AEB 6533
Assignment 2
Assuming a set of demand elasticities ( 1
ε
), a set of supply elasticities ( 1
η), equlibrium
prices ( 1
p
), and equlibrium quantities ( 1
q) in country 1:
1
1
1
1
1.50 0.75 3.50
0.61 0.50 5.70
0.50 0.05 1000
0.04 0.30 750
p
q
−
ε= =
−
−
η= =
−
along with a set of demand elasticities ( 2
ε
), a set of supply elasticities ( 2
η), equlibrium
prices ( 2
p
), and equlibrium quantities ( 2
q) in country 1:
2
2
2
2
1.25 0.80 3.25
1.08 0.75 6.00
0.60 0.07 1250
0.09 0.40 500
p
q
−
ε= =
−
−
η= =
−
the social welfare can be expressed as:
11 11 12 12
21 21 22 22
11 1 11 11 1 11
,,,,
,,,
22 2 22 22 2 22
11 11
112
12 12
11
max 22
11
22
,,
sdsd
sdsd
ddd sss
qqqq
qqqq
ddd sss
ds
dsd
ds
aq q Aq bq q Bq
aq q Aq bq q Bq
qqq
qqq
qq
′′
+−+ +
′′
+−+
===
21 21
2
22 22
11 11 21 21
12 12 22 22
,
.. 0
0
ds
s
ds
dsd s
dsd s
q
q
qq
st q q q q
qqqq
=
−+− =
−+− =
where
()
()
()
()
11
11
22
22
0.00478 0.00796
14.252 33.119 0.00796 0.0259
0.007097 0.001577
4.780 15.140 0.001577 0.025684
0.02753 0.07342
74.372 203.658 0.07342 0.21178
0.004
3.296 11.864
aA
bB
aA
bB
−−
==
−−
=− =
−−
==
−−
=− − =
457 0.00195
0.00195 0.030853
.
1. Compute the gradient of the objective function at the original solution.
2. What is the nullspace for the constraints?
3. Compute the projected gradient at the original solution.
4. Is the current solution optimal?
5. Are the second-order necessary conditions met?
