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cooperativebargainingredux.pdf
Intra-household distribution II: Cooperative bargaining
Econ 339
ECON 339: intra-household distribution II: bargaining
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Sources
• http://www.dictionaryofeconomics.com/article?id=p de2008_F000290&edition=current&q=household%2 0bargaining&topicid=&result_number=2#sec2.1
(New Palgrave Dictionary of Economics, online edition)
ECON 339: intra-household distribution II: bargaining
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Cooperative? bargaining?
• "Cooperative" - individuals will achieve efficient outcomes - on joint ppf
• "bargaining" - allocation depends on relative strengths, outside options, "threat points“
• Usual threat point: best could do if single
(consistent with marriage market analysis)
ECON 339: intra-household distribution II: bargaining
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Method?
• Two individuals: Jane and Fred • Step one: what can Jane and Fred accomplish as
singles? – Depends on individual ppf’s*, and individual
preferences – Can determine individual utilities, if single
– Optimal choices if single determine their
respective “threat points” in the bargaining game within marriage
*Recall model of “who does what” in household
ECON 339: intra-household distribution II: bargaining
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Method II • Step 2: possibilities as couple
• From individual ppfs, can derive joint ppf between household and market produced goods.
• Can derive a “utilities possibility set” that reflects combinations of Jane’s utility and Fred’s utility that are feasible given the combined resources.
ECON 339: intra-household distribution II: bargaining
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Utility possibility frontier
• Suppose Jane has control of all household resources, and can choose time allocations for herself and Fred to produce the combination of {H,C} giving her the maximum possible utility, ignoring Fred: label this utility level
• Same exercise for Fred gives
• Now suppose Jane is constrained to provide Fred with some positive level of utility. Given control over household resources, what is the max utility Jane can obtain?
• Trace out upf by considering higher and higher levels of utility for Fred.
Max
Jane U
Max
Fred U
ECON 339: intra-household distribution II: bargaining
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upf
• Upf will have negative slope – trading off one person’s utility against the other’s (given efficient use of household resources and (some) private goods)
• Shape will depend on individual preferences and technology…
• Nothing lost here in considering linear upf.
• Where is threat point relative to upf? Why?
ECON 339: intra-household distribution II: bargaining
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Fred’s utility
0
ECON 339: intra-household distribution II: bargaining
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Jane’s utility singleJU
single F
U
Bargaining threat point? Best each party can do if don’t reach agreement: i) Single (divorce, never marry) or ii) Non-cooperative solution within
marriage.
Each gets at least this much in solution.
Upf and bargaining outcome
• Where will households end up?
– 1. each gets at least threat point utility – otherwise, why marry/stay married?
– 2. division of surplus: most common approach is the “Nash bargaining solution”
• Note: this focusses on the outcome of bargaining, not on the process.
ECON 339: intra-household distribution II:
bargaining 9
Nash bargaining solution
• Nash’s approach: what are desirable properties of solution to the bargaining problem?
– Efficient: on the upf
– Outcome does not change if rescale the utilities in a linear manner
– If the utilities possibility set changes such that some outcome that would never be chosen is removed, outcome is not affected
– symmetry
ECON 339: intra-household distribution II: bargaining
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Outcome satisfying all four properties?
Unique: Characterized by utility pair that maximizes the “Nash product”: product of the surpluses
• Exponents reflect “bargaining strengths”
– Nash’s original model assumed equal bargaining strengths
Extension; exponents need not be equal, if clear inequality in bargaining power (as opposed to threat points)
ECON 339: intra-household distribution II: bargaining
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0.5 0.5 max( ) ( )
threat threat
j j f f U U U U
Fred’s utility
0
ECON 339: intra-household distribution II: bargaining
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Jane’s utility singleJU
single F
U
Nash bargaining solution: i) On upf, where ii) Iso-product curve is tangent to upf
sin sinChoose ( , )to max( - )( - )
on the feasible set. Solution is
gle gle J F J J F F
U U U U U U
Nash bargaining solution: • Calculating the solution with linear ppf (text pp74-5)
• 1. With a linear ppf, the sum of the utilities of the two individuals is constant – that is, Suppose k = 100, then
• 2. Solution is the pair that maximizes the Nash product
• Can rewrite this problem as choosing to maximize
• If Nash product=0
. j f
U U k
( , ) j f
U U
single single ( )( ) subject to 100
j j f f j f U U U U U U
100 f j j
U k U U
j U
single single ( )(100 )
j j j f U U U U
single single , or 100 ,
j j f j f U U U U U
• If each partner’s utility in the solution > their utility if single, the Nash product has a positive value.
• Example: suppose the highest utility each partner could obtain if single is = 15.
• (Notice: 15+15 < 100, so the threatpoint is inside the upf for marriage. The marriage surplus=100-30 = 70.)
• Then the problem is to find the utility level for J that maximizes the function
• Solution .
• Each partner receives their threatpoint utility (=15) plus ½ the surplus from marriage.
2 2
( 15)((100 ) 15) ( 15)(85 )
85 15 15 85 100 15 85
j j j j
j j j j j
U U U U
U U U U U
50, so 100 50 j f j
U U U
Notes on the math on the previous page
• 1. Notice that I did not bother solving the 15x85 the value of this is irrelevant to the solution: it is a constant, and the value of that maximizes the rest of the expression will also maximize the entire expression.
• 2. I used basic calculus to find the value of J’s utility that maximizes the Nash product. (The value of the Nash product itself is not important (or even interesting).) If you don’t know calculus, you can use another method: see http://www.youtube.com/watch?v=wgn5x0Tm4CE
• (if you look at this, you will see that the formula he uses is the one derived from the calculus solution)
j U
• each partner obtains their threatpoint utility,
plus
a share of the surplus
- share depends on bargaining power;
- if identical bargaining weights, each gets
1/2 of surplus
ECON 339: intra-household distribution II: bargaining
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Effects of changing threatpoint I?
• Suppose Fred’s threatpoint utility increased, and Jane’s threatpoint utility decreased by the same amount. This leaves the upf unchanged, and there is also no change in the marriage surplus.
• Effects? • 1. Fred has higher utility in marriage → Jane has lower utility in marriage (since sum of surplus has not changed) 2. Each still gets threatpoint utility + half of surplus. Individuals have incentives to make threatpoint utility as high as possible.
ECON 339: intra-household distribution II:
bargaining 17
• Suppose now that F could obtain utility = 20 if single, and the best J could do if single is utility = 10. .
• Keep the sum of the utilities possible within marriage = 100, so the marriage surplus is still 100 – 30 = 70.
• With these threatpoint utilities, the solution to the Nash bargaining problem is the value of that maximizes
• Again, each partner obtains their threatpoint utility plus ½ of the marriage surplus (=70/2=35).
j U
2
( 10)((100 ) 20) ( 10)(80 )
90 15 85 U 45, 55
j j j j
j j j f
U U U U
U U
U
Effects of changing threatpoint II?
• Suppose Jane’s threatpoint utility increased, with no change in upf or Fred’s threatpoint utility. Effects?
• 1. bargaining surplus decreases
• 2. Jane has higher utility in marriage
→Fred has lower utility in marriage
Individuals have incentives to make threatpoint utility as high as possible.
ECON 339: intra-household distribution II: bargaining
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Fred’s utility
0
ECON 339: intra-household distribution II: bargaining
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Jane’s utility singleJU
single F
U
Nash bargaining solution: i) On upf, where ii) Iso-product curve is tangent to upf
sin sinChoose ( , )to max( - )( - )
on the feasible set. Solution is
gle gle J F J J F F
U U U U U U
Outcome of cooperative bargaining
• Is this fixed throughout marriage?
• What variables will affect outcome? How?
- welfare payments to divorced mothers
- laws on division of marital property
- support payments
• Alternative threatpoint: non-cooperative equilibrium within marriage
ECON 339: intra-household distribution II: bargaining
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