Assignment #4

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Chapter 7

Medical Care Production and Costs

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The Short-Run Production Function of the Representative Medical Firm

Assumptions of short-run production:

Medical firm produces a single output of medical services, q

Only two medical inputs exist: nurse-hours, n, and a composite capital good, k

Quantity of capital is fixed at some amount

The Short-Run Production Function of the Representative Medical Firm

Medical firm faces an incentive to produce as efficiently as possible

Medical firm possesses perfect information regarding the demands for its product

The Short-Run Production Function of the Representative Medical Firm

Production function:

Identifies different ways nurse-hours & capital can be combined

To produce various levels of medical services

The Short-Run Production Function of the Representative Medical Firm

Each level of output produced by -

Several different combinations of the nurse and capital inputs

Each combination – assumed to be technically efficient – maximum amount of output that is feasible given the state of technology

The Short-Run Production Function of the Representative Medical Firm

Law of diminishing marginal productivity

At first, total output increases at an increasing rate

After some point, it increases at a decreasing rate

Total product curve, TP

Depicts total output produced by different levels of the variable input, holding all other inputs constant

Figure 7.1 - The Total Product Curve

Nurse-hours (n)

Δq

Δn

The total product curve shows that output initially increases at an increasing rate from 0 to n1 nurse-hours, then increases at a decreasing rate from n1 to n2 nurse-hours, and finally declines after n2 nurse-hours as the medical firm employs more nurse-hours.

Quantity of

medical

services

(q)

Diminishing marginal productivity provides the reason why output fails to expand at an increasing rate after n1 nurse-hours.

Marginal and Average Products

Marginal product

Change in total output associated with a one-unit change in the variable input

MPn= Δ q/Δn

Magnitude of the marginal product of a nurse-hour reveals - Additional quantity of medical services produced by each additional nurse-hour

Marginal and Average Products

Marginal product

Initially, increasing marginal productivity

MPn is positive and increasing

Next, diminishing marginal productivity sets in

MPn is positive but decreasing

Next, MPn = 0

When total product is maximized

After, MPn is negative

Figure 7.2 - The Marginal Product Curve

Nurse-hours (n)

Marginal

product of

nurse-hours

(MPn)

MPn

Marginal productivity first increases with the number of nurse-hours because of synergy and labor specialization and then falls because of the fixed input that exists in the short run.

The marginal product of an additional nurse-hour is found by dividing the change in output by the change in the number of nurse-hours and is measured by the slope of the total product curve.

Marginal and Average Products

Average product, APn

Total quantity of medical services divided by the total number of nurse-hours

APn = q/n

Average quantity of medical services produced within an hour

Derived from the total product curve

Figure 7.3 - Deriving the Average Product Curve from Total Product Curve

Nurse-hours (n)

Quantity of

medical

services

(q)

Nurse-hours (n)

Average

product of

nurse-hours

(APn)

APn

The average product of a nurse-hour is found by dividing total output by the total number of nurse-hours and can be derived by measuring the slope of a ray emanating from the origin to each point on the total product curve. Average productivity first increases with the number of nurse-hours and then declines because of increasing and then diminishing marginal productivity.

Marginal and Average Products

Average product, APn

Initially increases

Reaches a maximum

Then decreases

Law of diminishing marginal productivity

Marginal and Average Products

Marginal and average product curves

MP curve cuts AP curve at its maximum point

MP is above AP whenever AP is increasing

MP is below AP whenever AP is declining

Marginal and Average Products

At some point in the production process:

The additional nurse becomes less productive due to the constraint imposed by the fixed input

Marginal productivity

Influences the average productivity of the team of nurses

Figure 7.4 - Relation between Marginal and Average Product Curves

Nurse-hours (n)

Marginal

and average

products of

nurse-hours

MPn

APn

Average productivity rises when marginal productivity exceeds average productivity. Average productivity falls when marginal productivity lies below average productivity. Marginal productivity equals average productivity when average productivity is maximized.

Elasticity of Input Substitution

More than one variable input

Substitution between any two variable inputs

Degree of substitutability depends on technical and legal considerations

Elasticity of substitution

Percentage change in the input ratio divided by the percentage change in the ratio of the inputs’ marginal productivities, holding constant the level of output

Elasticity of Input Substitution

Elasticity of substitution, σ

Ii(i=1,2) - Quantity employed of each input

MP2/MP1 - ratio of marginal productivities

Marginal rate of technical substitution

Rate at which one input substitutes for the other in the production process, at the margin

Elasticity of Input Substitution

Elasticity of substitution, σ

Percentage change in the input ratio that results from a 1 percent change in the marginal rate of technical substitution

If σ = 0

Variable inputs cannot be substituted

If σ = ∞

Variable inputs are perfect substitutes in production

A Production Function for Hospital Admissions

Case-mix-adjusted hospital admissions = f(Physicians, nurses, other non-physician staff, hospital beds, X)

Marginal product for each input

Positive, declined in magnitude with greater usage

Law of diminishing marginal product

Nurse input - The most productive input

A Production Function for Hospital Admissions

Each input - a substitute for the others

Substitution elasticities between

Physicians and nurses - 0.547

A 10% increase in the marginal productivity of a doctor causes a 5.47% increase in the ratio of nurses to doctors

Physicians and beds - 0.175

Nurses and beds - 0.124

Hospital policy makers can avoid some of the price (wage) increase in any one input by substituting with the others

Short-Run Cost Theory of the Representative Medical Firm

Total cost = Explicit + Implicit costs

Explicit costs

Market transaction

Wage payments to the hourly medical staff, electric utility bills, and medical supply expenses

Accountants

Implicit costs

Opportunity costs of using any resources the medical firm owns

The Short-Run Cost Curves of the Representative Medical Firm

Short-run total cost, STC

Variable input nurse-hours - n

Fixed input capital - k

Level of medical output - q

Input prices - Assumed fixed

w- hourly wage for a nurse

r- rental or opportunity cost of capital

The Short-Run Cost Curves of the Representative Medical Firm

Short-run total costs of production, STC

Depend on quantities and prices of inputs employed

Total variable cost of production, STVC

Wage rate times the number of nurse-hours

Total wage bill

Responds to changes in the level of output

Total fixed costs of production, STFC

Rental price times the quantity of capital

Does not respond to changes in output

The Short-Run Cost Curves of the Representative Medical Firm

Calculating short-run total cost

Identify necessary number of nurse hours, n

Or each level of medical output

STVC = w × n

STC = STFC + STVC

Relation between STC and TP

When TP is increasing at an increasing rate

STC – increasing at a decreasing rate

When TP increasing at a decreasing rate

STC - increasing at an increasing rate

Figure 7.5 - The Short-Run Total Cost Curve

Quantity of

medical services (q)

STC

Cost of medical

services

STC2

w × n=STVC

r × k=STFC

STC first increases at a decreasing rate up to point q1 and then increases at an increasing rate with respect to producing more output. STC increases at an increasing rate after q1 because of diminishing marginal productivity.

The short-run total cost, STC, of producing medical services equals the sum of the total variable, STVC, and fixed costs, STFC.

The Short-Run Cost Curves of the Representative Medical Firm

Fixed costs

Occur in the short-run (operating period)

Levels of some inputs are fixed

Variable costs

All inputs are variable during the long run or planning period

Short-Run Per-Unit Costs of Production

Short-run marginal costs, SMC

Change in total costs associated with a one-unit change in output

Short-Run Per-Unit Costs of Production

Short-run average variable costs, SAVC

Short-run total variable costs, STVC, divided by the quantity of medical output

Short-Run Per-Unit Costs of Production

Short-run marginal variable costs

Inversely related to marginal product of labor

Short-run average variable costs

Inversely related to average product of labor

Maximum points on marginal product curve

Minimum point on marginal variable cost curves

Short-Run Per-Unit Costs of Production

Maximum points on average product curve

Minimum point on average variable cost curves

Figure 7.6 - Relation between the Per-Unit Product and Cost Curves

Nurse-hours (n)

Marginal

and average

products of

nurse-hours

MPn

APn

Quantity of medical services (q)

Cost of medical

services

SMC

SAVC

Short-run marginal cost, SMC, equals the change in total costs brought on by a one-unit change in output. Short-run average variable cost, SAVC, equals short-run total variable cost divided by total output. SMC and SAVC are inversely related to marginal and average productivity. For example, marginal costs decline as marginal productivity increases.

Short-Run Per-Unit Costs of Production

Short-run average fixed cost, SAFC

Total fixed costs, STFC divided by the level of output, q

SAFC = STFC/q

As medical services, q, increases

SAFC declines

Figure 7.7 - Relation among Short-Run Marginal, Average Variable, and Average Total Costs

Quantity of medical services (q)

Cost of medical

services

SMC

SAVC

SATC

SMC0

SATC0

SAVC0

SAFC0

Short-run average total cost, SATC, equals the sum of short-run average variable cost, SAVC, and short-run average fixed cost, SAFC. Hence, SAFC is reflected in the vertical distance between the SATC and SAVC curves at each level of output. SMC cuts both of the average cost curves at their minimum points. SMC lies above the SAVC and SATC curves when they are rising and below them when they are falling.

Factors Affecting the Position of the Short-Run Cost Curves

Prices of the variable inputs increase

Cost curves shift upward

Prices of the variable inputs decrease

Cost curves shift downward

Better quality of care

Cost curves shift upward

More severe patient case-mix

Cost curves shift upward

Excessive amounts of the fixed inputs

Cost curves shift upward

Factors Affecting the Position of the Short-Run Cost Curves

STVC = f(output level, input prices, quality of care, patient case-mix, quantity of the fixed inputs)

These factors can explain cost differentials among medical firms in the same industry

Output - where the medical firm operates along the cost curve

The other factors affect the location of the curve

Estimating a Short-Run Cost Function for Hospital Services

Studies by Cowing and Holtmann (1983) revealed:

STVC = f(q1,q2,q3,q4,q5,w1,w2,w3,w4,w5,w6,K,A)

qi(i=1,5) - quantity of one of five different patient services

Emergency room care, medical-surgical care, pediatric care, maternity care, and other inpatient care - measured in total patient days

wj(j=1,6) - one of six different variable input prices

Nursing labor, auxiliary labor, professional labor, administrative labor, general labor, and material and supplies

Estimating a Short-Run Cost Function for Hospital Services

STVC = f(q1,q2,q3,q4,q5,w1,w2,w3,w4,w5,w6,K,A)

K - single measure of the capital stock (measured by the market value of a hospital)

A - fixed number of admitting physicians in the hospital

Estimating a Short-Run Cost Function for Hospital Services

Economies of scope

Result from the joint sharing among related outputs of resources

Exist if the joint cost of producing two outputs is less than the sum of the costs of producing the two outputs separately

Estimating a Short-Run Cost Function for Hospital Services

Cowing and Holtmann (1983) results:

Short-run economies of scale

Limited evidence for economies of scope with respect to pediatric care and other services

Support for diseconomies of scope with respect to emergency services and other services

Estimating a Short-Run Cost Function for Hospital Services

Cowing and Holtmann (1983) results:

Short-run marginal cost of each output

Declined and then became constant over the levels of output observed in their study

Estimate short-run elasticities of input substitution between all pairs of variable inputs

Estimating a Short-Run Cost Function for Hospital Services

Substantial degree of substitutability between

Nursing and professional workers

Nursing and general workers

Nursing and administrative workers

Professional and administrative labor

The Cost-Minimizing Input Choice

Minimize TC(q0)=wR × RN + wL × LPN

subject to

q0=f(RN, LPN)

q0 – amount of medical services

TC – total cost

Variable inputs

RN – registered nurses

LPN- licensed practical nurses

Input prices

wR – hourly wage for RNs

wL – hourly wage for LPNs

The Cost-Minimizing Input Choice

Efficient mix of RNs and LPNs:

MPRN/wR=MPLPN/wL

Marginal product to price ratio is equal for both RNs and LPNs

The last dollar spent on RNs generates the same increment to output as the last dollar spent on LPNs

No other combination of RNs and LPNs is better

The Cost-Minimizing Input Choice

Optimal combination of doctors, doc and nurses, n:

MPdoc/wdoc=MPn/wn

Studies by Jensen and Morrisey (1986) revealed:

Estimated shadow price of a doctor

Implicit costs of approximately $7,012 per year from granting admitting privileges to the marginal physician

Long-Run Costs of Production

In the long-run

All inputs, including capital, can be changed

Long-run average total cost curve, LATC

Derived from a series of short-run cost curves

Different sizes

Choose the SATC or size that minimizes the average cost

Envelope curve

Figure 7.8 - Short-Run Average Cost Curves & Long-Run Planning Curve

Quantity of medical services (q)

Cost of medical

services

SATC1(k1)

SATC2(k2)

SATC3(k3)

LATC

All inputs are variable in the long run. SATC1, SATC2, and SATC3 represent the cost curves for small, medium, and large facilities, respectively. If decision makers choose the efficiently sized firm for producing output in the long run, a long-run average total cost, LATC, can be derived from a series of short-run average total cost curves brought on by an increase in the stock of capital. The U shape of the LATC reflects economies and diseconomies of scale.

Long-Run Cost Curves

Long-run average total cost curve, LATC

U-shaped

Initially declines, reaches a minimum, and eventually increases

Long-run economies and diseconomies of scale

Long-Run Cost Curves

Long-run economies of scale

Average costs fall as a medical firm gets physically larger due to specialization of labor and capital

Increasing returns to scale

An increase in all inputs results in a more than proportionate increase in output

Constant returns to scale

Horizontal LATC curve

Long-Run Cost Curves

Diseconomies of scale

Result when the medical firm becomes too large

Average costs increase as a medical firm gets physically larger

Decreasing returns to scale

An increase in all inputs results in a less than proportionate increase in output

Shifts in Long-Run Average Cost Curve

Increase in prices of medical inputs

LATC shifts upward

Quality

Cost-saving technology – LATC shifts downward

Higher quality of care – LATC shifts upward

Patient case-mix

More severe patient case-mix – LATC shifts upward

Long-Run Cost Minimization and the Indivisibility of Fixed Inputs

Assumption

All inputs can be costlessly adjusted upward or downward

Capital inputs

Not easily changed

Medical firms may adjust slowly

Not produce in long-run equilibrium

Operate with excess capital relative to a long-run equilibrium point

Figure 7.9 - Long-Run Disequilibrium of the Medical Firm

Dental output (q)

Cost of production

SATC1

SATC2

LATC

A firm may not operate in long-run equilibrium because of the sizeable costs of adjusting to a sharp change in demand. For example, assuming that the dental clinic is initially producing in long-run equilibrium at q0 and output sharply falls to q1, it may take time for the dental clinic to downsize its capital facility. As a result, the dental clinic may operate with costs, point b, that are higher than that predicted by long-run equilibrium, point a.

Long-Run Cost Minimization and the Indivisibility of Fixed Inputs

Cowing and Holtmann (1983) derived:

Long-run total cost, LTC

LTC = STVC(q, w ,k)+ r × k

LTC = sum of (minimum) short-run total variable costs, STVC and capital costs

STVC - depends on: quantity of output, q, the wage rate, w, and the quantity of capital, k

Long-Run Cost Minimization and the Indivisibility of Fixed Inputs

Necessary condition for long-run cost minimization

ΔSTVC/Δk = -r

Variable cost savings realized from substituting one more unit of capital

Must equal the rental price of capital in long-run equilibrium

Marginal benefits and costs of capital substitution should be equal

Long-Run Cost Minimization and the Indivisibility of Fixed Inputs

If estimated ΔSTVC/Δk is nonnegative

Medical firms are over-employing capital

Cost of capital substitution outweighs its benefit in terms of short-run variable cost savings

Long-Run Cost Minimization and the Indivisibility of Fixed Inputs

Cowing and Holtmann (1983), results

“Average” hospital in their New York sample

Too much capital and too many physicians

Hospitals could reduce their costs by limiting the amount of capital and controlling the number of physicians.

Neoclassical Cost Theory and the Production of Medical Services

Neoclassical cost theory

Assumptions

Perfect certainty

Firms produce as efficiently as possible

Perfect information

Short-run or long-run costs of producing a given level of output can be determined by observing the relevant point on the appropriate cost curve

Neoclassical Cost Theory and the Production of Medical Services

Misleading when applied to medical firms because:

Some medical firms

Are not-for-profit entities or are reimbursed on a cost-plus basis or both

No incentive to minimize costs

May operate above a given cost curve

Medical firms - uncertain demand for their services

May produce with some amount of reserve capacity just in case an unexpected large increase in demand occurs

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