ECO3020F 2024: TUTORIAL TWO [45 marks]
Question 1 [24 marks]
Consider two gold miners, Miner A and Miner B, who share a gold mine and face Hardin’s
Tragedy of the Commons. The two miners are identical with utility functions given by uA
and uB below, where g represents kilograms of gold extracted from the mine and h represents
the hours the miners spend mining.
1 A 2
uA = g A (h )
2
1 B 2
uB = g B (h )
2
The production of gold is given by identical production functions, g A and g B below, in which
> 0, and µ > 0.
g A = hA [ µ(hA + hB )]
g B = hB [ µ(hA + hB )]
In addition, each miner has a value function, v A and v B below, with their degree of altruism
toward the other miner given as 0 1
v A = uA + u B
v B = uB + uA
1. Mathematically show that Miner A’s best response function is given as:
, B hA
1 UA nA 8 µ nA h
A
y At 4B
nA 8 µ nA hB hA 2 4B
2 hB y M hath hb
nA 8 M hath B ha
B ha duh B 0
84 0 8 21h A_uh β 0
B nA it 2M 014h
8 Uh 0mn's
14B
A's Bra n
8 30 th 1 0
2 hA hB 4,1 1,0 2 E 1
sin
jointunity
It IT
2nA 30nA hA 10 hB
nA 3ft
0 0
uncoordinated nashequilibrium
30 30 hB
nA
2ha 30 ha hA 12 hB
nA 24B
A'SBRF
I
users
É 10
9
I 1
A hours nA
Question 1 [24 marks]
Consider two gold miners, Miner A and Miner B, who share a gold mine and face Hardin’s
Tragedy of the Commons. The two miners are identical with utility functions given by uA
and uB below, where g represents kilograms of gold extracted from the mine and h represents
the hours the miners spend mining.
1 A 2
uA = g A (h )
2
1 B 2
uB = g B (h )
2
The production of gold is given by identical production functions, g A and g B below, in which
> 0, and µ > 0.
g A = hA [ µ(hA + hB )]
g B = hB [ µ(hA + hB )]
In addition, each miner has a value function, v A and v B below, with their degree of altruism
toward the other miner given as 0 1
v A = uA + u B
v B = uB + uA
1. Mathematically show that Miner A’s best response function is given as:
, B hA
1 UA nA 8 µ nA h
A
y At 4B
nA 8 µ nA hB hA 2 4B
2 hB y M hath hb
nA 8 M hath B ha
B ha duh B 0
84 0 8 21h A_uh β 0
B nA it 2M 014h
8 Uh 0mn's
14B
A's Bra n
8 30 th 1 0
2 hA hB 4,1 1,0 2 E 1
sin
jointunity
It IT
2nA 30nA hA 10 hB
nA 3ft
0 0
uncoordinated nashequilibrium
30 30 hB
nA
2ha 30 ha hA 12 hB
nA 24B
A'SBRF
I
users
É 10
9
I 1
A hours nA
