ECON30020 Mathematical Economics Tutorial 2. More Logic and Proofs
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ECON30020 Mathematical Economics
Tutorial 2. More Logic and Proofs
Question 1. Involuntary unemployment
Consider a perfectly competitive labour market in which the demand and supply functions for labor as functions of the wage w, denoted D(w) and S(w) respectively, are such that the market clearing (or equilibrium) wage w* is positive and induces a positive employment level Q* = D(w* ) = S(w* ). Assuming that D(w) is strictly decreasing in w and S(w) is strictly increasing, show that imposing a minimum wage w > w* induces excess supply (or involuntary unemployment) and reduces employment below Q* .
Answer:
At w , demand for labour will be D(w ) which is less than Q* = D(w* ) because D is strictly decreasing. Labour supply will be S(w ), which is larger than Q* = S(w* ) because S is strictly increasing. Hence, the employment level will be D(w ) < Q* ; firms are able to hire as many workers as they want.
Unfortunately, not all workers will be able to find jobs. Involuntary unemployment (the excess supply of labour) will be
S(w ) 一 D(w ) = [S(w ) 一 Q*] + [Q* 一 D(w )] > 0.
minimum wage in labour market
S(w), supply of labour
w
w*
D(w), demand for labour
unemployment
Question 2. Marriage market
Consider four women (Oprah, Prim, Rita, Sophie) and four men (Austin, Boris, Conan, Dishi), who have the following preferences:
Oprah: Boris > Austin > Dishi > Conan
Prim: Dishi > Conan > Austin > Boris
Rita: Austin > Dishi > Conan > Boris
Sophie: Boris > Austin > Dishi > Conan
Austin: Prim > Sophie > Oprah > Rita
Boris: Rita > Oprah > Sophie > Prim
Conan: Prim > Rita > Oprah > Sophie
Dishi: Sophie > Oprah > Rita > Prim
Assume that all men and women are heterosexual and (once married) monogamous.
(a) Find a matching by using Deferred Acceptance Algorithm (DAA) (Gale-Shapley algo-
rithm) with men proposing.
(b) Find a matching by using Gale-Shapley algorithm with women proposing.
(c) Compare (a) and (b). When are women better off? When are men better off?
Answer:
(a) Men proposing DAA.
Stage 1:
Women |
proposed by |
provisionally keeps |
rejects |
Oprah |
|
|
|
Prim |
Austin, Conan |
Conan |
Austin |
Rita |
Boris |
Boris |
|
Sophie |
Dishi |
Dishi |
|
Stage 2:
Women |
proposed by |
provisionally keeps |
rejects |
Oprah |
|
|
|
Prim |
|
Conan |
|
Rita |
|
Boris |
|
Sophie |
Austin |
Austin |
Dishi |
Stage 3:
Women |
proposed by |
matched with |
Oprah |
Dishi |
Dishi |
Prim |
|
Conan |
Rita |
|
Boris |
Sophie |
|
Austin |
Note that this matching is stable.
(b) Women proposing DAA.
Stage 1:
Men |
proposed by |
provisionally keeps |
rejects |
Austin |
Rita |
Rita |
|
Boris |
Oprah, Sophie |
Oprah |
Sophie |
Conan |
|
|
|
Dishi |
Prim |
Prim |
|
Stage 2:
Men |
proposed by |
provisionally keeps |
rejects |
Austin |
Sophie |
Sophie |
Rita |
Boris |
|
Oprah |
|
Conan |
|
|
|
Dishi |
|
Prim |
|
Stage 3:
Men |
proposed by |
provisionally keeps |
rejects |
Austin |
|
Sophie |
|
Boris |
|
Oprah |
|
Conan |
|
|
|
Dishi |
Rita |
Rita |
Prim |
Stage 4:
Men |
proposed by |
matched with |
Austin |
|
Sophie |
Boris |
|
Oprah |
Conan |
Prim |
Prim |
Dishi |
|
Rita |
Note that this matching is stable.
(c) Comparison: as you can see from the table, women are weakly better off when they are proposing and men are weakly better off when men are proposing
People |
match (man prop.) |
choice # |
match (woman prop.) |
choice # |
Oprah |
Dishi |
3rd |
Boris |
1st |
Prim |
Conan |
2nd |
Conan |
2nd |
Rita |
Boris |
4th |
Dishi |
3rd |
Sophie |
Austin |
2nd |
Austin |
2nd |
Austin |
Sophie |
2nd |
Sophie |
2nd |
Boris |
Rita |
1st |
Oprah |
2nd |
Conan |
Prim |
1st |
Prim |
1st |
Dishi |
Oprah |
2nd |
Rita |
3rd |
2022-06-18