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if necessary.

The author would like to note the work of Savkin, M. [9] as this study has probably the closest relation to the issues discussed below. The main difference between the works with lies in the analytical tools selected for modelling, Savkin approaches the agency problem from the perspective of signalling, identifying different (in respect of intrinsic motivation) types of agents, and different types of principals (corporate culture and environment). While the author uses psychological game theory in attempts to model the general process of establishment of incentive schemes that is suitable for analysis of different types of initial internal motivation. This approach, however, does not prevent discourage the use of signalling analysis, in fact, the model presented in Part 3, is advised to be perceived as general, and is suitable for modification in order to use more sophisticated analytical techniques.

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3. The model

The model, as noted above, is the product of works on agency theory, psychological game theory and theories of motivation. It, however, deviates in some notions from the original frameworks developed in sources presented above. In particular, the definition of psychological games implies that the payoffs of the actors are affected by beliefs on other players‟ strategy choice, while in this model the definition is not strictly followed, and is stretched to the case when, the beliefs, that influence the payoffs, are about some psychological variable, rather than the choice of strategy5. However, in general, the model does not violate the finding in related literature.

3.1. The Set-up

The framework of the model is based on the standard moral hazard problem model that can be found in most sources on agency theory (a compact and sufficient set up can be found in Symeonidis [5]).

For the sake of the model, we will perceive the agent as a manager – a human being with both extrinsic and intrinsic motives for action, and the principal as an organization, caring only about monetary benefits.6 In fact, we can set the general form of the utility functions as follows:

	(1)	A : U A W V C D
		P : U p	B

5This, of course is a flaw of the model, however, the author believes, that the psychological game theory can be developed further to account for cases, like this model proposes, or alternatively, the presented model can be adjusted to fit into the framework of psychological games in its strict definition.

6Without loss of generality it is safe to assume, that the owner or multiple owners have no intrinsic sources of satisfaction, with the exception for intangible assets, such as brand image or reputation (see below), from the P-A relationship.

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where W, V are the expected utilities from monetary rewards and intrinsic satisfaction of the agent respectively; C, D – the expected monetary costs directly related to work (such as transport cost, costs of moving, etc.) and the expected disutility of effort and other non-monetary costs of the agent7, respectively; Π – is the expected monetary benefit to the principal as a result of successful project8 And B is the agent‟s remuneration.

Within this general setup the model can be presented through a series of simple psychological games that can further be generalized to more complicated cases. For the moment, we will assume the following simple utility functions:

(2)	W w, x(e) b, A ;		V x(e) , A ;	C 0;	D e ;
	x(e) , ;	B b, P
Thus:
(3)		U A u w,b, , A , x(e), (e)
		UP	u ,b, P , x(e)

Variables are defined as follows:

w – agent‟s fixed wage; b – the determines the actual bonus which

endogenous variable (set by the principal) that is a function of beliefs and b9; this bonus is the

7It is also possible to assume, that utility of effort can be positive, for instance, for the creative jobs, the process of which is interesting and brings satisfaction, this case can be accounted for by allowing the disutility of effort function to include negative values in its codomain.

8Π – can be a function of profit (for private owner(s)) or the market value of the firm (for multiple owners of a public company), thus accounting for intangible assets and their value.

9This notion might seem puzzling, however, it has the following form in order to distinguish the effect of beliefs.

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result-contingent part of agent‟s remuneration; θ – intrinsic value of success to the agent10; e – agent‟s level of effort, x(e) – probability of project success as a function of agent‟s effort; ρ – project payoff in case of success; A ; P - sets of agent‟s and principal‟s beliefs, respectively.

In the games, described below, we will generally follow some assumptions, allowing to present the model in as simple a form as possible without loss of generality:

the principle and the agent are the only two players and have no other relationship alternatives, or all the agents and principals are identical

the agents receive either a zero fixed wage, or the fixed wage is identical across all principals, thus for the sake of the games the fixed wage factor is redundant

the agent is assumed to be risk-averse, while the principle is considered to be risk-neutral11

the level of effort is not observable and is not monitored, thus the incentive schemes are output contingent

the project in case of success pays the principal a payoff of ρ and 0 otherwise

games will generally consider up to the second order beliefs, i.e. up to one‟s beliefs about beliefs of the counteragent on one‟s set of strategies or some variable

10This variable is closely related to Valence in Vroom‟s expectancy theory (see source [6]). It is also possible to introduce a variable that would denote the disutility of failure.

11It is generally true, as a rational principal would generally seek to diversify risk by investing in multiple projects (see Douma and Schreuder [4]).

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3.2. The Game Form

3.2.1. Game 1

Let us consider a very simple decision-making game, the principal sets the parameter that defined the manager‟s bonus (b)12 and the agent chooses between high level of effort (e=1) and low level of effort (e=0). The level of effort is unobservable. The probability of project success is much higher in case of high effort than in case of low effort (x(1)>>x(0)). The intrinsic motivation of the agent is presented as a product of his second order beliefs and the value of success. Defining the variables as:

m Ep principal ' s belief about agent ' s ;

(4)

m EA m agent ' s belief about principal ' s belief of his

we, then, get , A m . For simplicity let us assume , m, m - all belong to the interval [0,1] and (1) 1; (0) 0 . So it is good for the agent not only to value success as much as possible, but also to believe, that the principal believes in his high intrinsic motivation, i.e. believes that the success is strongly valued by the agent, while for the principal it is good, if the agent exerts high effort as long as the net monetary benefit to him is positive, it is also good for principal to believe that the agent is highly motivated as the bonus announced in the contract is a share of b depending on principal‟s belief of agent‟s motivation: (1-m)b – the higher the intrinsic motivation expected, the lower the size of the bonus, to avoid crowding out of intrinsic motivation as predicted by Frey [13] or Benabou and Tirole [1]. Think of b as of some variable that helps to establish the bonus size, given the set of beliefs which in the marginal case, when m m 0

12 Assume, that by setting no bonus, the principal may at least guarantee a low level of effort from the agent, which he is obliged to do for his fixed wage rate according to labor legislation.

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is the bonus size for an unmotivated agent. The game in the normal form is simple and presented below13:

e=1

set b

P A

e=0

x(1) (1		) b		(1);	x(1) (1 m) b
	m		m

x(0) (1		) b		;	x(0) (1 m) b
	m		m

Fig. 1 – Game 1. At final nodes are the expected payoffs of the agent and after the semicolons – those of

The participation constraint (in some sources the „individual rationality constraint‟) for this model is assumed to be satisfied a priori, since the reservation wage is paid regardless of the project results and effort level. The incentive compatibility constraint [IC] for the game is as follows:

(5) x(1) (1 m) b m (1) x(0) (1 m) b m

Solving (5) for b:

x(1) x(0) 1 m b x(1) x(0) m (1)

	x(1) x(0)			(1)
b			m
	x(1)	x(0) 1
					m

13 According to Geanakoplos, Pearce and Stacchetti [2] – this is a summary form of the game with only two orders of beliefs, while in an original psychological game these orders/dimensions are usually higher, however, as the authors claim, this information should be sufficient to obtain a Nash Equilibrium.

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b 0 b 0 , thus:

from the payoff functions, we observe that for the principal

(1)

(6)

b

m

1

x(1) x(0) 1

m

m

Proposition 1 – A positive bonus is only announced only if it can satisfy the incentive compatibility constraint. Proof: assume the incentive compatibility constraint is not satisfied, then x(1) (1 m) b m (1) x(0) (1 m) b m and the agent will play low effort strategy (e=0), resulting in a lower payoff for the principal. However, the principal can guarantee at least e=0 by setting b=0; thus by setting a positive bonus that does not satisfy the incentive compatibility constraint, the principal only decreases his expected payoff: x(0) x(0) 1 m b b 0 . As a result, any bonus that

does not satisfy the incentive compatibility constraint should equal to 0.

A further constraint (non-binding)14 on b is derived from the payoffs of the principal:

(7)x(1) 1 m b x(0)

Let us call (7) the „bonus compatibility constraint‟ [BC]. The intuition behind this constraint is that, the bonus payment should be compatible with the utility maximization problem for the principal, i.e. the net expected payoff with a non-zero bonus stated in a contract should be at least as high as a payoff that can be achieved with certainty when no bonus is proposed, since as mentioned above, the principal is risk neutral. From (7) we derive the expression for b:

x(1) 1 m b x(1) x(0)

14 As noted above with b=0 the principal may ensure that at least e=0 is played, but the agent can still play e=1, if highly motivated

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(8)	b	x(1) x(0)
		x(1)		1 m

Proposition 2 – A positive bonus is only announced if it satisfies the bonuscompatibility constraint to the principal. Proof: assume that b determined in (6) does not satisfy the bonus compatibility constraint, then, x(1) 1 m b x(0) , then the principal will prefer to set a bonus of 0 and receive an expected payoff of at least x(0) , since the principal is risk-neutral and cares only about the expected payoff.

Equilibria.

In equilibrium m m =a [2], where a in this game is a constant defined on [0;1]. With this condition we can rewrite (6) and (8) as follows:

			a2			(1)
(9)	b
			1 a		x(1) x(0)			1 a
(10)	b	x(1) x(0)
			x(1)			1 a

3.2.2. Stylized facts and possible equilibria.

Since the expectations and the utility of success are continuous functions, equilibria will exist for any set of θ and beliefs as long as beliefs are coherent, i.e. as long as all the parameters are equal.

Given the equilibrium condition we can further analyze the model graphically. Let us firstly assign some values to the variables in the model. First let us assume that effort significantly affects the probability of success: x(1) 0.9; x(0) 0.15 ; and disutility of

effort	is	lower	than	the	probability	effect	of	effort:
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(1) 0.6 0.9 0.15 x(1) x(0) assume the project payoff of 1: 1 . In the graph15 the dotted line is the bonus compatibility constraint (10), and the shaded area below is the area, where it is satisfied; the thick line is the optimal b derived in the incentive compatibility constraint (9).

5	b
4.5
4
3.5
3						BC
2.5
2
1.5
1									IC
0.5										a
0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1

Fig. 2 – BC and IC as functions b(a) with the parameters set above

In this case the BC is satisfied for all points on IC, so for all a from 0 to a little less than 0.9 b is positive and some bonus payment will be stated in the contract, resulting in the equilibrium, where the agent exerts high effort. For a higher than that level, no bonus would be state in the contract, since the agent is highly motivated and the principal anticipates that, while the agent also believes that he is expected to be highly motivated. The intuition behind the graph with these parameters is consistent with common sense: for low a the agent is not motivated to exert high effort, as it disutility of effort, which needs to be compensated for, while as a grows and the project success becomes

15 Graphs are drawn with Advanced Grapher 2.2

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increasingly valuable for the agent, and the beliefs are coherent, the intrinsic utility partly compensates for the disutility of effort, thus requiring a smaller b and, consequently, a smaller bonus16. At some level b becomes 0, and, since b 0 , as there is no way the principal can state a negative bonus, so all negative b from the incentive compatibility result in b=0. In the case above the BC is met and the IC has a more or less sensible form, however, the parameters that we set earlier, need not necessarily follow the pattern proposed. In fact, they may be sufficiently different from those proposed, to change the behavior of either constraint.

Consider the bonus compatibility constraint first. Let us observe the effect of the project payoff magnitude ρ on the BC constraint. Obviously, an even higher ρ would result in an upward shift in BC, affecting the above result in no way, thus changes are expected in cases when ρ is lower. Lowering ρ results in downward shift in BC, which if sufficient will make some part of IC lie outside the BC. For that part of IC the bonus required for providing a credible incentive for the agent to exert high effort is no longer beneficial for the principal. Therefore, the principal will state a zero bonus along that part of IC (see Fig. A3 in appendix). Again, the intuition in this case does not contradict reality: the lower the payoff of the project the higher is the probability that the person considering it will be willing to incur additional costs (bonus) to improve the probability of its success. An interesting conclusion, that can be drawn in this case, is that, when the payoff of the project is small, agent‟s with higher intrinsic motivation a required for this project to be promoted by the principal (i.e. for him to set a remuneration for high effort). The effect of the magnitude of project payoff is probably the least ambiguous in the model.

16 Recall that b is some parameter, proportionally related to bonus, but is not exactly the bonus itself, which is determined as (1-m)b, so even though b(a) might be increasing in a for some medium levels of a the bonus in the contract is not necessarily increasing (see figure A2 in appendix).

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