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Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University of Bristol, RAND, & NBER Rodrigo Moreno-Serra Imperial College London Carol Propper Imperial College London & University of Bristol Human Capital Research Collaborative Fall Conference Federal Reserve Bank of Minneapolis October 14,15, 2010 1

Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

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Page 1: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Death by Market Power:Reform, Competition and Patient Outcomes

in the British National Health ServiceMartin Gaynor

Carnegie Mellon University, University of Bristol, RAND, & NBERRodrigo Moreno-SerraImperial College London

Carol PropperImperial College London & University of Bristol

Human Capital Research Collaborative Fall ConferenceFederal Reserve Bank of Minneapolis

October 14,15, 2010

1

Page 2: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

INTRODUCTION

Page 3: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Motivation

1. Health care reform is happening in most developed countries: U.S., U.K., Germany, Netherlands, Belgium, Israel, Australia,...

2. Market-oriented approaches to health care are an important reform model outside U.S. a. Lack of strong research evidence w.r.t. quality.b. Price not relevant in most systems outside the

U.S., or for U.S. Medicare program (20% of U.S. spending).

3

Page 4: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Motivation, cont’d.

3. In U.S. consolidation in markets has led to questions about functioning of markets in health care.a. How well are markets working?b. Could further consolidation promoted by health

reform be harmful?4. Need to add to knowledge about conditions under

which markets might work.a. Quality, cost, prices (U.S.).

5. Little information about impacts on children.

Page 5: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Our Contribution1. Most empirical evidence on the impact of competition

on quality uses variation in market structure across existing hospital markets. a. We’d like to randomly assign hospitals to varying degrees

of competition.b. Clearly not the case.

2. We exploit a policy “experiment” (NHS) to assess the impact of competition.a. Examine a range of hospital outcomes (measures of

quality, quantity/access, spending).3. Find that lower market concentration is associated

with higher quality without a commensurate increase in expenditure.

Page 6: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

The NHS Reforms

1. 1997- buyers and sellers operated under co-operation and negotiated annual budgets on price and volume.a. Patients had little choice of hospital.b. Quality not contracted on (except waiting times).

2. Policy change initiated in 2003, put in place in 20063. GoaI – to promote competition among hospitals

Page 7: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

NHS Reforms, cont’d.

4. Key elementsa. ‘Choose and Book’ – patients must be offered choice of 5

hospitals.b. Payment by Results (PbR) - movement from negotiated

to fixed prices (HRGs - similar to U.S. DRGs). i. PbR accounts for almost 70% of activity.

c. Reward /Penalties for Performance.i. Foundation Trust Status – Retain net income.ii. Poor performance – management replacment, closure, merger.

Page 8: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Expected Effects of the Reform

1. Expected effectsa. ‘Choose and Book’ – increase elasticity of demand facing

hospitals.b. PbR – change conduct .

i. Hospitals paid for activity.ii. Focus on quality since prices fixed.

2. Do hospitals have incentives to respond?a. Not for profit, annual budget constraint.b. Poor financial & clinical performance heavily penalised.c. PbR system is very highly geared.

i. Levels of prices key.

Page 9: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

What We Do

1. Exploit policy change in NHS 2006 to undertake difference in difference analyses .

2. Use time periods before/after reform and variation in market concentration.a. Before/After: 2003/2007b. Concentration: More/Less Concentrated

3. Post-policy a hospital in a less concentrated market faces greater exposure to the policy.

21 i

ni sHHI

Page 10: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Data

1. HES (Hospital Episode Statistics) data from the NHS (http://www.hesonline.nhs.uk/Ease/servlet/ContentServer?siteID=1937)a. Data on all admissions to NHS hospitals in England.

i. Standard hospital discharge data set: diagnoses, procedures, patient characteristics, location,etc.

ii. ~13 million records per yeariii. We use data on ~160 short term general hospitals per

yearb. We use hospital level data for 2 years – 2003/04,

2007/08c. Used to construct measures of concentration and

some outcome measures

Page 11: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Data, cont’d.

2. Measures of quality and performancea. Some calculated from HES data (e.g. in-hospital

deaths within 28 days of admission for various treatments, deaths in all locations after AMI admission, LOS)

b. Some derived from official data on hospital performance (e.g. waiting times data, CQC data)

3. Data from administrative sourcesa. NHS staffing datab. Small area characteristics (wages, mortality)

Page 12: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Measures of Concentration

1. HHIs for hospitals based on patient flows

1. Built up from small area (~ 7000 persons) patient flows to hospitals

a. Calculate HHIs at MSOA level for 2003 and 2007b. Use all non-emergency admissionsc. Allow market to be whole country

2. Aggregate to hospitals based on share of patient flows to hospital from each MSOA

21 i

ni sHHI

Page 13: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

WHAT THE RAW DATA SHOW

Page 14: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Raw Data

1. Did mortality rates go up (more) in more concentrated markets after the reform?

2. Did concentration change (before/after reform)?

3. Did demand change post-reform?

Page 15: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

The Paper While Standing on One Foot28 Day AMI Mortality Rate and HHI

Page 16: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Still on One Foot 28 Day All Causes Mortality and HHI

Page 17: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Decrease in Concentration

Page 18: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Levels and Changes in Concentration by Location

Page 19: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Did Demand Change Post Reform?

1. Examine changes in patterns of patient care seeking 2003-07 by:a. Quality of Hospitals (top vs. bottom quartiles of

AMI mortality rate in 2003)b. Exposure to competition (bottom vs top quartile

of HHI in 2003)

Page 20: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Better Hospitals are Attracting More Patients

AMI mortality rate (2003) Bottom quartile Top quartile

2003 2007

% change (2003-

07) 2003 2007

% change (2003-

07)

Number of elective admissions

33,985 38,274 12.6% 41,398 45,132 9.0%

Average distance travelled by patients

11.4 11.7 2.4% 10.0 10.1 1.1%

Share of patients bypassing nearest hospital

0.37 0.39 5.4% 0.45 0.43 -4.4%

Number of hospitals

33 33 32 32

Page 21: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Hospitals More Exposed to Policy are Attracting More Patients

Market concentration: HHI (2003)Low (bottom quartile) High (top quartile)

2003 2007

% change (2003-

07) 2003 2007

% change (2003-

07)Number of elective admissions

21,757 26,924 23.8% 55,253 61,049 10.5%

Average distance travelled by patients

8.1 8.3 2.3% 15.5 15.5 0.5%

Share of patients bypassing nearest hospital

0.45 0.46 2.2% 0.47 0.47 0.0%

Number of hospitals 41 41 40 40

Page 22: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

REGRESSION ANALYSIS

Page 23: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Econometric Strategy

1. Exploit policy change in NHS 2006 to undertake difference in difference analyses .

2. Use time periods before/after reform and variation in market concentration.

3. Identification from cross sectional and time series variation.4. Policy effect = Market Concentration* Policy On (2007).

a. Parameter δ in regression.

qit = + I(t=2007) +I(t=2007)*HHIit + HHIit + Xiti +it

Page 24: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Econometric Issues

1. There may remain concerns over endogeneity of concentration + patient heterogeneity.

2. Control for patient heterogeneity with observables.a. Patient age, sex, severity (Charlson index) .b. Local area health, income.c. Include hospital fixed effects.

3. Replace actual HHI with a measure of market structure based on factors unrelated to quality or unobserved patient heterogeneity.a. Predicted from estimated model of hospital choice.

Page 25: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Regression Results

1. Mortality, Waiting Times2. Quantity, Expenditure

25

Page 26: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

DiD Estimates of Market Structure on Outcomes and Waiting Times

(1) (2) (3) (4) (5) (6) (7)

28 day AMI

mortality rate (in-

hospital, ages 55+)

30 day AMI

mortality rate

(on or after discharge,ages 35-

74)

28 day all causes

mortality rate (in-

hospital)

28 day mortality

rate (in-

hospital, excluding

AMI)

MRSA bacterae

mia rate

Patients waiting

3 months or more

Attendances

spending less than

4 hours in A&E

DiD coefficient 0.246*** 0.313** 0.069** 0.066** -0.110 0.078 -0.005

(0.084) (0.116) (0.027) (0.028) (0.118) (0.167) (0.011)

Hospitals 133 133 162 162 161 162 150

Observations 250 250 323 323 318 323 299

Page 27: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Estimated Effect of the Policy1. Hospitals in less concentrated markets had significantly lower

mortality rates post-reform than those in more concentrated markets.a. The policy “worked.”

2. 10% fall in HHI associated with a 2.46% reduction in in-hospital AMI mortality rate.

3. 1/3rd of a percentage point at mean AMI mortality rate (13.2%).4. Similar to #s from previous work.

a. Kessler and McClellan (2000) i. Change from top to bottom quartile of HHI leads to 3.37 percentage point

decrease in AMI death rate. Our equivalent #: 3.61.b. Cooper et al. (2010)

i. 1 s.d. change leads to 0.3 percentage point reduction in AMI death rate. Our #: 0.33.

27

Page 28: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Policy Impacts on LOS, Admissions, and Expenditure

28

(1) (2) (3) (4) (5) (6)

Length-of-stay and admissionsExpenditure and

productivity

Mean length-of-

stay(days)

Total admissions(number)

Elective admissions

(share of total)

Non-elective

admissions (share of

total)

Operating expenditure

(£1,000)

Operating expenditure

(£1,000) per admission

DiD coefficient 0.254*** -0.012 -0.005 -0.001 0.007 0.014

(0.059) (0.031) (0.017) (0.024) (0.072) (0.074)

Hospitals 162 162 162 162 162 162

Observations 323 324 324 324 319 319

Page 29: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Did the Policy Matter?1. Benefits from the observed change in market

structure post reform.a. 3,354 life years saved = £227 million (=$350mil).

2. Cost of being in a concentrated market compared to being in a less concentrated one.a. An HHI of 2,000 less (=one s.d.) implies a saving of £3.7

billion (=$5.7 bil).3. NHS budget is £100 billion (=$154 bil).

a. Impact is 0.2% of NHS budget.b. Immediate impact is small, but we only value deaths

averted and longer term impact of reducing concentration considerably larger.

Page 30: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Conclusions1. Robust evidence that under a regulated price regime,

within two years a pro-competitive policy resulted in:a. an improvement in clinical outcomes, as measured in death

rates,b. reduction in length of stay,c. no increase in expenditures.

2. Conclude: policy appears to have saved lives and did not (measurably) increase costs.a. “Competition saves lives; monopoly kills.”

3. Competition can be an important mechanism for enhancing the quality of care.

4. This may have a significant impact on children’s health.a. Vulnerable population.b. Future research.

Page 31: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Additional slides

Page 32: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Competition and Quality in Health Care1. Theory

a. Regulated prices – competition increases quality (if price > marginal cost; e.g., regulated airline models).

i. Quality elasticity of firm’s demand increasing in the # of firms.ii. Competition increases management effectiveness, thereby increasing

quality.b. Market determined prices – anything can happen.

2. Evidence is mixeda. Regulated prices

i. Medicare – competition improves quality (e.g., Kessler & McClellan, 2000)

b. Market determined pricesi. U.S. private markets – not so clear (Volpp et al., 2003)ii. U.K. 90s reforms – competition reduced quality (Propper et al, 2008)

3. Little evidence from policies designed to introduce competition

Page 33: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Tests of Difference in Difference Assumptions

1. If we find a relationship between mortality and market structure is it due to the policy or to pre-existing differences between hospitals with different market structures?

2. We examine:a. bivariate associations between the observed baseline conditions and the

subsequent four year change in the HHI;i. Admissions, AMI admissions, doctors, clinical staff, area mortality rate, case mix,

Index of Multiple Deprivation, Charlson Index.b. bivariate associations between the initial levels of mortality and the

subsequent changes in market structure.i. In-hospital AMI mortality, 30 day AMI mortality, In-hospital all-cause mortality.

3. If the change in HHI is associated with pre-existing differences this may indicate that hospitals that differ in HHI growth may also differ in unobserved factors.

4. None of the associations significantly different from zero.

33

Page 34: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Predicted HHI

1. Predicted HHIs from predicted patient flows from estimated MNL model of hospital choice.a. a la Kessler and McClellan (2000).b. Choice in MNL model depends on:

i. hospital characteristics (size, teaching status), ii. differential distance from patient’s MSOA centroid to

hospital,iii. patient characteristics (age, sex, level of co-morbidity).

c. Choice set – all hospitals within 100km extended to ensure that there is always a first and second choice within hospital type (size, teaching) with minimum of 50 admissions.

Page 35: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Robustness Checks1. Results may be driven by pre-existing differences

between hospitals that are correlated with market structure.a. Placebo test using 2001 as before policy and 2003 as

after policy – insignificant.2. Estimation using only pre-policy variation in market

structure.a. HHI*2007 – significant.

3. Add further controls for patient heterogeneity; income shock from PbR; local area economic conditions (male wage, ambulance speeds).a. DiD estimates still significant, magnitudes almost

unchanged.

35

Page 36: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Hospitals Used by GPs and Distances Travelled by Patients

Page 37: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Tests of DiD Assumptions

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

CovariateTotal

admissions

AMI admissions (ages 55+)

Doctors (share of clinical staff)

Qualified clinical staff

(share of clinical staff)

Area standardized

mortality rate

Case mix

Index of Multiple Deprivation (average for

patients’ areas of residence)

Charlson index

(average for admissions at the hospital)

28 day AMImortality rate (in-hospital, ages 55+)

30 day AMI mortality rate(on or after discharge, ages 35-74)

28 day all-cause mortality

rate (in-hospital)

Coefficient -0.624 -0.051 5.329 -10.233 -1.695 -0.001 81.732 -1.080 -1.808 47.847

(0.642) (0.086) (8.792) (7.947) (1.936) (0.003) (72.197) (4.751) (6.623) (42.840)

P-value for Wald test 0.129

Observations 162 151 161 161 162 162 162 162 130 130 162

Page 38: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Robustness tests(1) (2) (3)

28 day AMI mortality rate

(in-hospital, ages 55+)

28 day all-causemortality rate (in-hospital)

Mean length-of-stay

(days) Robustness test

1. Baseline 0.246*** 0.069** 0.254***

(0.084) (0.027) (0.059)

Observations 250 323 323

2. Placebo DiD test for 2001-2003 -0.047 0.005 -0.036

(0.077) (0.027) (0.047)

Observations 250 309 309

3. Using time invariant pre-reform HHI level (2003) 0.216*** 0.066** 0.245***

as market structure measure (0.079) (0.028) (0.059)

Observations 250 323 323

4. Controlling for the Charlson index 0.246*** 0.067** 0.239***

(0.084) (0.027) (0.060)

Observations 250 323 323

5. Controlling for the Index of Multiple Deprivation 0.278*** 0.067** 0.263***

(0.085) (0.029) (0.061)

Observations 250 323 323

6. Controlling for surpluses/deficits 0.242** 0.076** 0.229***

(0.093) (0.030) (0.066)

Observations 236 302 302

7. All hospitals (weighted by number of admissions) 0.138** 0.069*** 0.261***

(0.069) (0.024) (0.061)

Observations 299 323 323

8. Using levels of the dependent variable and HHI 0.170** 0.069*** 0.197***

(implied elasticity)

Observations 250 323 323

9. Controlling for income (male wage in area) 0.247*** 0.061** 0.258***

(0.086) (0.029) (0.062)

Observations 248 319 319

10. Controlling for the share of urgent ambulance calls 0.238**

responded within eight minutes (0.100)

Observations 233

Page 39: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Magnitude of Effects

39

28 day mortality rate (all causes)

Panel A - Observed magnitudes

Average number of admissions (2003/04) 63,094Average number of deaths (2003/04) 1,135.1Average mortality rate (2003/04) (%) 1.799%Average number of admissions (2007/08) 72,558Average number of deaths (2007/08) 1,053.5Average mortality rate (2007/08) (%) 1.452%Average change in deaths (2003-07) (positive = deaths averted) 81.5Average decrease in predicted HHI (2003/04-2007/08) -118

Panel B - Continuous HHI: magnitudes implied by estimated coefficient (summary stats refer to HHI 03)

Baseline coefficient (elasticity) (%) 0.069

Scenario 1: Average decrease in HHI (Policy impact)Implied counterfactual percentage increase in the mortality rate 2007/08 per hospital (for elasticity calculated at mean HHI) 0.2%Total lives saved for the whole sample of hospitals (N = 162) 327Total number of years of life saved for the whole sample of hospitals 3,354Total savings in £million (value of year of life = £60,000/p.a.) £201

Scenario 2: One standard deviation increase in HHI 03 (= increase of 1928 units, from 4,353 to 6,281 in the whole sample)Implied counterfactual percentage increase in the mortality rate 2007/08 per hospital (for elasticity calculated at mean HHI) 3.1%Total lives saved for the whole sample of hospitals (N = 162) 5,336Total number of years of life saved for the whole sample of hospitals 54,771Total savings in £million (value of year of life = £60,000/p.a.) £3,286

Page 40: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

No change in patient type except for IMD for good hospitals

40

Change 2007-2005

(1) (2) (3) (4) (5) (6)

Elective admissions

Number of MSOAs

(electives)

Mean distance travelled

(electives)

Mean IMD

ranking Charlson

index Number of diagnoses

Mean waiting time 56.434* 0.694** 0.004* 13.836*** 0.00002 0.001 (elective admissions) (33.680) (0.295) (0.002) (3.061) (0.00013) (0.001)

Overall quality of services 1165.859 21.762** 0.078 207.347** -0.00030 0.024 (score) (897.260) (9.294) (0.076) (95.395) (0.00519) (0.031)

In-hospital mortality rate -942.177 14.816* -0.175* 14.927 -0.01229* -0.005 (all causes) (1255.291) (8.718) (0.103) (91.709) (0.00638) (0.050)

In-hospital mortality rate -91.980 -1.035 0.001 26.533 0.00086 0.002 (AMI) (227.052) (1.029) (0.011) (21.365) (0.00131) (0.005)

Teaching hospital status 1234.181 20.211 -0.098 335.378** -0.00041 -0.034

(2088.132) (17.763) (0.133) (143.230) (0.01348) (0.042)

Note: Index of Multiple Deprivation (over all patients), where patients in the most deprived locality in the year are attributed the ranking of 1 and higher values are attributed to patients living in less deprived areas.

Page 41: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

Competitive hospitals in 2005 not attracting observably sicker/different patients but are attracting more patients

41

Change 2007-2005

(1) (2) (3) (4) (5) (6)

Elective admissions

Number of MSOAs

(electives)

Mean distance travelled

(electives)

Mean IMD

ranking Charlson

index Number of diagnoses

Level of HHI -483.360 -2.620 0.032 41.200 0.002 -0.003

(505.402) (2.788) (0.025) (43.781) (0.002) (0.015)

Indicator for bottom 3156.728** 18.499* -0.084 -139.657 0.001 0.004 quartile of HHI (1513.533) (10.949) (0.096) (162.770) (0.008) (0.056)

Number of hospitals 162 162 162 162 162 162 Note: Index of Multiple Deprivation (over all patients), where patients in the most deprived locality in the year are attributed the ranking of 1 and higher values are attributed to patients living in less deprived areas.

Page 42: Death by Market Power: Reform, Competition and Patient Outcomes in the British National Health Service Martin Gaynor Carnegie Mellon University, University

42

Predicted HHIs based on predicted patient flows based on MNL model of hospital choice

2

1 1

1 1 1 1 1

ˆ ˆˆ ˆ ˆ,ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ ˆ, 1 ,k k k

K Jkj jk

j k kk jj k

n n nn J

j ij k ij k kj jk iji i j i i

n nHHI HHI HHI

n n

n n n n n