Economics 350 A01 Mathematical Economics 1: An Introduction to Static Methods
Fall 2009 First Midterm Examination
Answer all parts of each of the following seven questions. All questions are of equal value. This exam accounts for either 0% or 50% of your final grade depending on whether your second midterm score is higher or lower respectively than your score on this exam. This exam paper has two pages. Time allowed: 45 minutes. 1. Consider the sets A = {x ∈ R : 0 < x < 1} and B = {x ∈ R : 1 < x < 2} . (a) What is A ∩ B ? (b) What is A ∪ B ? (c) Let C = B . What is A ∩ C ?
2. A consumer has income m and faces prices p1 and p 2 for goods x1 and x 2 respectively. No other goods are available. (a) Illustrate the budget set for this consumer. (b) Is this budget set strictly convex? Explain your answer. 3. Consider the set A = {x ∈ R : 0 ≤ x < 1} . Answer “true” or “false” to each of the following statements. (a) A is bounded. (b) A is closed. (c) inf A = 0 (d) A has a maximum. (e) A is convex.
4.(a) Find the Euclidean distance between the ordered pairs (2,1) and (4,2). (b) Find the Euclidean distance between the ordered triples (2,1,3) and (4,2,5).
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5. Consider the following function:
y = ax + b
(a) Express this function in implicit form. (b) Graph the function for a > 0 and b > 0 . (c) On your graph, identify the set A = { y ∈ R : 0 ≤ y ≤ ax + b, − b ≤ x ≤ 0} a 6. Consider the following function:
y = 10 x
(a) Graph this function. (b) What type of function is this? (c) Write down the inverse function and graph it. (d) Evaluate log10 (10 x ) .
7.(a) Let x = λx ′ + (1 − λ ) x ′′ where λ ∈ (0,1) . Complete the following statement. A function f is strictly convex iff ______________. (b) Consider the function
y = x2
Show that this function is strictly convex.
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Answer Guide
1.(a) A ∩ B = φ (b) A ∪ B = {x ∈ R : 0 < x < 2} − {1} (c). C = {x ∈ R : x ≤ 1, x ≥ 2} . Then A ∩ C = A 2.(a) See Figure 1. (b) It is convex; the set contains all convex combinations of points within the set. It is not strictly convex.
x2
m p2
m p1
x1
Figure 1
3.(a) True. (A is bounded above and below). (b) False. (c) True. (d) False. (e) True.
4.(a) d = (2 − 4) 2 + (1 − 2) 2 = 5
(b) d = (2 − 4) 2 + (1 − 2) 2 + (3 − 5) 2 = 3
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5.(a) y − ax − b = 0 = 0 (b) and (c) See Figure 2.
y
b
−b a
x
Figure 2
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6.(a) See Figure 3. (b) An exponential function. (c) x = log10 y . See Figure 4. (d) log10 (10 x ) = x
y
1
x
Figure 3
5
x
1
y
Figure 4
7.(a) f ( x ) < λf ( x ′) + (1 − λ ) f ( x ′′) (b) RHS = λ ( x ′) 2 + (1 − λ )( x ′′) 2 LHS = (λx ′ + (1 − λ ) x ′′) 2 = λ2 ( x ′) 2 + (1 − λ ) 2 ( x ′′) 2 + 2λ (1 − λ ) x ′x ′′ .
RHS − LHS = λ (1 − λ )[( x ′) 2 + ( x ′′) 2 − 2 x ′x ′′] = λ (1 − λ )( x ′ − x ′′) 2 > 0 . Thus, f is strictly convex.
See Figure 5.
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y
λf ( x′) + (1 − λ ) f ( x′′)
f (x )
x′
x
x′′
x
Figure 5
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