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Production and Cost

© Allen C. Goodman, 2015

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Production and Cost of Health Care

We want to look at 4 basic aspects.- Substitution between Inputs

- Cost Function, and Economies of Scale and Scope

- Use of Economies of Scale and Scope in Cost Function Studies

- Measurement of Efficiency

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Traditional Health Technology Analysis

• Although it would seem obvious from economics that we can substitute inputs for each other, health professions don’t always agree.

• Consider the discussion of need in health care resources.• Most important early work that employed a medical

determination of health manpower needs was a study by Lee and Jones (1933) -- we reference it some in Chapter 16. We discussed it more in earlier editions.

• Their method calculated the # of physicians necessary to perform the needed # of medical procedures. The needed # of medical procedures, in turn, was based on the incidence of a morbidity (illness) in the population.

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Traditional Health Technology Analysis (2)

Consider Condition A, which strikes 1 percent of the population in a given county.

Suppose further that its treatment requires 6 hours of physician time, and that there are 250,000 residents in the county.

How many physicians are needed, if a physician works 2,000 hours per year?

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Traditional Health Technology Analysis (3)

a. 250,000 persons x (1 morbidity/100 persons) = 2,500 morbidities.

b. 2,500 morbidities x (6 hours/morbidity) = 15,000 hours

c. 15,000 hours x (1 physician/2,000 hours) = 7.5 physicians.

Suppose the county currently has 7.5 physicians, and that the county's population is projected to rise from 250,000 to 400,000.

Without any adjustment for the morbidity rate, or for the technology of care, need would be projected to rise to (400,000/250,000) x 7.5, or 12.0 physicians.

If the projected (actual) total is less than 12.0 then a projected (actual) shortage is said to exist.

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Severe assumptions

• Even if we supposed that there are only two factors of production, physicians L and some amount of capital or machinery K, the Lee-Jones approach tends to ignore the possibilities for substitution between inputs.

Presumes:

a. there is no substitution of other inputs for physician inputs, and,

b. there is no projected technological change in the production of health care services.

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c. there is a single, unique answer to the question of how many medical procedures are appropriate given the illness data for a population,

d. and prices and costs of various inputs are safely ignored.

and additionally:

e. manpower provided to the public will be demanded by the public or otherwise paid for,

f. medical doctors are the appropriate body of people to determine population needs.

Severe assumptions.

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Reviewing Elasticity of substitution, .

= the % change in the factor input ratio, brought about by a 1% change in the factor price ratio.

K

L

K/L1

K/L2

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Production FunctionsSeveral different types of production functions. The typical Cobb-

Douglas production function for capital and labor can be written as:

Q = A L K

It turns out that there is a property of the Cobb-Douglas function that

= 1. What does this mean? This gives an interesting result that factor shares stay constant. Why?

s = wL / rK

s = (w/r) (L/K)

Increase in (w/r) means that (L/K) should fall. With matching 1% changes, shares stay constant.

1% 1%

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Consider C.E.S. production function with capital and labor. Q = A [K + (1-) L] R/.If profits are: = pQ - rK - wL, when we substitute in for the quantity relationship, we get:

Differentiating with respect to L and K, we get: /L = A(R/) (1-) L-1 [K + (1-) L] (R/)-1 - w= 0 /K = A(R/) K-1 [K + (1-) L] (R/)- 1 - r= 0

Simplifying, we get:(1-)/ (K/L)1- = w/r

Production Functions

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Production Functions

Redefine k = K/L, and = w/r, so:

[(1-)/] (k)1- = Now, differentiate fully. We get:

[(1-)/] (1-) (k)- dk = d, or:

dk/d = [/(1-)] [1/(1-)] (k). Multiplying by /k, we get the elasticity of substitution, or:

= 1/(1-).

What does a Cobb-Douglas function look like? What do others look like?

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Cost Functions and Production

A Short Primer on Estimating Cost Functions

We start with the premise that Q = f (x), where x refers to inputs.

Economic theory has moved us to the “cost function” in which we minimize costs subject to a given level of production.

C = pixi = C (Q, p).

where xi are inputs, p are factor costs.

It turns out that we can estimate production elasticities of substitution between any two variable inputs i and j, as:

ij = C Cij / Ci Cj (subscripts refer to partial derivatives)

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Two fundamental problems

Two fundamental problems.

You need measures of factor costs if you want to make it complete.

If you’re looking over time, input price effects may account for large portions of the increasing costs.

Even with static models, however, you have problems. Not including factor prices implies that they do not matter. This is similar to assuming that the substitution elasticity is zero.

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Cost minimization may occur in the short run or in the long run, depending on how variable the factors are.

It turns out that hospitals employ VERY FIXED capital. Let K represent fixed inputs. Variable cost function is:

Cv = Cv (Q, p, K) = xi pi.

To be in LR eq’m you must also satisfy

C = Cv (Q, p, K*) = xI pI + pkK*, and hence:

Cv / K = pk.

May or may not hold.

Most people tend to net out capital costs and try to estimate short run cost functions with respect to variable inputs.

Two fundamental problems

GeometricallyLet K be the fixed level

of capital.

Inputs X, output Y.

SR Cost will be higher than LR costs everywhere except at K.

Only at K* will marginal product ratio equal the factor price ratio.

K

X

K

Factor cost ratio

c(Y1) < src(Y1) c(Y2) = src(Y2) c(Y3) < src(Y3)

Y3

Y2

Y1

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Jensen and Morrissey (1986)

They try to estimate a production function for hospital care. From this production function, they trace isoquants, and calculate substitution elasticities. They have a number of labor inputs, as well as “beds” a measure of capital inputs.

They look at a hospital producing output Q, adjusted for case-mix, using

admitting physicians L and

non-physician labor and capital K.

L is a fraction of S, size of the medical staff, so:

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Output is a function of capital and labor services:

Q = Q (K, L) (1)

Labor is a function of the fraction of the staff that is physicians.

L = (K, N) S (2)

where N is a set of market area conditions such as the availability and attractiveness of competing hospitals in the area.

Jensen and Morrissey (1986)

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Substituting (2) into (1) Q = Q (K, S, N).

They estimate a translog production function for annual cases treated Q:

ln Q = 0 + i ln Xi + i ln Xi2 + ij ln Xi ln Xj.

This leads to:

MPi = i (Q/Xi), where

i = i + 2i (ln Xi) + ij (ln Xj) = output elasticity

Jensen and Morrissey (1986)

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From the translog, we can develop ij as:

ij = (i + j)/(i + j + 2ij - 2ij /i -2ji/j).

where i is an output elasticity.

For a cost minimizing firm, ij is the percentage change in the input ratio of Xi and Xj per 1% change in the ratio of input prices.

If ij > 0, i and j are substitutes in production.

If ij < 0, i and j are complements in production.

Jensen and Morrissey (1986)

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CrossTerms

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Substitution Elasticities

Substitution Elasticities

Among the substitution elasticities, we have:

- Between medical staff relative to nurses = 0.547

SOME

- Between nurses and residents = 2.127;

CONSIDERABLE

- Even between labor and capital = .124 - .211

A LITTLE BIT

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Measuring Efficiency

• If we’re right on an isoquant, then we must be efficient.

• If we’re not, then we must be inefficient. How hard could that be?

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Inputs

Output

White line is best feasible. But we can’t use plain old OLS

Problems

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Frontier MethodsEfficient is efficient

We want to identify the difference between “bad luck” and inefficiency.

qi = f(Xi) ui vi, (6')

vi = random shock, and ui = multiplicative efficiency term.

ui = qi/f(Xi)vi ui [0, 1].

If:

qi = Xji bj + ej, then:

ei = ui + vi,

where ui are one-sided, non-negative, and vi ~ N (0, 2

v).

Normal vi$

Disturbance

Truncatednormal ui

Efficiency

q

True

We’ll see thisin another context in a couple of weeks

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Inputs

Output

White line is best feasible. But we can’t use plain old OLS

Data Envelopment AnalysisNon-parametric

Non-parametric

Usually uses some sort of linear programming algorithm

Usually uses some sort of linear programming algorithm

Data Envelopment Analysis

• Cooper, W.W., Seiford, L.M., Tone, K., 2007. Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References, and DEA Solver Software, 2nd ed. Kluwer Academic Publishers: Hingman, MA.

Efficiency

Inputs (X)

Outputs (Y)

Eff = 1

Eff < 1

DEA• Consider 2 inputs.• Collect a lot of data

Capital, K

Labor, L

• Micro theory tells us that some are more efficient than others.

• DEA allows us to use linear programming methods to find the best envelope.

DEA• Those on the frontier are

considered efficient.

Capital, K

Labor, L

• Those off the frontier are considered inefficient.

• There are lots of measures related to lengths of the rays that indicate how inefficient they are.

DEA Math – 1

• A fractional formulation for the case of s outputs, m inputs, and n Decision Making Units where the– y terms represent output levels, – the x terms represent input levels, and – the u and v terms represent the weights

associated with outputs and inputs respectively

DEA Math – 2

Efficiency1 =

1 11 1

1 11 1

s s

r r r rr rm m

i i i ii i

out weight output u y

inp weight input v x

s

Max: Z = Σ ur yro Weighted outputs r=1

m

Subject to: Σ vixio = 1 Input fractions

i=1

m s

-Σvixij + Σuryrj ≤ 0 i=1 r=1

j = 1, …..n; ur, vi ≥ 0 for all r, i.

For all formulations, x represents inputs and y represents outputs

Variables

• Input variables for the analysis are: 1. Total Community Hospitals, 2. Total full time employees, 3. Total part time employees, 4. total beds

• Output variables are 1. Inpatient days,2. Inpatient surgeries, 3. births, 4. emergency outpatient, 5. other outpatient.

Butler/Goodman (2011)Efficiency v. Number of Hospitals - 3 years

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

0 50 100 150 200 250 300 350 400 450

Number of Hospitals

Eff

icie

ncy

DEA-94

DEA-01

DEA-08

California

Texas

Efficiency at State level for 1994, 2001, and 2008

Thoughts

• Higher population, more beds More efficient• More hospitals/county Less efficient• Investor owned More efficient• Not For Profit More efficient but effect is 1/3

that of Investor owned. Gov’t hospitals are other category.

• Major improvement in efficiency from 1994 to 2001; No change from 2001 to 2008. BBA of 1997 may be big explanation

Issues with DEA

1. Frontier relies on outlying observations. DEA assumes no measurement error or random variation of output.

2. DEA is sensitive to number of input and output variables you need lots of observations.

3. DEA measures relative efficiency among the sample collected it’s possible that greater efficiency could be achieved.

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How easy is it to measure efficiency?

• Newhouse (1994) says “not easy.”• Argues there are problems with:

– unmeasured inputs– case-mix controls– omitted or improper inputs

• All of these could look like inefficiency, when they are not.

References:Burgess, James, Chapter 32 in Elgar readings

Newhouse, Joseph P. “Frontier Estimation: How useful a tool for health economics?” Journal of Health Economics 13 (1994): 317-22.