Solutions for Foundations of Mathematical Economics
c
⃝2001 Michael Carter
All rights reserved
0-1 1 𝑥1
-1
1
𝑥2
Figure 1.1: The relation {(𝑥, 𝑦):𝑥2+𝑦2=1}
1.10 The sample space of a single coin toss is {𝐻, 𝑇 }. Thesetofpossibleoutcomesin
three tosses is the product
{𝐻, 𝑇 }×{𝐻, 𝑇 }×{𝐻, 𝑇 }=(𝐻, 𝐻, 𝐻),(𝐻, 𝐻, 𝑇 ),(𝐻, 𝑇, 𝐻),
(𝐻, 𝑇, 𝑇 ),(𝑇,𝐻,𝐻),(𝑇,𝐻,𝑇),(𝑇,𝑇,𝐻),(𝑇,𝑇,𝑇)
A typical outcome is the sequence (𝐻, 𝐻, 𝑇 ) of two heads followed by a tail.
1.11
𝑌∩ℜ
𝑛
+={0}
where 0=(0,0,...,0) is the production plan using no inputs and producing no outputs.
To see this, first note that 0is a feasible production plan. Therefore, 0∈𝑌. Also,
0∈ℜ
𝑛
+and therefore 0∈𝑌∩ℜ
𝑛
+.
To show that there is no other feasible production plan in ℜ𝑛
+, we assume the contrary.
Thatis,weassumethereissomefeasibleproductionplany∈ℜ
𝑛
+∖{0}. This implies
the existence of a plan producing a positive output with no inputs. This technological
infeasible, so that 𝑦/∈𝑌.
1.12 1. Let x∈𝑉(𝑦). This implies that (𝑦, −x)∈𝑌. Let x′≥x. Then (𝑦, −x′)≤
(𝑦, −x) and free disposability implies that (𝑦, −x′)∈𝑌. Therefore x′∈𝑉(𝑦).
2. Again assume x∈𝑉(𝑦). This implies that (𝑦, −x)∈𝑌. By free disposal,
(𝑦′,−x)∈𝑌for every 𝑦′≤𝑦, which implies that x∈𝑉(𝑦′). 𝑉(𝑦′)⊇𝑉(𝑦).
1.13 The domain of “<”is{1,2}=𝑋and the range is {2,3}⫋𝑌.
1.14 Figure 1.1.
1.15 The relation “is strictly higher than” is transitive, antisymmetric and asymmetric.
It is not complete, reflexive or symmetric.
2