Another concern you will have as a manager is what output you gain for each successive increase in input. Is the impact (or return) constant or does it vary with the overall size of the service?
Returns to a factor indicates the addition to output by increasing only one factor.
Returns to scale, on the other hand, indicates the addition to output attained by increasing all factors by the same proportion. People talk about returns to a factor in the short term because at least one factor is fixed. Returns to scale is a long-run phenomenon because only in the long term can all factors be increased proportionally.
Increasing returns to scale are explained in the same way as increasing returns to factor – by improved division of labour and increased specialization. In the long term there is less constraint on specialization because the quantity of all factors is variable. Hence there may be increasing returns to scale over ranges where there were decreasing returns to a factor.
Decreasing returns to scale are usually explained by the difficulties involved with managing and coordinating all the decisions that need to be made to run a large organization.
Activity 8.3
Compare the total product associated with one vehicle and one nurse with that for two vehicles and two nurses and so on. Are there constant returns to scale? What might explain the pattern you observe?
Feedback
The output added by one nurse and one vehicle is 15 visits (see Table 8.5). If you add another nurse and another vehicle then you get 15 more visits. Over this range you can say that there are constant returns to scale. As you add another nurse and another vehicle you get an additional 70 visits. This means there are increasing returns to scale.
If you do so again you get even more (80 visits) but afterwards the extra product from one nurse and one vehicle is only 20 visits. Therefore there are decreasing returns to scale. This may be explained by the geographical spread of the population and the fact that the population and therefore output is limited.
Table 8.5 Returns to scale
Input combinations Total product Extra product
1, 1 15 15
2, 2 30 15
3, 3 100 70
4, 4 180 80
5, 5 200 20
Now continue with your analysis of the long-term options for the community nursing service.
Activity 8.4
1 In Table 8.6 finish filling in the total cost associated with each input combination using the factor prices mentioned in Activity 8.1 – ignoring the operationally ineffi- cient combinations. Also calculate the average cost for each combination and enter it between the parentheses.
2 What is the lowest cost for the following output levels: 20, 30, 100, 180, 200?
Feedback
1 Total cost is equal to the number of nurses multiplied by £30 plus the number of vehicles multiplied by £25. For two nurses and two vehicles this is (2 × 30) + (2 × 25) =
£110. Average cost is total cost divided by output (output is given in Table 8.3). For two nurses and two vehicles this is 110 ÷ 30 = £3.67. The full set of total and average costs is shown in Table 8.7.
2 There are three input combinations that give an output of 30: 2N2V, 4N1V and 10N.
These have costs of £110, £145 and £300 respectively. Therefore the lowest cost method of producing 30 visits is two nurses and two vehicles. The other minimum cost points are given in Table 8.8. Every point in this table is economically efficient because at these points output cannot be increased without incurring extra cost. Or, put another way, they are situations where cost cannot be reduced without reducing the total product.
Table 8.6 Total cost (average cost) in £ per day Vehicles Nurses
1 2 3 4 5 6 7 8 9 10
0 30 60 90 120 150 180 210
(6) (6) (6) (6) (6.82) (7.5) (8.08) ( ) ( ) ( )
1 55 85 115 145 175 205 235
(3.67) (4.25) (4.6) (4.83) (5) (4.56) (3.92) ( ) ( ) ( )
2 110 140 170 200 230 260
(3.67) (2) (1.7) (1.54) (1.53) (1.53) ( ) ( ) ( )
3 165 195 225 255
(1.65) (1.3) (1.25) (1.28)
4 220 250
(1.22) (1.28)
5 275
(1.38)
Another concern you will have as a manager is how the average cost of a nurse visit varies with the size of the service. In other words, are there economies (or disecono- mies) of scale? Generally, economies of scale (i.e. average costs falling as output increases) are explained by three factors:
• increasing returns to scale;
• falling factor prices with increased scale;
• spreading of fixed costs.
Likewise, diseconomies of scale (i.e. increasing average costs with respect to output) are explained by:
• decreasing returns to scale;
• increasing factor prices with increased scale.
Table 8.9 summarizes the relationship between cost and output in the long run compared to the short run. Returns to scale have already been discussed. Factor prices might decrease with scale because the firm is able to negotiate a better deal with the suppliers of inputs – for example, the vehicle supplier might give dis- counts for multiple purchases. On the other hand, factor prices might increase with scale perhaps because the factor becomes harder to find. For example, it might be harder to find more than three nurses available for work in the local area and Table 8.7 Total cost (average cost) in £ per day (solution)
Vehicles Nurses
1 2 3 4 5 6 7 8 9 10
0 30 60 90 120 150 180 210 240 270 300
(6) (6) (6) (6) (6.82) (7.5) (8.18) (8.57) (9.31) (10)
1 55 85 115 145 175 205 235 265 295 325
(3.67) (4.25) (4.6) (4.83) (5) (4.56) (3.92) (3.53) (3.28) (3.25)
2 110 140 170 200 230 260 290 320 350
(3.67) (2) (1.7) (1.54) (1.53) (1.53) (1.61) (1.68) (1.75)
3 165 195 225 255
(1.65) (1.3) (1.25) (1.28)
4 220 250
(1.22) (1.28)
5 275
(1.38)
Table 8.8 Long-run costs
Total product Input combination Total cost (£) Average cost (£)
20 2 nurses, 1 vehicle 85 4.25
30 2 nurses, 2 vehicles 110 3.67
100 3 nurses, 3 vehicles 165 1.65
180 4 nurses, 4 vehicles 220 1.22
200 6 nurses, 3 vehicles 255 1.28
therefore employing more will involve advertising or perhaps increased pay to lure nurses away from their current employment. For Figure 8.3, however, factor prices are assumed to be stable and the relationship between cost and output is caused by returns to scale alone.
It was stated that in the long run all factors are variable. However, there may be costs that are fixed with respect to output even in the long run. For example, marketing costs or research and development costs may be entirely unconnected to output. Therefore as output is increased, these costs are spread over a larger number of units.
Figure 8.3 Short-run average cost
Table 8.9 Causes of trends in average cost with respect to total product
Short run Long run
Average cost falls as output rises (economies of scale)
1 Increasing returns to factor (specialization)
1 Increasing returns to scale (specialization)
2 Spreading of fixed costs 2 Decreasing factor prices 3 Spreading of fixed costs Average cost rises as output
rises (diseconomies of scale)
Decreasing returns to factor 1 Decreasing returns to scale (i.e. bureaucracy) 2 Increasing factor prices
Activity 8.5
1 For which input/output combination are average costs at their lowest?
2 In Figure 8.3, plot the long-run average cost curve alongside the short-term curve (from Activity 8.1). Does it display economies of scale? Why?
3 Consider what would happen if factor prices, such as the price of vehicles or of nurses, were to change.
Feedback
1 The output level with the lowest average cost (from Table 8.8) is 180 visits per day using four vehicles and four nurses. This is described as scale-efficient because at this output level the cost per visit is minimized. Therefore the size of the operation is optimal.
2 Figure 8.4 shows the long-run average cost plotted next to the short-run average cost curve that was plotted in Activity 8.1. The long-run curve has the same shape as the short-run curve – U-shaped. At higher levels of output the long-run cost curve is below the short-run curve. This is because the use of two vehicles is an inefficient technique for producing this level of output. The U-shaped long-run cost curve implies economies of scale initially followed by diseconomies of scale for higher levels of output.
3 If the vehicle costs were higher, say £35 per day, then all input combinations with one or more vehicles would be more costly. The greater the number of vehicles the more would be the additional cost. A rise in the cost of vehicles, if the rise is large enough, will cause a cost-efficient provider to substitute from vehicles to nurses. For example, Figure 8.4 Long-run and short-run average cost
Table 8.10 shows the cost of producing 180 visits a day using different input combin- ations. It shows the effect of an increase in the price of vehicles from £25 to £35. At £25, four nurses and four vehicles will be the least-cost method of producing 180 visits.
However, at £35 five nurses and three vehicles will be the least-cost method of pro- ducing 180 visits. Hence the increase in vehicle price causes a substitution from vehicles to nurses. The change from £25 to £35 shifts the long-run average cost curve upwards.
The substitution means that costs do not increase as much as they would have if the input levels had remained unchanged.
A rise in the price of nurses, on the other hand, will cause a cost-efficient provider to substitute nurses for vehicles. A country where the price of labour is high relative to the price of capital is likely to have more capital-intensive production. A country where the price of labour is low relative to the price of capital is likely to have more labour intensive production.
What do you think are the implications of the above cost analysis for community nursing service resource allocation? You can look at this question from two per- spectives. First, the perspective of the producer (or provider) and second, the perspective of the service as a whole.
As far as the producer goes, the analysis does not tell us very much about the appropriate output level, although it does tell us which input combinations are economically efficient for each output level. The actual output level has to depend on the budget as well as the cost. If the budget of the community nursing service is
£110 a day then they will be able to make 30 visits a day – see Table 8.8. Alter- natively, if the budget is £275 a day then they will be able to produce 200 patient visits each day.
Summary
You have learned how the production function describes the relationship between inputs and output (or product) and the cost function describes the relationship between output and cost. You went on to learn about marginal outputs and econ- omies of scale. In the short run this is because of increasing returns to factor (due to increased specialization) and the spreading of fixed costs. In the long run this is because of increasing returns to scale (due to increased specialization) and because input prices might decrease as output rises. Eventually average cost may get larger – diseconomies of scale. Finally you saw that when average costs are at their long-run minimum, the provider is not only technically efficient and economically efficient but also scale-efficient.
Table 8.10 Total cost of 180 visits (£)
Input combination Cost if vehicle price is £25 Cost if vehicle price is £35
4 nurses, 4 vehicles 220 260
5 nurses, 3 vehicles 225 255
10 nurses 300 300