and its constraint!
yang dikerjakan :
T/F = All
essay : 1, 5, 6
