AN OPTIMIZATION MODEL OF UNIVERSITY BEHAVIOUR
In this section we summarize the results of an analysis of the impact of various funding modalities on the behaviour of universities.^{[1]}
We assume that every university at every stage maximizes its probability of survival. The control variables are the tuition fee and teachers' pay.
In simulation experiments using this optimization model we compare the university's behaviour and the overall market supply in the system for various different funding modalities. We will test the common assumption that the introduction of tuition fees leads to a larger supply of university places and to higher teachers' pay.
In the experiments we compare three university funding modalities which are comparable in terms of the overall amount universities receive, but different in terms of the allocation of these funds within the system. The funding modalities are the following:
• tuition fees only
• a combination of tuition fees and subsidies,
• no tuition fees, i.e. subsidies from donors (e.g. the government] only.
In the case of subsidies we assume that the subsidy amount is directly proportional to the number of students, i.e. the donor sets a subsidy per student Operating costs are also proportional to the number of students, while maintenance costs are proportional to the (exogenous] capacity of the university.
The management of the university seeks a strategy that (with regard to its competitors] minimizes its risk of extinction. We assume that the biggest risks of extinction faced by nonprofit university organizations are:
a) excessively low teachers' pay (by comparison with the competition] — this increases the likelihood that key teachers will quit and go to work for other universities, which, in turn, may cause loss of accreditation;
b) excessively high teachers' pay — this increases expenditure, thereby increasing the likelihood that the university will go under for financial reasons;
c) a relatively high tuition fee (by comparison with the competition)^{[2]} — this may deter students, causing the university to lose money because it is not operating at full capacity;
d) a very low tuition fee — this may increase the number of applicants above the university's capacity and represents an opportunity cost (which economically rational decisiontakers must always take into consideration).
In cases b), c), and d) the university's profit deteriorates, thereby increasing its probability of extinction due to financial loss or due to loss of willingness of the owner (the donor, e.g. the government) to finance an excessively (by comparison with the competition) lossmaking operation.
By maximizing its probability of survival the university avoids risks a), b), c), and d) to the optimal extent At the same time it must keep an eye on the competition. For example, if teachers' pay increases in the system, it must respond by raising pay or else it will face an increased risk of extinction due to loss of accreditation. Likewise, an increase in the average tuition fee in the system will give every university scope to increase its tuition fee the following year.
The university's optimization task each year must therefore respect:
a) the tuition fees of its competitors last^{[3]} year,
b) the teachers' pay of its competitors last year,
c) its own (mainly capacity) constraints,
d) any changes to the subsidy regulations.
The university therefore for mulates its decisionmaking problem only in response to past outcomes and to the decisions of its competitors and (where relevant) the donor.
The appropriate instrument for modelling the decisionmaking process described above would seem to be a simple system of optimization submodels for each university, with the outcomes of optimization of one submodel influencing the other submodels in the next iteration step. We assume that average variables are taken into consideration, i.e. the university makes decisions when its tuition fee or teachers' pay is above or below the average in the system.
In its present for m, the model assumes that the donor's decision regarding the parameters of the subsidy regulations is exogenous. In the model simulation experiments, any reflection of this decision must therefore be limited to analysing the sensitivity of the model (i.e. to observing the impact of changes in selected exogenous parameters on the optimum or equilibrium in the system).
We assume that every university knows:
• the average tuition fee in the system in the past period,
• the average teachers' pay in the system in the past period,
• the number of teachers at all universities in the system in the past period,
• the number of students at all universities in the system in the past period,
• the total student demand in the given year for the system as a whole,
• the marginal teachers' pay — when pay is at this or a lower level, the university's probability of extinction as a result of teachers quitting is 100 per cent,
• the donor's (government's) subsidy per student,
• the university's fixed costs (proportional to its capacity),
• other costs per student.
We assume two threats to the university:
• the threat of insolvency, where expenditures exceed revenues (or revenues plus reserve fund),
• the threat of loss of accreditation as a result of (elite, accreditationenabling) teachers quitting.
The university cannot choose unilateral extremes. So, for example, while extremely low salaries (by comparison with the competition) will reduce the need for funds (and thereby reduce the threat of insolvency), they will lead to teachers quitting and thereby greatly increase the risk of extinction as a result of loss of accreditation. If the university gets part of its revenues from tuition fees, it again holds that the extreme strategy is not profitable — a sharp increase in the tuition fee may more than proportionally reduce the number of people applying to study at the university and thereby reduce its revenues. Even offering the incentive of a low tuition fee with the aim of making up revenue by attracting a large number of students will not necessarily benefit the university, as it is possible that the increase in student numbers will not cover the loss of tuition fee revenue.
As in other chapters we will assume that the probability of survival is proportional to the margin relative to the boundary of the extinction zone and can therefore be quantified using the Pareto probability distribution. In this chapter we will stay with the firstorder Pareto probability distribution (see Chapter 1).
The university sets its control variables in such a way as to minimize its probability of extinction due to a decrease in a key variable below its boundary of certain extinction. The key variables are:
• the university's revenue R for the threat of insolvency,
• the number of teachers b for the threat of loss of accreditation.
The boundaries of certain extinction are:
• R = C + F (where C denotes costs and F the reserve fund) for the threat of insolvency,
• b = bmin for the threat of loss of accreditation.
The control variables are:
• the tuition fee q,
• teachers' pay m.
The timevariant variables (updated in each iteration step) are:
• the number of students at the individual universities si,
• the number of teachers at the individual universities bi,
• the average tuition fee in the university education market Q,
• the average teachers' pay in the system M.
The timeinvariant parameters (identical in all iteration steps) are:
• the capacity (maximum possible number of students) of the ith university ki
• the total number of teachers B,
• the total study demand (total for all universities) D,
• the prescribed building and equipment maintenance costs δ relative to one (not necessarily filled) study place at the university,
• the donor's (government's) subsidy per student s.
The university chooses its control variables (tuition fee and teachers' pay) in such a way that its decision gives it the maximum probability of survival (of all the feasible alternatives).
We assume that the feasible alternative levels of the tuition fee and teachers' pay do not differ by more than 10% from the average for the whole system in the past period. The university is in a situation of uncertainty and is "feeling its way”, so its (theoretically rational) strategy is incorporated into its choice to that extent only. The university therefore regards a greaterthan10% deviation of the tuition fee or teachers' pay from the systemwide average as a sui generis risk. This condition will limit the possibility of having a "walltowall" strategy, which is unfavourable as regards the speed of achieving equilibrium in the system.
The sets of feasible values for teachers' pay Фm and for the tuition fee Фq (identically for all universities) are therefore the 21member sets
The iterative process continues until the average tuition fee and average teachers' pay in the system repeats itself or differs by less than a specified (low) value e (say the resolving power for teachers/study applicants) from one step to the next. Thereafter, no university has any incentive to change its choice from the previous step and equilibrium has been reached in the system.
The flow chart for this algorithm is shown on the following two pages.
We will now summarize the main conclusions from the computational experiments. Table 6 gives an overview of the main outputs (at system equilibrium, i.e. in the final iterative step, which is practically no different from the penultimate step). The final column in Table 6 shows the difference compared to the mixed variant combining government funding and tuition fees.
Table 6: Summary of results of computational experiments for comparison of alternative university funding modalities
Funding variant 
University number: 
Total 
Versus mixed 

1 
2 
3 
4 

No. of students 
Mixed 
597 
737 
733 
743 
2 810 
 
Fees only 
599 
780 
778 
781 
2 938 
+128 

State only 
556 
754 
1100 
1200 
3 610 
+800 

No. of teachers  university demand 
Mixed 
68 
85 
99 
108 
360 
 
Fees only 
66 
80 
90 
99 
335 
25 

State only 
56 
76 
110 
125 
367 
+7 

Tuition fee 
Mixed 
2.8 
2.8 
2.8 
2.8 
2.8 
 
Fees only 
7.3 
7.3 
7.3 
7.3 
7.3 
+4.5 

State only 
0 
0 
0 
0 
0 
 

Teachers' pay 
Mixed 
10.8 
10.8 
10.8 
10.8 
10.8 
 
Fees only 
10.4 
10.4 
10.4 
10.4 
10.4 
0.4 

State only 
10.5 
10.5 
10.5 
10.5 
10.5 
0.3 
Figures 26, 27, and 28 show the evolution of the number of students and the demand for teachers for each variant.
Figure 26: The path for tuition fee funding only (horizontal axis: iterative step number)
Figure 27: The path for combined funding (horizontal axis: iterative step number)
Figure 28: The path for government funding only (horizontal axis: iterative step number)
The experiments revealed that the number of iterative steps needed to stabilize the system is lowest for the liberal "tuition fee only" variant. This, however, is surprising only at first glance — both instruments for "finetuning” the choices of the individual universities (tuition fees and teachers' pay) are available to the full extent, whereas for the "government subsidy only" variant one of these instruments disappears, and in the mixed "tuition fee plus government subsidy" variant these instruments affect only part of the revenue.
Another piece of information to emerge from computational experiments not presented here is that the results are independent of the width of sets Фm, Фq, determining the permitted "force" of the iterative step. This is because the optimum was always substantially closer to the past average values for the whole system than to 10% of those average values (as used in the presented calculations).
The results of the model comparisons for the resulting variables for the system as a whole are worthy of note.
As regards the number of study applicants satisfied, the least advantageous variant is the one combining tuition fees and government funding. The highest number of applicants is satisfied (at roughly the same costs) by the purely statefunded system. This is because of the risk of extinction, which of course is naturally much lower in the pure government funding variant.
Demand for teachers is logically lowest in the liberal "tuition fee only” variant and highest for pure government funding. Note, however, that the mixed variant "saves" relatively little compared to pure government funding — as far as demand for teachers is concerned it is much closer to the government funding variant than to the liberal variant
The tuition fee is very sensitive — it is three times lower in the mixed variant, where around half the funding comes from the state and half from tuition fees, than in the liberal "tuition fee only" variant.
Rather surprisingly it turned out that teachers' pay is virtually the same. Hence, the expectations of university lecturers that their pay will rise drastically after the introduction of tuition fees may be very misplaced.
 [1] For detailed results of these experiments, see Hlaváček, J.: Dynamický model soustavy univerzit. WP IES No. 90. Praha: Fakulta sociálních věd UK, 2005, or Reichlová, N., Cahlík, T., Hlaváček, J., Švarc, P.: Multiagent Approaches in Economics. In Mathematical Methods in Economics, edited by L. Lukáš, 77–98. Plzeň: Západočeská univerzita, 2006, or Cahlík, T., Hlaváček, J., Marková, J.: Školné či dotace? (Simulace s modely systému vysokých škol). Politická ekonomie 56, 1(2008): 54–66.
 [2] Only, of course, if the university system is partly or fully funded from tuition fees. For the “centralist” case (subsidy funding only) risks c) and d) are not relevant.
 [3] It would be better (from the point of view of the highest possible probability of survival) for the university to know not only the past, but also the future strategy of its competitors. We do not assume this; as in the standard theory of the firm we assume that information of this type is confidential and unavailable to competitors.