Let us allow a university operating under the centralist "government subsidy only” variant to charge a tuition fee as well. This will lead to an increase in its supply of study places and to an increase in its probability of survival, i.e. the number of students will increase (the expected number, i.e. the optimum number multiplied by the probability of survival). How much this will be constraining in terms of optimal (from the university's perspective) filling of the university's capacity is a matter of demand. We model the supply of the university by including the additional tuition fee revenue (compared to the first of the experiments described above).

Figure 29: Supply of the university (expected number of students) the versus tuition fee (mixed variant: the university's revenue includes a government subsidy)

The function is growing and strictly concave. A reduction of the government subsidy would naturally shift the whole curve to the right. The complete abolition of the subsidy would lead to the supply function in Figure 30:

Figure 30: Supply of the university (expected number of students) versus the tuition fee ("no government subsidy" variant)

In the mixed variant the university receives a subsidy per student from the donor (the government). For the government subsidy allocation decision it is useful to construct a function showing the number of students (the expected number, i.e. the optimum number multiplied by the survival probability) as a function of the subsidy per student This is again a supply function, since the subsidy per student in the "government subsidy only” variant represents marginal revenue, i.e. the output price. For the first university in the experiments described above the supply function has the shape depicted in Figure 31.

Figure 31: Supply of the university (expected number of students) versus the subsidy per student ("no tuition fee" variant; subsidy revenue only)

If we take into account neither the average tuition fee nor teachers' pay in the system in the past period, we view the university's supply simply as the product of its capacity and its probability of survival. The supply function is the relation between supply s and marginal revenue MR, i.e. the sum of the tuition fee and the government subsidy per student.

Figure 32: Supply of the university (expected number of students) versus the marginal revenue MR

Like in the standard theory of the firm, therefore, the university supply function is zero up to a certain "price" for its service (the shutdown point) and positive, growing and strictly concave above that point. This holds for the last three supply functions given above (Figures 30, 31, and 32). For the first supply function (Figure 29) the high government subsidy enables the university to survive even with a zero tuition fee, hence this function is positive, growing and strictly concave over its entire domain.

As in the standard theory of the firm, the supply function here describes the subjectively optimal situation (at a particular value of its explanatory variable] of all the situations lying in the set of feasible situations.^{[1]} Taking into account the parameters in the system in the previous period (Figures 29, 30, and 31) puts a stricter constraint on this set of feasible situations by comparison with the last-mentioned supply function (Figure 32), as it is not admissible to deviate from the previous parameters by more than the specified margin. The suitability of using the first or second method to describe supply (i.e. of using the narrower or wider set of feasible situations) will depend on the type of problem being solved.

[1] The analogy in the theory of the firm is the set of technically feasible situations, i.e. the set of production situations not lying above the production function.

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