A Rational Model for Earnings Benchmarks
Luca Ashok
Stanford University
Lashok1@stanford.edu
September, 2013
Abstract
I present a rational model of earnings management as an explanation for why the market assigns
a premium for firms achieving key earnings benchmarks. I study an infinite period model where
in each period firms are probabilistically chosen to have either high or low discretion. Firms with
high discretion have the option to either manipulate earnings downward, and have the
manipulation reverse next period, or to do nothing at all. I show that, under certain conditions,
equilibrium exist where achieving a benchmark earns a market premium and the market assigns
the premium rationally.
1
1. Introduction
Benchmarks play an important role in managers’ decision making. Managers are
typically concerned with meeting analyst forecasts, meeting earnings of prior periods, and
avoiding losses (Moehrle, 2002).
In many instances managers have been found to manipulate earnings to reach key
benchmarks. Brown (2001) finds that over time firms have shifted towards meeting or just
beating analyst forecasts, implying an increase in earnings management. Kasznik and McNichols
(2002) find that the market assigns a premium to firms that meet analyst’s expectations, giving
managers incentives to reach these benchmarks. Burgstahler and Dichev (1997) find a large
amount of firms that just beat last period’s earnings or report a small positive gain. They also
find unusually low frequencies of firms reporting just below last period’s earnings or small
losses, implying that firms manage earnings when close to these benchmarks. Barth et al. (1999)
find that the market rewards firms even more for achieving consecutive periods of increased
earnings. Managers also have incentives to achieve benchmarks in the form of CEO pay.
Matsunaga and Park (2001) find that CEO bonus payments incentivize managers to meet both
analyst forecasts and earnings from prior periods.
The accounting literature has studied three main ways that managers manipulate earnings
to reach these benchmarks: Accrual manipulation, expectations management, and real activities
manipulation (Doyle et al. 2013). My model focuses only on accrual manipulation, specifically
when a firm lowers earnings in the current period through accounting reserves so that it can
report higher than normal earnings in future periods.
2
In my setting each firm is probabilistically assigned a firm value and an amount of
discretion in every period. The firm value of each firm is drawn from a finite, discrete
distribution and each firm’s discretion can either be low or high. High discretion firms can
choose to exercise an option where they misreport their earnings one lower than their true firm
value. In exchange, the firm can misreport their earnings one higher in the following period.
Firms that choose to exercise this option trade off the difference between their “true”
market value (the market value if they report truthfully) and the market value of the report one
lower in exchange for a higher report in the future. Therefore firms will choose to exercise if and
only if the difference between the market value of a truthful report and the report one lower is
sufficiently low.
I try to show the existence of equilibrium where the market puts a premium on firms that
report above a certain threshold. High discretion firms just above the benchmark will choose to
exercise since the difference between their truthful report and the benchmark report is
sufficiently low. These high value firms making the benchmark earnings report raise the
expected firm value for that specific report maintaining the market premium.
The main characteristic of the model that allows for this result is that firms cannot predict
their types in the next period. Since they only have the potential option of lowering earnings now
so that they can increase them next period, the firms which misreport higher earnings will be
spread out evenly throughout all firm types. Meanwhile only firms which find themselves at an
opportune point to exercise (firms just above a benchmark) will choose to misreport lower
earnings. This pattern of earnings management allows for market premiums on firms which
reach a benchmark.
3
The rest of the paper is organized as follows. Section 2 discusses related theoretical
literature. Section 3 presents the model and solves for specific equilibrium. Section 4 discusses a
numerical example. Section 5 discusses limitations of the model. Section 6 discusses empirical
implications of the model and Section 7 concludes. Appendix A contains proofs.
2. Related theoretical literature
Guttman, Kandel, and Kadan (2006) offer an alternative explanation for a discontinuity in
the distribution of earnings around a key benchmark. In their model they have a more standard
earnings management setting similar to Fischer and Verrecchia (2000). They find that a
discontinuity in earnings may be due to a special form of equilibrium where there is a pooling
interval. A key difference in their model is that managers face a personal cost for misreporting
that increases in the amount of earnings management taking place. Their results are driven by
investor’s beliefs that all firms in the pooling interval will make the same report. Managers trade
off the higher stock price from increasing their report against the personal cost of earnings
management.
In my setting the result is driven from a different process. Here only high discretion firms
can manipulate, and all firms that manipulate do so by the same amount and without personal
cost to the manager. Managers that choose to exercise trade off a loss in current period stock
price in exchange for a future gain in stock price.
4
3.1 The Model
Each period firms are assigned randomly to a firm value ∈ {0,1, … , 𝑁 − 2, 𝑁 − 1}
according to a probability distribution 𝑑. The market is risk neutral and cannot distinguish
between two firms making the same report. Therefore it assigns a value 𝑀(𝑥) =
𝐸[𝑓𝑖𝑟𝑚 𝑣𝑎𝑙𝑢𝑒|𝑟𝑒𝑝𝑜𝑟𝑡 𝑥] 𝑓𝑜𝑟 𝑥 ∈ {0,1,2 … 𝑁}.
Firms are assigned high or low discretion according to the following function:
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝐻𝑖𝑔ℎ 𝐷𝑖𝑠𝑐𝑟𝑒𝑡𝑖𝑜𝑛 = {
𝑝
0
𝑖𝑓 𝑓𝑖𝑟𝑚 𝑑𝑖𝑑 𝑛𝑜𝑡 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 𝑙𝑎𝑠𝑡 𝑝𝑒𝑟𝑖𝑜𝑑
𝑖𝑓 𝑓𝑖𝑟𝑚 𝑑𝑖𝑑 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑒 𝑙𝑎𝑠𝑡 𝑝𝑒𝑟𝑖𝑜𝑑
Firms with high discretion can choose to report one lower than their true value in
exchange for an automatic reversal the next period (they must report one higher than their true
value the next period). 𝑁 is greater than four and there is a discount rate 𝑟.
Each high discretion firm faces a choice to either report truly or report lower in exchange
for a higher report the next period. The expected value of exercising the option minus the
expected value of not exercising the option is the following (for a firm of type x):
𝑁−1
− 𝑀(𝑥) + 𝑀(𝑥 − 1) − 𝜀
𝑁(1 + 𝑟)
Where 𝜀 is the expected gain from having the potential of high discretion next period.
If this value is greater than zero, the firm will exercise its option, otherwise it will report
truthfully. Another way to state this is:
𝑁−1
A firm x will exercise if and only if 𝑀(𝑥) − 𝑀(𝑥 − 1) < 𝑁(1+𝑟) − 𝜀
5
𝑁−1
Notice that 𝑁(1+𝑟) − 𝜀 is a constant. Therefore this is the constant threshold value of 𝑀(𝑥) −
𝑀(𝑥 − 1) that determines whether or not firm x will exercise.
3.2 Equilibrium Conditions
1) 𝑀(𝑥) = 𝐸[𝑓𝑖𝑟𝑚 𝑣𝑎𝑙𝑢𝑒|𝑟𝑒𝑝𝑜𝑟𝑡 𝑥] 𝑓𝑜𝑟 𝑥 ∈ {0,1,2 … 𝑁}.
𝑁−1
2) 𝑀(𝑥) − 𝑀(𝑥 − 1) < 𝑁(1+𝑟) − 𝜀 𝑓𝑜𝑟 𝑎𝑙𝑙 ℎ𝑖𝑔ℎ 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑖𝑜𝑛 𝑓𝑖𝑟𝑚𝑠 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑖𝑛𝑔
𝑁−1
3) 𝑀(𝑥) − 𝑀(𝑥 − 1) > 𝑁(1+𝑟) − 𝜀 𝑓𝑜𝑟 𝑎𝑙𝑙 ℎ𝑖𝑔ℎ 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑖𝑜𝑛 𝑓𝑖𝑟𝑚𝑠 𝑛𝑜𝑡 𝑒𝑥𝑒𝑟𝑐𝑖𝑠𝑖𝑛𝑔
3.3 Equilibrium with a single benchmark
Suppose we have an equilibrium with a single benchmark Y ∈ {0,1 … 𝑁} such that the
market assigns a premium for that benchmark and for no others. (We will also assume 𝑌 > 1.) I
will define a premium as the market valuing the report for more than the actual report made. I
will solve for the conditions (𝑝, 𝑁, 𝑟, 𝑑) such that this type of equilibrium can be sustained.
This can only happen in equilibrium when firms of type 𝑌+1 with high discretion choose
to report down. It will suffice to show that the equilibrium price 𝑀(𝑌) > 𝑌, and that the only
high discretion firm that will exercise the option to report lower is the 𝑌+1 firm.
I will assume that only high discretion firms of type 𝑌+1 will exercise and then verify
that this is in fact an equilibrium.
6
Given the behavior of each type of firm, we can deduce the market value for each type of
report. There are five different types of reports to calculate. The highest report (𝑁+1), the lowest
report (0), the benchmark report (𝑌), the report just above the benchmark (𝑌+1), and all the rest.
𝑀(𝑁 + 1) = 𝑁, since only firms of type 𝑁 can report 𝑁+1
𝑀(0) = 0, since only firms of type 0 will report 0
Let 𝑃 be the proportion of firms that will be reporting higher this period because they exercised
last period.
𝑃=
𝑀(𝑌) =
𝑝 ∗ 𝑑(𝑌 + 1)
1 + 𝑝 ∗ 𝑑(𝑌 + 1)
(1 − 𝑃)𝑑(𝑌)𝑌 + 𝑝(1 − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1) + 𝑃𝑑(𝑌 − 1)(𝑌 − 1)
(1 − 𝑃)𝑑(𝑌) + 𝑝(1 − 𝑃)𝑑(𝑌 + 1) + 𝑃𝑑(𝑌 − 1)
𝑀(𝑌 + 1) =
𝑃𝑑(𝑌)𝑌 + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1)
𝑃𝑑(𝑌) + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1)
𝑀(𝑥) =
𝑃𝑑(𝑥 − 1)(𝑥 − 1) + (1 − 𝑃)𝑑(𝑥)𝑥
𝑃𝑑(𝑥 − 1) + (1 − 𝑃)𝑑(𝑥)
Next we check all the possible values of 𝑀(𝑥) − 𝑀(𝑥 − 1):
There is a general form and then four different forms when x is close to Y or to 0.
The general form is:
𝑀(𝑥) − 𝑀(𝑥 − 1)
=
𝑑(𝑥)(1 − 𝑃)𝑥 + 𝑑(𝑥 − 1)𝑃(𝑥 − 1)
𝑑(𝑥)(1 − 𝑃) + 𝑑(𝑥 − 1)𝑃
−
𝑑(𝑥 − 1)(1 − 𝑃)(𝑥 − 1) + 𝑑(𝑥 − 2)𝑃(𝑥 − 2)
𝑑(𝑥 − 1)(1 − 𝑃) + 𝑑(𝑥 − 2)𝑃
7
The four corner cases are:
𝑀(1) − 𝑀(0) =
𝑑(1)(1 − 𝑃) + 𝑑(0)𝑃 ∗ 0
𝑑(1)(1 − 𝑃)
−0 =
𝑑(1)(1 − 𝑃) + 𝑑(0)𝑃
𝑑(1)(1 − 𝑃) + 𝑑(0)𝑃
𝑀(𝑌 + 2) − 𝑀(𝑌 + 1)
=
𝑑(𝑌 + 2)(1 − 𝑃)(𝑌 + 2) + 𝑑(𝑌 + 1)𝑃(𝑌 + 1)
𝑑(𝑌 + 2)(1 − 𝑃) + 𝑑(𝑌 + 1)𝑃
−
𝑃𝑑(𝑌)𝑌 + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1)
𝑃𝑑(𝑌) + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1)
𝑀(𝑌 + 1) − 𝑀(𝑌)
=
𝑃𝑑(𝑌)𝑌 + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1)
𝑃𝑑(𝑌) + (1 − 𝑝(1 − 𝑃) − 𝑃)𝑑(𝑌 + 1)
−
(1 − 𝑃)𝑑(𝑌)𝑌 + 𝑝(1 − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1) + 𝑃𝑑(𝑌 − 1)(𝑌 − 1)
(1 − 𝑃)𝑑(𝑌) + 𝑝(1 − 𝑃)𝑑(𝑌 + 1) + 𝑃𝑑(𝑌 − 1)
𝑀(𝑌) − 𝑀(𝑌 − 1)
=
(1 − 𝑃)𝑑(𝑌)𝑌 + 𝑝(1 − 𝑃)𝑑(𝑌 + 1)(𝑌 + 1) + 𝑃𝑑(𝑌 − 1)(𝑌 − 1)
(1 − 𝑃)𝑑(𝑌) + 𝑝(1 − 𝑃)𝑑(𝑌 + 1) + 𝑃𝑑(𝑌 − 1)
−
𝑑(𝑌 − 1)(1 − 𝑃)(𝑌 − 1) + 𝑑(𝑌 − 2)𝑃(𝑌 − 2)
𝑑(𝑌 − 1)(1 − 𝑃) + 𝑑(𝑌 − 2)𝑃
In order to have such an equilibrium we need the following conditions to hold:
1) 𝑀(𝑌) > 𝑌
𝑁−1
2) 𝑀(𝑌 + 1) − 𝑀(𝑌) < 𝑁(1+𝑟)
𝑁−1
3) 𝑀(𝑥) − 𝑀(𝑥 − 1) > 𝑁(1+𝑟) − 𝜀
∀𝑥 ≠𝑌+1
8
PROPOSITION 1: There exists an equilibrium with a single benchmark for uniform 𝑑 and for
any 𝑝. A sufficient condition for such an equilibrium is:
1 − 𝑝(1 − 𝑃)(1 + 2𝑃)
𝑁−1
<
< 1−𝑃
2
2
1 − 𝑝 (1 − 𝑃)
𝑁(1 + 𝑟)
4. Numerical Example
1) There are 5 firm types: -2, -1, 0, 1, and 2.
2) There are 6 possible reports: -2, -1, 0, 1, 2, and 3.
3) The probability of having high discretion is .5, and there is no rate of return.
The market value of -2 will be -2 since only firms of type -2 will report -2.
10
1
12
The market value of -1 will be: −1 ∗ 11 + −2 ∗ 11 = − 11
1
10
5
11
The market value of 0 will be: (−1 ∗ 11 + 0 ∗ 11 + 1 ∗ 11 ) 16 =
5
1
The market value of 1 will be: (1 ∗ 11 + 0 ∗ 11)
1
11
6
10
The market value of 2 will be: 1 ∗ 11 + 2 ∗ 11 =
=
1
4
5
6
21
11
The market value of 3 will be 2 since only firms of type 2 will report 3.
Next we check if all the choices made are in fact rational:
4
The expected gain from reporting one more than true value in the next period is 5. The
1
expected gain from doing nothing is 10 times the gain from reporting down when you are of type
7
1 and of high discretion. The loss from reporting down when you are of type 1 is 12.
9
So this will be in fact the rational choice if
4
7
1
4
7
− 12 > 10 ∗ (5 − 12) or if
5
4
7
− 12 > 0.
5
Which is in fact the case.
4
Finally we must check that no other type will lose less than 5 from reporting down one.
This is in fact the case.
A possible way to interpret this example is a firm that has either negative, slightly
negative, zero, slightly positive, or positive earnings. The gap in market price is smallest between
a report of zero and slightly positive gain. Therefore firms with a slightly positive gain and high
discretion will choose to exercise their option and report zero earnings for a chance to report
higher in the next period. Because so many firms with a true value of slightly positive gain are
actually reporting zero earnings, the market price for firms which report zero earnings is above
zero and sufficiently high to maintain the equilibrium.
5. Limitations of the Model
The model leaves out many details both for simplicity and tractability. For example the
distribution of firm values is assumed to be uniform and the market is assumed to be risk neutral.
Some of the other details left out are more worrisome, however.
The model assumes that the market cannot distinguish between firms making the same
report. This is assumed for tractability so that the market can simply price the firm based on its
report without having to condition on the past. This does not line up perfectly with reality since
in practice the market is generally able to connect past reports with current ones for any firm.
10
If the market were able to condition on the past, for example if it could look back and see
every firm’s report last period, it would be able to pick out the firms which were more likely to
have exercised in the last period thus reducing the benefit of exercising the option. This could
either weaken or completely reverse the results.
Another worrisome limitation is that firms are forced to immediately reverse the
manipulation. In practice firms can choose to reverse their manipulations at a time which they
most need it. This may not qualitatively affect the results of the model. For example if 𝑁 is large
and there is no reason to wait a period since it will be too unlikely to be in a better position next
period. Alternatively, if the discount rate is high then firms may almost never want to wait for
the next period.
Finally, firms with high discretion only have the option to first misreport down and then
misreport up the next period. In reality, we might expect firms with high discretion to also have
the capability of doing the opposite. We might expect them to be able to first misreport up and
then misreport down. If it were equally easy for a firm to do so (misreport up then down), then
the market premium would vanish. One possible justification of the model is that if it were
significantly easier for firms to first misreport down and then up, the results of the model would
not change qualitatively.
6. Empirical Implications of the Model
The model predicts that if there is a discontinuity around a single earnings benchmark, it is
the product of firms just above the benchmark creating an accounting reserve and reporting low.
This would correspond to the following empirical predictions:
11
1) A larger frequency of reports at or just above an earnings benchmark (relative to a
scenario with no earnings management).
2) A drop in frequency of reports somewhere above and close to the earnings benchmark
(relative to a scenario with no earnings management).
3) Low discretionary accruals for firms at or just above an earnings benchmark.
Suppose the benchmark is report 𝑌. The first two points come from the fact that a
relatively large amount of firms of type 𝑌+1 will be exercising and reporting 𝑌 instead. Of
course this is the result of a discrete distribution and real earnings are more easily described in
terms of continuous distributions. Another way to summarize the first two points is to say that
the model predicts, relative to a scenario with no earnings management, for an interval just above
the earnings benchmark, a shift in earnings towards the benchmark.
The third point is perhaps the most counterintuitive. The common story is that firms just
below an earnings benchmark manage earnings through increased discretionary accruals to reach
the benchmark. You would then expect relatively high discretionary accruals for firms just at or
above an earnings benchmark. Burgstahler and Dichev (1997) were looking for this effect but
instead found the opposite. They found that “other accruals” were lower for the small profit
group than for the small loss group. This finding is consistent with my model.
7. Conclusion
This paper presents a theoretical explanation for why in equilibrium both firms and
investors may care whether or not a benchmark is reached. This result comes from firms just
above a benchmark that manage earnings and report lower so that they can report higher in future
12
periods. I show that if we assume the distribution of firm types is uniform, for any given
probability of high discretion there exist equilibrium where there is a single benchmark and the
market assigns a premium to firms which reach that benchmark.
Intuitively, one might expect that firms that end just above a certain benchmark via
earnings management would have reached that benchmark by boosting earnings. In this paper,
however, I present a model where the opposite takes place: firms find themselves just above a
benchmark as a result of managing earnings down. This provides a counterintuitive empirical
prediction that firms that just meet or beat a key earnings benchmark may have particularly low
discretionary accruals.
In future research I hope to relax many of the strong assumptions in the model to see if the
results still hold and such equilibrium are still possible.
13
REFERENCES
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Increasing Earnings” Journal of Accounting Research 37 (1999): 387-413
BROWN, L. “A Temporal Analysis of Earnings Surprises: Profits versus Losses” Journal of
Accounting Research 39 (2001): 221–241
BURGSTAHLER D. AND I. DICHEV “Earnings management to avoid earnings decreases and
losses” Journal of Accounting and Economics 24 (1997): 99-126
DOYLE J., J. JENNINGS, AND M. SOLIMAN “Do managers define non-GAAP earnings to
meet or beat analyst forecasts?” Journal of Accounting and Economics 56 (2013): 40-56
FISCHER P. AND R. VERRECCHIA “Reporting Bias.” The Accounting Review 75 (2000):
229-245
GUTTMAN I., O. KADAN, AND E. KANDEL “A Rational Expectations Theory of Kinks in
Financial Reporting” The Accounting Review 81 (2006): 811-848
KASZNIK, R. AND M. MCNICHOLS “Does Meeting Earnings Expectations Matter? Evidence
from Analyst Forecast Revisions and Share Prices” Journal of Accounting Research 40
(2002): 727–759
MATSUNAGA S. AND C. PARK “The Effect of Missing a Quarterly Earnings Benchmark on
the CEO's Annual Bonus” The Accounting Review 76 (2001): 313-332
MOEHRLE, S. R. “Do Firms Use Restructuring Charge Reversals to Meet Earnings Targets?”
The Accounting Review 77 (2002): 397-413
14
Appendix A
PROPOSITION 1:
With 𝑑 uniform, we have:
𝑀(𝑥) − 𝑀(𝑥 − 1) = 1
𝑀(1) − 𝑀(0) = 1 − 𝑃
𝑀(𝑌 + 2) − 𝑀(𝑌 + 1) = 1 − 𝑃 +
𝑃
1 − 𝑝(1 − 𝑃)
𝑀(𝑌 + 1) − 𝑀(𝑌) =
1 − 𝑝(1 − 𝑃)(1 + 2𝑃)
1 − 𝑝2 (1 − 𝑃)2
𝑀(𝑌) − 𝑀(𝑌 − 1) =
𝑝(1 − 𝑃) − 𝑃
+1+𝑃
1 + 𝑝(1 − 𝑃)
𝑀(1) − 𝑀(0) < 𝑀(𝑥) − 𝑀(𝑥 − 1)
𝑀(1) − 𝑀(0) < 𝑀(𝑌 + 2) − 𝑀(𝑌 + 1)
We can write 𝑃 in terms of 𝑝 and 𝑁
𝑃=
𝑝/𝑁
1 + 𝑝/𝑁
𝑀(𝑌) − 𝑀(𝑌 − 1) =
𝑝(1 − 𝑃) − 𝑃
+1+𝑃
1 + 𝑝(1 − 𝑃)
𝑝
𝑁
𝑝(1 − 𝑃) − 𝑃 =
𝑝−
𝑝 >0
1+𝑁 1+𝑁
𝑝
This implies:
𝑀(𝑌) − 𝑀(𝑌 − 1) > 1 + 𝑃 > 𝑀(1) − 𝑀(0)
Need to show:
𝑀(𝑌 + 1) − 𝑀(𝑌) < 𝑀(1) − 𝑀(0)
1 − 𝑝(1 − 𝑃)(1 + 2𝑃)
− (1 − 𝑃) < 0
1 − 𝑝2 (1 − 𝑃)2
15
1−
𝑝
1 + 3𝑝/𝑁
𝑝 ( 1 + 𝑝/𝑁 )
1+𝑁
1
−
𝑝 2
𝑝 <0
1−(
)
1
+
𝑝
𝑁
1+𝑁
2
3𝑝
1+ 𝑁
𝑝
𝑝
(1 −
𝑝(
𝑝 )) (1 + 𝑁) − 1 + (
𝑝) < 0
1+𝑁 1+𝑁
1+𝑁
𝑝
2
3𝑝
1+ 𝑁
𝑝
𝑝
(1 + ) − 𝑝 (
𝑝 )−1+(
𝑝) < 0
𝑁
1+𝑁
1+𝑁
𝑝 3
𝑝 2
𝑝
3𝑝
(1 + ) − (1 + ) − 𝑝 (1 + ) (1 + ) + 𝑝2 < 0
𝑁
𝑁
𝑁
𝑁
𝑝
𝑝 2
𝑝
3𝑝
(1 + ) − 𝑝 (1 + ) (1 + ) + 𝑝2 < 0
𝑁
𝑁
𝑁
𝑁
1
𝑝 2
𝑝
3𝑝
(1 + ) − (1 + ) (1 + ) + 𝑝 < 0
𝑁
𝑁
𝑁
𝑁
1 2𝑝 𝑝2
4𝑝 3𝑝2
+ 2+ 3−1−
− 2 +𝑝 <0
𝑁 𝑁
𝑁
𝑁
𝑁
Want to show this is true for all 𝑝, 𝑁
𝑓(𝑝, 𝑁) =
1 2𝑝 𝑝2
4𝑝 3𝑝2
+ 2+ 3−1−
− 2 +𝑝
𝑁 𝑁
𝑁
𝑁
𝑁
𝜕𝑓
2
2𝑝 4 6𝑝
= 2+ 3− − 2+1
𝜕𝑝 𝑁
𝑁
𝑁 𝑁
𝜕𝑓
> 0 ∀𝑝, 𝑁 > 4
𝜕𝑝
So suffices to show that:
𝑓(1, 𝑁) < 0 ∀𝑁
𝑓(1, 𝑁) =
1
2
1
4
3
+ 2+ 3−1− − 2+1
𝑁 𝑁
𝑁
𝑁 𝑁
𝑓(1, 𝑁) = −
3
1
1
− 2 + 3 < 0 ∀𝑁
𝑁 𝑁
𝑁
So equilibrium is always possible. Just need 𝑟, 𝑁 such that:
16
𝑀(𝑌 + 1) − 𝑀(𝑌) <
𝑁−1
< 𝑀(1) − 𝑀(0)
𝑁(1 + 𝑟)
1 − 𝑝(1 − 𝑃)(1 + 2𝑃)
𝑁−1
<
< 1−𝑃
2
2
1 − 𝑝 (1 − 𝑃)
𝑁(1 + 𝑟)
Note that:
𝑀(𝑌) > 𝑌
Since 𝑃 < 𝑝 ∀𝑁
17