Transcript Extensive Form - London School of Economics

Prerequisites
Almost essential
Welfare: Efficiency
Adverse selection
Frank Cowell: Microeconomics
August 2006
Non-convexities
MICROECONOMICS
Principles and Analysis
Frank Cowell
Introduction
Frank Cowell: Microeconomics
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What are non-convexities?
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Concerned with production…
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drop the convenient divisibility assumption
potentially far-reaching consequences
Approach:
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…awkward name
…crucial concept
start with examination of economic issues
build a simple production model
examine efficiency implications
consider problems of implementation and policy
Terms other than “non-convexities” sometimes used…
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…not always appropriately
but can give some insight on to the range of issues:
Other terms…?
Frank Cowell: Microeconomics
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“Increasing returns”
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“Natural monopoly”
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but issue arises regardless of market form…
… not essentially one of industrial structure
“Public utilities”
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but increasing returns everywhere are not essential
but phenomenon is not necessarily in the public sector
None of these captures the concept exactly
We need to examine the economic issues more closely…
Non-convexities
Overview...
Frank Cowell: Microeconomics
The issues
The nature of
non-convexities
Basic model
Efficiency
Implementation
Issues: the individual firm
Frank Cowell: Microeconomics
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Consider supply by competitive firms
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If there are lots of firms
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average supply is approximately continuous
so we can get demand=supply at industry level
If there is in some sense a “natural monopoly”
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upward-sloping portion of MC curve
supply discontinuous if there is fixed cost
perhaps very large fixed cost?
perhaps MC everywhere constant/falling?
no supply in competitive market?
In this case….
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how does the firm cover costs?
how “should” the firm behave?
how can it be induced to behave in the required manner?
Issues: efficient allocations
Frank Cowell: Microeconomics
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Related to the issue discussed for firm
Concerns implementation through the market
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Relationship between CE and efficiency
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non-convexities seen an aspect of “market failure”?
consider reason for this…
…and a solution?
fundamental to welfare economics
examine key questions of implementation
First a simple example of how it works…
Implementation through the market
Frank Cowell: Microeconomics
Uh(xh) = Uh(x*h)
f
q2
h
x2
Ff(qf) = 0
q*f

p1
—
p2
f
q1``
 all f and h optimise at these prices such that
…for all pairs of goods MRS = MRT= price ratio
Production possibilities
Firm f max profits
U contour
h min expenditure

p1
—
p2
x*h
x1`h`
now for the two
key questions.
Frank Cowell: Microeconomics
Efficiency and the market: key
questions
1.
Is a competitive equilibrium efficient?
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2.
Can an arbitrary Pareto-efficient allocation be supported
by a competitive equilibrium?
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Yes if all consumers are greedy, there is no hidden information,
and there are no externalities
Yes if all consumers are greedy, there is no hidden information,
there are no externalities and no non-convexities
If there are non-convexities the equilibrium price signals
could take the economy away from the efficient allocation
Non-convexities
Overview...
Frank Cowell: Microeconomics
The issues
Back to the
firm….
Basic model
Efficiency
Implementation
A model of indivisibility (1)
Frank Cowell: Microeconomics
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Take simplest model of production:
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The indivisibility:
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a single output (q)…
…from a single input (z)
A fixed amount of input required before you get any output
Otherwise production is conventional
q = f(z − k) , z ≥ k
f(0) = 0, fz(∙) > 0, fzz(∙) ≤ 0
q = 0, z < k
Given a required amount of output q > 0…
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minimum amount of z required is:
f−1(q) + k
A model of indivisibility (2)
Frank Cowell: Microeconomics
q
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Case 1
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z
0
k
q
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Case 2
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0
The minimum input
requirement
fz(∙) > 0, fzz(∙) < 0
Attainable set
z
k
The minimum input
requirement
fz(∙) > 0, fzz(∙) = 0
Attainable set
A model of indivisibility (3)
Frank Cowell: Microeconomics
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Suppose units of input can be bought for w
What is cost of output q?
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clearly C(w, 0) = 0 and
C(w,q) = v(w,q) + C0, for q > 0,
where variable cost is v(w,q) = wf−1(q)
and fixed cost is C0 = wk
Therefore:
marginal cost: w / fz(f−1(q))
 average cost: wf−1(q) / q + C0 / q
 In the case where is f a linear function
 f−1(q) = aq
 marginal cost: aw
 average cost: aw + C0 / q
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Marginal cost is constant or increasing
Average cost is initially decreasing
A model of indivisibility (4)
Frank Cowell: Microeconomics
p
Case 1
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q
0
p
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Case 2
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0
Average cost
Marginal cost
Supply of competitive firm
q
Average cost
Marginal cost
Supply of competitive firm
“Natural Monopoly”
Frank Cowell: Microeconomics
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Subadditivity
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Natural monopoly
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C(w, q + q) < C(w, q) + C(w, q)
Apply the above inequality…
C(w, 2q) < 2C(w, q)
And for any integer N > :1
C(w, Nq) < NC(w, q)
Cheaper to produce in a single plant rather than two
identical plants
But subadditivity consistent with U-shaped average cost
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Does not imply IRTS
Non-convexity: the economy
Frank Cowell: Microeconomics
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Now transfer this idea to the economy as a whole
Use the same type of production model
An economy with two goods
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Good 1. A good with substantial setup costs
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Good 2. All other goods
Assume:
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Rail network
Gas supply system
Electricity grid
a given endowment of all good 2
good 1 is not essential for survival
Consider consumption possibilities of two goods x1, x2
Fundamental non-convexity (1)
Frank Cowell: Microeconomics
Endowment of good 2
Fixed set-up cost to produce good 1
Possibilities once fixed-cost has
been incurred
x2
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x°
Attainable set is shaded
area + “spike”
Endowment point x° is
technically efficient
x1
0
Fundamental non-convexity (2)
Frank Cowell: Microeconomics
Endowment of good 2
Fixed set-up cost to produce good 1
Possibilities once fixed-cost has
been incurred
x2
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x°
 MRT is everywhere
constant
Again endowment point
x° is technically efficient
x1
0
Non-convexities
Overview...
Frank Cowell: Microeconomics
The issues
An extension of
the basic rules of
thumb
Basic model
Efficiency
Implementation
Competitive “Failure” and
Efficiency
Frank Cowell: Microeconomics
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Characterisation
problem:
Implementation
problem:
Requires a modification of
first-order conditions
Involves intervention in, or
replacement of, the market
Usually achieved through
some “public” institution or
economic mechanism
Efficiency: characterisation
Frank Cowell: Microeconomics
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Two basic questions:
Should good 1 be produced at all?
If so, how much should be produced?
The answer depends on agents’ preferences
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assume…
…these represented by conventional utility function
…all consumers are identical
Method:
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use the simple production model
examine efficiency in two cases…
…that differ only in representative agent’s preferences
Efficiency characterisation: case 1
Frank Cowell: Microeconomics
Attainable set as before
Reservation indifference curve
Indifference map
Point where MRS=MRT
Efficient point
x2
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x°
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Attainable set is shaded
x′
area + “spike”
 In this case MRS=MRT
x1
0
is not sufficient
 Utility is higher if x1 = 0
Efficiency characterisation: case 2
Frank Cowell: Microeconomics
Attainable set as before
x2
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 Indifference map
 Consumption if none of good 1 is
produced
 The efficient point
x°
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x′
x1
0
Non-convexities
Overview...
Frank Cowell: Microeconomics
The issues
The market and
alternatives
Basic model
Efficiency
Implementation
•Full information
•Asymmetric information
Efficiency: implementation
Frank Cowell: Microeconomics
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Move on from describing the efficient allocation
What mechanism could implement the allocation?
Consider first the competitive market:
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Then consider a discriminating monopoly
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Assume given prices…
…profit-maximising firm(s)
Allow nonlinear fee schedule
Then consider equivalent regulatory model
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Maximise social welfare…
… by appropriate choice of regulatory régime
Frank Cowell: Microeconomics
Nonconvexity: effect of the
competitive market
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x2
Efficient to produce where
MRS=MRT
Iso-profit-line
x°
Profit-maximisation over the
attainable set

x′
p1
—
p2
x1
0
Frank Cowell: Microeconomics
Nonconvexity: efficient fee
schedule
Efficient to produce at x'
x2
MRS=MRT
Fixed charge
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x°
Variable charge
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x′
p1
—
p2
x1
0
Implementation: problem
Frank Cowell: Microeconomics
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Situation
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Competitive “solution”:
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Clearly inefficient…
…monopoly would force price of good 1 above MC
Discriminating monopoly
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Firms maximise profits by producing x1 = 0 at these prices.
Goodbye Railways?
Simple monopoly:
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U(x′) > U(x°) : x′ is optimal
Prices at x′ given by MRS
A combination of fixed charge…
…plus linear variable charge
How to implement this?
Implementation: analysis
Frank Cowell: Microeconomics
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Set up as a problem of regulating the firm
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Regulator can:
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observe quantity of output
grant a subsidy of F
F is raised from consumers
through non-distortionary taxation?
Criterion for regulator
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produces output q of good 1
values denominated in terms of good 2
a measure of consumer welfare
the firm’s profits
Take case where regulator is fully informed
Regulation model: the firm
Frank Cowell: Microeconomics
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There is a single firm – regulated monopoly
Firm chooses output q, given
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The firm’s revenue is given by
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R = p(q) q + F
Firm’s profits are
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price-per unit of output p(q) allowed by regulator
fixed payment F
costs C(q)
P = R  C(q)
Firm seeks to maximise P subject to regime fixed by
regulator
Regulation model: the regulator
Frank Cowell: Microeconomics
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Regulator can fix
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But, given the action of the firm
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revenue is R = p(q) q + F
choosing q to max profits
…fixing p(∙) and F is equivalent to fixing
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price per unit p
fixed payment to firms F
firm’s output q
firm’s revenue R
So transform problem to one of regulator choosing (q, R)
Regulation model: objectives
Frank Cowell: Microeconomics
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Assume consumers are identical
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take a single representative consumer
consumes x1 = q
Assume zero income effects
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so take consumer’s surplus (CS) as a measure of welfare
q
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Note properties of CS(∙):
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CSq(q, R) = p(q)
CSR(q, R) =  1
Social valuation taken a combination of welfare and profits:
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CS(q, R) = ∫0 p(x) dx  R
V(R, q) = CS(q, R) + b [R  C(q)]
b<1
Note derived properties of V(∙):
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Vq(q, R) = p(q)  bCR(q)
VqR(q, R) =  1 + b
Regulation model: solution
Frank Cowell: Microeconomics
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Problem is choose (q, R) to max V (q, R) subject to R  C(q) ≥ 0
Lagrangean is
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If “*” denote maximising values, first-order conditions are
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1− b + l* = 0
p(q*)  bCR(q*) − l* Cq(q*) = 0
Clearly l* = 1− b > 0 and from the FOCs
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Vq(q*, R*) − l*Cq(q*) = 0
VR(R*, q*) + l* = 0
l* [R  C(q*)] = 0
Evaluate using the derivatives of V:
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V(q, R) + l [R  C(q) ]
R* = C(q*)
p(q*) = CR(q*)
So the (q* , R *) programme induces a zero-profit, efficient outcome
Provisional summary
Frank Cowell: Microeconomics
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Characterisation
problem:
Implementation
problem:
supplement the MRS = MRT
rule by a "global search" rule
for the optimum.
Set user prices equal to marginal
cost
Cover losses (from fixed cost)
with non-distortionary transfer
Don't leave it to the unregulated
market....
Non-convexities
Overview...
Frank Cowell: Microeconomics
The issues
Regulation…
Basic model
Efficiency
Implementation
•Full information
•Asymmetric information
The issue
Frank Cowell: Microeconomics
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By hypothesis there is only room for only one firm
The efficient payment schedule requires
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However, implementation of this is demanding
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A per-unit payment such that P = MC
A fixed amount required to ensure break-even
requires detailed information about firm’s costs
by hypothesis, there isn't a pool of firms to provide estimates
To see the issues, let’s take a special case
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Two possible types of firm
Known probability of high-cost/low-cost type
Low-cost type
Frank Cowell: Microeconomics
Preferences
x2
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Efficient to produce where
MRS=MRT
Amount of good 1 produced
x°F'
Efficient payment schedule
 q =x1' is the amount that

x'
p
0
x1
x1'
the regulator wants the
low-cost type to produce
F' is the (small) fixed
charge allowed to the lowcost type by the regulator
p is the variable charge
allowed to low-cost type by
the regulator (=MC)
High-cost type
Frank Cowell: Microeconomics
Preferences
x2
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Efficient to produce where
MRS=MRT
Amount of good 1 produced
x°F''
Efficient payment schedule
Essentially same story as
before
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But regulator allows the
x''
high-cost type the large
fixed charge F''
0
x1
x1''
Misrepresentation
Frank Cowell: Microeconomics
Production possibilities and
solution for low-cost type
Production possibilities and
solution for high-cost type
Outcome if low-cost type
masquerades as high-cost type
x2
High-cost type is allowed

x''

higher fixed charge than
low-cost type
x'
Low-cost type would like
to get deal offered to highcost type
0
x1
x1''
x1'
Second-best regulation: problem
Frank Cowell: Microeconomics
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Regulator is faced with an informational problem
Must take into account incentive compatibility
Design the regime such that two constraints are satisfied
Participation constraint
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Incentive compatibility constraint
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firm of either type will actually want to produce positive output
must at least break even
neither firm type should want to masquerade as the other…
…in order to profit from a more favourable treatment
each type must be allowed to make as much profit as if it were
mimicking the other type
Requires a standard adaptation of the optimisation problem
Second-best regulation: solution
Frank Cowell: Microeconomics
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Model basics
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low-cost firm is a-type – cost function Ca(∙)
high-cost firm is b-type – cost function Cb(∙)
probability of getting an a-type is p
objective is EV(q, R) = pV(qa, Ra) + [1−p]V(q, Rb)
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Regulator chooses (qa, qb, Ra, Rb) to max EV(q, R) s.t.
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Lagrangean is
Rb  Cb(qb) ≥ 0
Ra  Ca(qa) ≥ Rb  Ca(qb)
pV(qa, Ra) + [1−p]V(q, Rb)
+ l [Rb  Cb(qb) ]
+ m [Ra  Ca(qb)  Rb + Ca(qa) ]
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Get standard second-best results:
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type a: price = MC, makes positive profits
type b: price > MC, makes zero profits
Conclusion
Frank Cowell: Microeconomics
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May give rise to inefficiency if we leave everything to the
market
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So the goods may be produced in the public sector
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if there are non-convexities,,,
…separation result does not apply
but they are not “public goods” in the conventional sense
public utilities?
Could private firms implement efficient allocation?
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for certain goods – a monopoly with entrance fee
may be able to implement through pubic regulation
but may have to accept second-best outcome