This column again presents several interesting models of new institutional economics. First the repeated game of the prisoners is analyzed. Next a model is presented, which takes into account the search costs on the market. Also useful are models, which describe the maintenance of norms. And finally the principal-agent model is again discussed. The used abstractions of human nature are tested with insights from psychology.

Traditionally the economic science uses the anthropology of the homo economicus: a rational decider, who defends his own interest. Moreover, in the standard theory he has stable preferences, and he is omniscient. This image of man is controversial, also within economics itself, since the rise of behavioural economics. Therefore during the past half of a century various new models have been proposed, which make the homo economicus more human. A large part of these models is included in the new institutional economics. It notably takes into account the transaction costs, which inevitably emerge during human actions. Such additional costs make the situation significantly more complex.

For instance, the complete search process for the optimal transaction itself is accompanied by costs. Moreover, investments are needed in order to maintain norms and other institutions. Labour contracts and other economic agreements are not automatically observed, so that extra costs are needed for their execution. On the other hand, a positive point is that under suitable conditions the homo economicus is willing to cooperate, albeit merely for serving his own interest. Nowadays such properties are included in the economic models, and the present column discusses some important ones.

In the present economic science the game theory has established a prominent position. In various columns the *prisoner's dilemma* has been mentioned. For the sake of convenience, this game (transaction) is presented in the table 1. There are two actors A_{1} and A_{2}, who both can choose for cooperation or exploitation. They can not in advance make a binding agreement. The table 1 shows the rewards (b_{1}, b_{2}) for A_{1} and A_{2} in each situation^{1}. It is clear that cooperation is the best option for the actors as a collective. However, each actor separately must fear, that the other will exploit him. In this situation each actor wants to optimize his utility (outcome). Therefore both actors will finally choose exploitation, although they thus obtain a bad result as a collective.

A_{2} cooperates | A_{2} exploits | |
---|---|---|

A_{1} cooperates | 1, 1 | -1, 2 |

A_{1} exploits | 2, -1 | 0, 0 |

The collectively worst outcome (0, 0) is called the *Nash equilibrium*. For, no actor can unilaterally improve his situation. And the choice of both actors is called the *dominant* strategy. In practice the one-shot prisoner's dilemma often occurs. However, even more often both actors will engage in repeated actions, and that changes the character of the game^{2}. Namely, when the game is repeated, the actor must not optimize the one-shot outcome, but the outcome of all of his transactions together. It is interesting that A_{1} and A_{2} can use their transactions for mutually signalling their intentions. In the present situation it turns out that the *tit-for-tat* strategy yields good results. Both players start with cooperation, and in each subsequent transaction imitate the choice of the other in the preceding interaction.

It is immediately clear, that now both actors will always cooperate. Explotation by an actor would be foolish, because the other would imitate it. Incidentally, each actor could return to cooperation by again starting to do it. Explotation is no longer automatically the dominant strategy, because the tit-for-tat strategy can yield more for both actors. Under these circumstances the tit-for-tat strategy could become a social *norm*, which benefits all. Unfortunately the hallmark of an informal norm is, that not everybody observes it. Assume for the sake of convenience, that each actor separately and consistently chooses the tit-for-tat or exploitation strategy. One may calculate when obedience to the norm is profitable. Suppose that for each transaction the probability is π, that at least a further transaction will follow. The the total *expected* outcome is given by^{3}

(1) B_{k} = Σ_{j=1}^{∞} b_{k}(j) × π^{j-1}

When one of the two actors is an exploiter, then after the first transaction both actors will exploit. This yields 0, so that the total expected outcome equals those in the table 1. The outcome will be different only when two actors in the tit-for-tat group meet. For, then one will always have b_{k}(j) = 1. Therefore the series in the formula 1 obtains a simple form, namely 1 / (1 − π). The table 2 gives the outcomes of the repeated game of prisoners.

A_{2} cooperates | A_{2} exploits | |
---|---|---|

A_{1} cooperates | (1 − π)^{-1}, (1 − π)^{-1} | -1, 2 |

A_{1} exploits | 2, -1 | 0, 0 |

Apparently the tit-for-tat combination leads to the best B_{k}, as soon as π>½ holds. Now suppose that a fraction p of the actors belongs to the tit-for-tat group. In other words, p is the *trust*, which the actor has in the validity of the norm. Then, according to the table 2 the expected outcome for the tit-for-tat strategy is p / (1-π) + (-1) × (1-p), and for the exploitation strategy 2×p. The tit-for-tat strategy is at least equal in value, as soon as p / (1-π) + p − 1 ≥ 2×p holds. That is to say, one must have p ≥ 1/π − 1. This formula expresses the quality of the norm. A special case occurs, when an (almost) endless series of transactions is expected. For, then one has π=1, so that p≥0 must hold and therefore the actors with a tit-for-tat strategy perform on average as good as the rest, or better.

Remarkable is also the situation, where two actors do adhere to the tit-for-tat norm (that is to say, p=1), but on j=J+1 *by accident* one chooses exploitation. Suppose that the last interaction occurs on j=N (which is evidently unknown in advance). Tit-for-tat implies, that the reaction to the action of the other is lagged. Therefore, now for interactions j>J both actors will alternate exploitation and cooperation, which results in a series of outcomes (-1, 2) en (2, -1). Then one will have B_{k} = J×1 + (N − J) × ½ × (-1+2) = ½ × (N+J). The mistake of the one actor has caused a loss of no less than ½ × (N-J) for both. Apparently the other actor will benefit from forgiving the mistake, and continuing to cooperate. That would yield N-2 for him, and N+1 for his clumsy competitor. The tit-for-tat strategy would no longer by optimal^{4}.

This example of the prisoner's dilemma with repetition makes the essential assumption, that the actors have a fixed strategy. They are encouraged in their morals by the fact, that the moment of the last interaction is unknown. Only in this case the tit-for-tat strategy is interesting. For, suppose that j=N would with certainty be the last interaction^{5}. Assume furthermore, that the actors have yet to choose their norm of stategy. In the interaction j=N, exploitation becomes again the dominant strategy. However, now it follows, that the tit-for-tat strategy is also unsound for j=N-1. Therefore on j=N-1 the strategy of exploitation also dominates. Etcetera. Rational actors will already immediately at j=1 select exploitation as their norm. Apparently the tit-for-tat strategy is only rational for an *undetermined* repetition of interactions. Fortunately in reality the number of interactions is usually indeed indeterminate in advance.

In conclusion: as long as the number of interactions is indeterminate, a certain norm of behaviour will not develop naturally. It is true that for a sufficiently large p it is optimal and thus rational to adhere to the tit-for-tat morals. But when an actor prefers the morals of exploitation, then this is a stable situation, where cooperation will never occur. Nevertheless, experiments in laboratories show, that the test persons indeed exhibit some tit-for-tat behaviour. Incidentally, the situation can develop in a dynamic manner^{6}. Furthermore, in such experiments it turns out, that even in one-shot games or transactions the actors are sometimes inclined to cooperate. Apparently man is by nature not a *homo economicus*, who purely egoistically and rationally defends his own interest^{7}. Besides, certainly outside of the laboratories the individuals are socially embedded, and subjected to group pressure, so that it becomes rational for them to moderate their own egoism^{8}.

The famous economist R.H. Coase has emphasized the search costs, which must be made in order to find the best product-price combination on the market. The search naturally does save money, because in this manner the actor realizes a favourable exchange. This implies, that an optimal search-effort exists, where the benefits and the costs of the search are balanced. It is possible to describe this search behaviour by means of a mathematical model^{9}. Suppose that the distribution of prices p is uniform in the interval [0, p_{max}]. That is to say, the distribution function (probability density, frequency function) has the constant value f(p) = 1/p_{max}. Let F(p) = ∫_{0}^{p} f(ρ) dρ = p / p_{max} be the cumulative distribution function. It is the probability that the supply price is not larger than p. Apparently there is a probability of Pr(>p) = 1 − p/p_{max}, that the supply price is higher than p.

The search of the consumer requires demanding various offers (unless the first offer is a fortunate random hit). Namely, according as one has more offers, it becomes more probable that it includes a low offer. Suppose that the isolated demand of an offer leads to costs c, and that the consumer wants to buy q pieces of the product. Suppose that the lowest supply price of all those n offers is at most p_{0}(n). Unfortunately the consumer can not in advance calculate his final costs T, due to the random nature. But he can estimate the *expected* costs, namely E(T) = E(n×c + q×p_{0}) = n×c + q × E(p_{0}). The minimization of E(T) is only possible, when the relation between the *expected* E(p_{0}) and n is determined.

Assuming n offers, all offered prices are p_{0} or larger with a probability of Pr(>p_{0})^{n}. It follows immediately from this, that the situation, where at least one offered price is below p_{0}, occurs with a probability of 1 − Pr(>p_{0})^{n}. This probability Pr(p ≤ p_{0}) naturally corresponds again with a cumulative distribution-function, say G(n, p_{0}). The corresponding density-function g(n, p_{0}) is the derivative of G. Thus p_{0} has the expected value^{10}

(2) E_{n}(p_{0}) = ∫_{0}^{pmax} p_{0} × g(n, p_{0}) dp_{0} = p_{max} / (n + 1)

Now the consumer can determine, when demanding the next offer is no longer profitable. According to the formula 2 the expected costs E(T) are minimal for n = -1 + √(q×p_{max}/c). Therefore this is the number of offers, that is optimal for the consumer (see figure 1). This value n=n_{0} also determines, by means of the formula 2, which upper limit p_{0} of the supply price is desirable for a given c and q. There is truly an exchange possible between information costs and the sales price. The consumer may of course have bad luck, so that after n offers all supply prices are still above p_{0}. On the other hand, he may be lucky, and (almost) immediately obtain a low offer. In this case it is useless to continue the search. There are rules of thumb also for this situation^{11}.

For, let p_{0} be the unexpectedly low offer. Demanding another offer only makes sense, when it is expected that this transaction will lead to lower costs than q×p_{0}, so that E(T, another) = c + q × E(p, another) ≤ q×p_{0}. Now E(p, another) has two components. The next offer can lie above p_{0}, with a probability of Pr(>p_{0}) = 1 − p_{0}/p_{max}. In that case the sales price remains p_{0}. Or the next offer lies below p_{0}, and then one has E(p_{0}') = ∫_{0}^{p0} p × f(p) dp = p_{0}²/2. Combining the two components yields E(p, another) = p_{0} × (1 − ½×p_{0}/p_{max}). Insertion in the formula for E(T, another) leads to

(3) E(T, another) = c + q × p_{0} × (1 − ½ × p_{0}/p_{max})

Apparently another search offers good prospects, as long as one has c ≤ ½×q × p_{0}²/p_{max}. This model surprisingly shows, that the neoclassical paradigm can take into account the transaction costs. Here it must be noted, that the situation is special. For, the consumer knows in advance the probability distribution f(p) of the supply prices. This is also information, which undoubtedly came with a price for the consumer.

For instance, he may be a member of a consumer network, which provides him with information. According to the theory of social capital C_{s} such networks can indeed reduce the transaction costs. The members of a network pay a price, namely that they must maintain the collective norms of the network, by being obedient and by encouraging others to adapt. Besides, the members of the network will only cooperate, for instance by providing information, as long as the beneficiary feels obliged to return the favour at a later time. Such complex mechanisms can not be included in the model. The neoclassical paradigm does not (for the moment) contain formulas to describe social processes such as rights and duties. Therefore many still feel somewhat discomforted, when applying this paradigm.

At the end of the preceding paragraph is has been suggested, that each group has its own informal and formal institutions. These make the individual behaviour easier to predict, but also limit the freedom of the group members. Institutional influences are important, but their theoretical modelling is yet in its infancy. A previous column has described some of these early models. The present paragraph elaborates on this. First, consider the game of Tsebelis, which models the interaction between a group member A_{1} and an inspector A_{2}. The group member A_{1} can obey the group norm, or violate it. The inspector A_{2} can maintain the norm, or fail to do this (for the love of ease or another reason). The table 3 summarizes the typical outcomes of this game^{12}. The reader can check, that the outcomes are logical. For instance, as long as A_{1} obeys, it does not make sense for A_{2} to maintain.

A_{2} maintains | A_{2} does not | |
---|---|---|

A_{1} violates | 0, 1 | 1, 0 |

A_{1} obeys | 1, 0 | 0, 1 |

This game is special, because there does not exist a Nash equilibrium of behavioural strategies. For, each outcome (b_{1}, b_{2}) allows one of the actors to improve his situation by changing his behaviour. For instance, suppose that A_{1} obeys, and A_{2} does not maintain. Then A_{1} benefits from henceforth violating the norm. This situation is confusing for both actors. In principle they can create a Nash equilibrium by both choosing a so-called *mixed* strategy. For instance, A_{1} can choose to obey with a probability of p_{1}, and therefore to violate with a probability of 1 − p_{1}. In the same way A_{2} must choose to maintain with a probability p_{2}, and therefore to fail to do it with a probability of 1 − p_{2}. The matrix of probabilities is shown in the table 4.

A_{2} maintains | A_{2} does not | |
---|---|---|

A_{1} violates | 1-p_{1}, p_{2} | 1-p_{1}, 1-p_{2} |

A_{1} obeys | p_{1}, p_{2} | p_{1}, 1-p_{2} |

In such a situation with mixed strategies the outcomes are purely coincidental, so that the actors can merely estimate the *expected* outcomes E(b_{k}) for a series of interactions. Thus one finds E(b_{1}) = p_{1} × (p_{2}×1 + (1-p_{2})×0) + (1-p_{1}) × (p_{2}×0 + (1-p_{2})×1) = 2×p_{1}×p_{2} − (p_{1}+p_{2}) + 1. Now A_{1} obtains his best outcome for ∂E(b_{1})/∂p_{1} = 0, that is to say for p_{2}=½. In the same way one has E(b_{2}) = p_{2} × ((1-p_{1})×1 + p_{1}×0) + (1-p_{2}) × ((1-p_{1})×0 + p_{1}×1) = p_{1}+p_{2} − 2×p_{1}×p_{2}. Also A_{2} obtains his best outcome for ∂E(b_{2})/∂p_{2} = 0, that is to say for p_{1}=½. Apparently it is attractive for both actors to choose the mixed strategy with p_{1} = p_{2} = ½. The expected outcome in this optimum is (½, ½). The situation is comparable to the *best reactions* in an oligopoly of the Cournot type. An actor, who deviates from this, hurts himself and favours the other. An example: for p_{2}=¼ one has E(b_{1}) = ¾ − ½×p_{1}. Then A_{1} can choose p_{1}=0 and the outcomes are (¾, ¼).

Thanks to the mixed strategy now a Nash equilibrium can be realized. The two actors have created a stable situation in a rational manner. In addition, there is predictability, at least concerning the average behaviour. However, it would imply that the actors no longer use a pure-strategy norm, which intuitively feels undesirable. The human nature and cognition are not designed to apply morals in an opportunistic manner. This illustrates that indeed sometimes the instrumental rationality is less appealing than value rationality. Nevertheless, in practice the mixed strategy yet often occurs. Consider the inspection of passenger tickets in the public transport, which is never more than taking regular samples. Yet the intention is also here, that free riding is discouraged, and that the passengers internalize the norm .

The internalization of institutions is a fascinating theme, because in this way the individual changes his utility function. The Dutch economist P. Frijters has made a first attempt to include this phenomenon in the utility function. His colleague B.M.S. van Praag has found empirical formulas for the preference drift, where the individual adapts the utility to his reference group. Nonetheless the majority of the models assumes a fixed utility function. Then the individual is simply a rational decider. Consider again an actor A_{1}, who expects an outcome b_{1} of a transaction. This yields him a utility v(b_{1}). Suppose that he can increase the outcome with Δb_{1} by violating a norm. This would make the transaction more useful. Unfortunately for A_{1} there is an inspector, who can punish violations with a fine s. Just like in the game of Tsebelis the probability of maintenance equals p_{2}. Then the *expected* utility U of A_{1} due to a violation becomes^{13}

(4) U = p_{2} × v(b_{1} + ρ×Δb_{1} − σ×s) + (1 − p_{2}) × v(b_{1} + Δb_{1})

The parameters ρ and σ have values between 0 and 1. That is to say, even in the case of maintenance it is conceivable that A_{1} yet appropriates a part of Δb_{1}. For instance, the inspector will sometimes not know the true size of Δb_{1}. Conversely, it is conceivable, that the fine s is not imposed completely. For instance, A_{1} can have good relations with lawyers. Or A_{1} simply underestimates the sanction s by a factor σ. Finally, assume that the actor A_{1} is neutral to risk. Then he will decide to obey the norm, as long as one has U ≤ v(b_{1}). Therefore the inspector must choose the parameters p_{2}, ρ and σ in such a way, that this condition is satisfied. It follows immediately from the formule 4, that A_{1} will obey as long as one has

(5) p_{2} ≥ (v(b_{1} + Δb_{1}) − v(b_{1})) / (v(b_{1} + Δb_{1}) − v(b_{1} + ρ×Δb_{1} − σ×s))

The formula 5 shows, that the inspector has several options. A reduced probability p_{2} can be compensated somewhat by increasing the fine s. Due to the neutrality of A_{1} with regard to risk it can be assumed that v(b) = b. Write the right-hand side of the formula 5 as p_{2}°. Then one has p_{2}° = 1/ (1 − ρ + σ×s / Δb_{1}), and the elasticity of p_{2}° for s can simply be calculated: (∂p_{2}° / ∂s) × (s / p_{2}°) = -p_{2}° × σ×s / Δb_{1}.

The formula 5 is useful, because it clearly shows the causal relations for maintenance. Nevertheless, it is difficult to apply in practice. For, each individual has his own utility function U. Some will not be risk-neutral, but avoid or search risk. Those who can bearly live from b_{1}, will have a strong desire to acquire Δb_{1}. Sometimes an offender can count on the admiration in his own subgroup^{14}. It turns out the children are more stimulated to internalize a norm, when the punishment s is mild instead of severe. Since the external coercion remains limited, they get the idea that obedience to the norm is a personal choice^{15}. And since the function U is so personal, this model has mainly been applied to statistical data of large groups, where the individual properties average out.

Furthermore, note that the fine s can also be immaterial. Think about loss of status, disapproval by the group, or even exclusion^{16}. In fact the immaterial punishment will even dominate within groups, because most group norms are informal. For, violations undermine the mutual cohesion, and therefore the existence of the group. Moreover, obedience leads to self-affirmation of the group members, so that conversely a violation can be accompanied by some *self*-punishment^{17}. The differences in power between the group members originate more from the social relations than from the material inequality. The immaterial sanctions naturally cause material losses in the long term, but the size of those losses is difficult to quantify. The rational-choice paradigm of the sociologist J.S. Coleman is a couragious attempt to model the exchange (substitution) of material and immaterial sanctions.

A core piece of the new institutional economics is the principal-agent problem. A well-known variant of this problem is the situation of an enterprise, which due to several accidental circumstances has a fluctuating production q per period of production. That is to say, q is distributed according to a probability density f(q). The variant assumes, that the yield q is completely spent on paying the wage sum w and the profit π (so q=w+π). Therefore the income of the participants in the enterprise is subjected to risk. Suppose that the workers (as agents) are risk-averse, whereas the entrepreneur (as principal) is risk-neutral. In this situation it is important, whether the entrepreneur is able to measure the effort e of the workers. When this is the case, then the wage can simply be fixed in a contract. But when the entrepreneur can not measure e, then he must include income incentives in the labour contract.

It is common in this variant to mathematically calculate the optimum of the entrepreneur and the workers. However, it is also possible to analyze the problem by means of an Edgeworth box, and that is the theme of this paragraph^{18}. Although the graphical method does not yield *new* knowledge, it does *deepen* the already existing insights of the mathematical method. The Edgeworth box is usually applied in the description of the exchange of goods. Here a special exchange will by analyzed, namely the distribution of the income risks. The application of the graphical method requires, that the probability density f(q) of the yield is simple. Only two yields are allowed, namely q_{H} with a probability of p_{H}, and q_{L} with a probability of p_{L} = 1 − p_{H}, where q_{H}>q_{L} holds. Then the income risk is q_{H} − q_{L}. It is advantageous for all, when the entrepreneur insures his workers for this risk, and in exchange receives a contribution^{19}.

The figure 2 shows the Edgeworth box for the insurance. Horizontally the incomes for q_{H} are displayed, and vertically the incomes for q_{L}. The origin of the coordinate system (w_{H}, w_{L}) of the workers is left below, and the values along the axes increase to the right and upwards. The origin of the coordinate system (π_{H}, π_{L}) of the entrepreneur is right above, and the values along the axes increase to the left and downwards. So the origin of the entrepreneur is the point (q_{H}, q_{L}) of the workers, and vice versa. Since the workers value security, they prefer the situation w_{H}= w_{L}. In the figure 2 this so-called certaintly line is shown as the line with an angle of 45°. For the sake of convenience the certainty line of the entrepreneur is also shown. These two lines coincide only, when q_{H}=q_{L} holds.

with observable effort

The expected wage of the workers is E(w) = p_{H}×w_{H} + p_{L}×w_{L}. A similar formula holds for the expected profit E(π). In the figure 2 the lines with a constant expected income (E(w)=a or E(π)=a with constant a) apparently have a slope of -p_{H}/p_{L}. Call these the iso-wage lines (for the workers) and the iso-profit lines (for the entrepreneur). Note that E(w) + E(π) = E(q), and this is fixed in the present situation. Therefore the iso-wage lines and the iso-profit lines coincide. In the figure 2 they are shown in green. The workers derive an utility v(w) from their wage, and the entrepreneur experiences an utility φ(π). Since the entrepreneur is neutral to risk, the indifference curves (φ(E(π)) = φ(a) = constant) of his expected profit match his iso-profit line E(π)=a. Therefore both are shown as green lines in the figure 2.

This is different for the workers. Consider an iso-wage line E(w)=a. In the intersection with the certainty line E(w) = w_{H} = w_{L} one has v(E(w)) = v(a). Yet the indifference curve v(E(w)) = v(a) of the expected wage does not coincide with this iso-wage line. For, when one moves over the iso-wage line, in such a manner that the distance to the mentioned intersection increases, then the difference w_{H} − w_{L} becomes larger and larger. This leads to dissatisfaction for the workers, which decreases their utility v. The workers want to have a material compensation, as it were, for the extra risk. The graphical expressions of this risk-aversion is the convex indifference curve v(E(w)) = v(a), which is shown in red in the figure 2.

The figure 2 also shows the consequences of the various risk preferences for the labour contract. Let the point b = (w_{H}, w_{L}) be the original contract. That contract is not optimal. For, the indifference curves of the workers and the entrepreneur, which intersect in b, separate outside the point of intersection. The largest distance between two indifference curves is reached on the certaintly line of the workers, namely the length of the line-piece a-d. The points a and d are also the points of contact of the indifference curves of the entrepreneur with those of the workers^{20}. In the point a the entrepreneur has increased his utility in comparison with the point b. The same holds for the workers in the point d. Each of these contracts insures the workers against wage fluctuations. The relation of power will determine which of these points will be selected^{21}.

Note that each contract in the yellow area of the figure 2 is better than the original contract in b. The relation between the principal and the agent becomes more complicated, when the principal must make costs in order to measure the effort of the agent. Suppose for instance, that the probability p_{H} depends on the effort e of the workers. That is to say, one has p_{H} = p_{H}(e), with evidently ∂p_{H}/∂e > 0 and thus ∂p_{L}/∂e < 0. The entrepreneur wants to stimulate the efforts, because they increase his expected profit E(π). Unfortunately he can not determine e from q, because each q_{H} or q_{L} is due to a random process. The attitude of the workers towards e is ambiguous, because they experience the effort as a burden. Let c(e) be the size of the burden (costs, dissatisfaction). In this situation their utility is given by

(4) u(w, e) = v(w) − c(e)

with variable effort

Thus the workers try to maximize w and minimize e. When E(q(e)) increases, then the entrepreneur can indeed pay higher wages w(e). The workers know the motives of the entrepreneur, and can exploit them by telling lies about their effort^{22}. This forces the entrepreneur to determine the circumstances, where the workers will truly (of their own free will) make an extra effort. His method is explained in the figure 3. The indifference curves v(w) of the figure 2 are also shown here. Assume that on these curves e is not experienced as a burden, so that c(e)=0 holds. Now suppose that the workers will decide to make an extra effort Δe. This will create new indifference curves, which in the figure 3 are presented in brown. The new indifference curves are steeper than the old ones, because Δe raises the ratio p_{H}/p_{L} and therefore the iso-wage lijn will be steeper.

The figure 3 shows two *points of intersection* of the old and new indifference curves, namely b and g. First consider the point b, defined by v(E(w)) = v(a) = u(E(w), Δe). The new indifference curve through b intersects the certainty line in the point m. Note that without the extra effort this point would correspond to a larger utility v(m). The formula 4 shows, that the difference v(m) − v(a) is exactly the burden c(e + Δe) of the extra effort. Apparently on the new indifference curves the workers receive a higher wage w ^{23}. Therefore the workers are tempted to state wrongly that they have made the effort Δe. Fortunately, this is different in the part of the new indifference curve u(E(w), Δe) = v(a) below b. For, this part is below the old indifference curve. Without the extra effort the points of this part have a utility v(w), which is lower than v(a). Now the workers are willing to voluntarily make the effort Δe.

The same argument can be applied for the point g, and for all other points on the dotted curve. On the right-hand side of the dotted curve the entrepreneur can be confident, that his workers will indeed supply the contracted e. In short, the entrepreneur must take care, that he does not offer labour contracts on the left-hand side of the curve b-g^{24}. This is elaborated in the figure 4.

with unobservable effort

Suppose, the point a represents the old labour contract without extra effort. Here the wage is w=a, so that the iso-profit line is given by E(π) = E(q) − a. This is also the indifference curve φ(E(π)) = φ(E(q) − a) = constant of the entrepreneur (green line). It intersects the certaintly line of the entrepreneur in the point n. In the situation with extra Δe the new iso-profit line and indifference curves of the entrepreneur become steeper. The blue indifference curve in the figure 4, through the points n and g, is an example of this. As already stated, it corresponds with a constant utility value of φ(E(q) − a). However, above the line a-n the *old* indifference curves have a value *below* φ(E(q) − a). Apparently Δe indeed favours the entrepreneur. Thanks to the extra effort better labour contracts are possible than a, namely the points between the old and new indifference curve.

The entrepreneur must focus on the better contracts on the right-hand side of the curve b-g, due to the opportunism (exaggerate the effort) of the workers. Moreover, the workers do not want to give up utility, so that those contracts must lie above the new indifference curve u(E(w), Δe) = v(a) of the workers. Thus one finds in the figure 4 the feasible improved contracts in the yellow area. The concrete choice of the contract again depends on the power relations. It is striking that the contracts no longer lie on the certainty line of the workers, so that they are now willing to bear some risk. The unobservable value of Δe blocks an exchange, which would yield the optimal distribution of the risk. Therefore on the curve b-g the indifference curves of the workers and the entrepreneur do not touch. Incidentally, the contracts do satisfy Pareto efficiency.

Another interpretation of the situation is, that the workers must get an extra reward in order to compensate them for their income risk. This clause in the contract is unfavourable for the entrepreneur, because he is risk-neutral and therefore would be willing to bear the risk of the workers against lower costs. The necessity to pay a premium to encourage the workers (the limitation of the feasible contracts due to the curve b-g) makes this spread of risk unfeasible. This would obviously change, when the workers become risk-neutral. These conclusions are found in the preceding column about the principal-agent problem with unobservable effort. Graphically oriented readers will undoubtedly prefer the derivation in the present column^{25}.

It is stressed again that this model sketches an abstract picture of the labour contract. In reality the entrepreneur disposes of many additional instruments in order to motivate his workers. These instruments are often based on some trust between the entrepreneur and his workers. The appreciation for justice and reciprocity are a part of human nature. These are called prosocial norms. Mutual obligations are exchanged for the long term. Nevertheless, your columnist still believes, that the homo economicus of the hedonistic wage-theory is a plausible anthropological model and that the presented principal-agent model contributes to the insight.

- According to p.172 in
*Rational-Choice-Theorie*(2011, Juventa Verlag) by N. Braun and T. Gautschi 2 is the Temptation, 1 the Reward, 0 the Punishment, and -1 the Sucker's payoff. In general one has a prisoner's dilemma, as soon as T > R > P > S holds. (back) - The following argument is copied from paragraph 1.8.2 in
*The economics of business enterprise*(2002, Edward Elgar Publishing, Inc.) by M. Ricketts. (back) - Note that the formula 1 does not contain a discount factor δ, or if so wished δ=1. Outcomes get an equal weight, despite the differing times of acquisition. On p.194 and further in
*Rational-Choice-Theorie*the case is also presented, where δ<1 holds. It turns out that then the tit-for-tat norm is only profitable, as long as δ is sufficiently large. For the situation in the table 1 this leads to the requirement δ>½.(back) - See for this argument p.192-193 and p.261 in
*Rational-Choice-Theorie*. There the table 1 is formulated more generally, with T_{k}=2, R_{k}=1, P_{k}=0, and S_{k}=-1. This general form requires the condition T_{k}> R_{k}> P_{k}> S_{k}. Then tit-for-tat behaviour in combination with an accident leads to B_{k}= J×R + (N − J) × (S+T)/2 = J × (R − ½×(S+T)) + N × (S+T)/2. A remarkable situation occurs, when 2×R < S+T holds (for instance R=¼ in table 1). Now S+T is so large, that an alternating exploitation and cooperation is profitable. Although here the tit-for-tat norm remains optimal, it is no longer a means to enforce cooperation. Apparently tit-for-tat behaviour as a weapon against exploitation requires, that 2×R > S+T holds. (back) - See for this induction argument for instance paragraph 7.1.2 in
*Rational-Choice-Theorie*. (back) - See p.342 in
*An introduction to behavioral economics*(2008, Palgrave Macmillan) by N. Wilkinson. Sometimes such experiments are organized in such a way, that the actors regularly change partners. Thus not the concrete individual is important, but the*average*individual, defined by the probability p. When p is small, then the outcomes are so bad, that sometimes yet the tit-for-tat norm gains ground, and p will rise. It has even been stated, that tit-for-tat behaviour (reciprocity) is furthered genetically by the evolution (p.343). When the choice for cooperation yields an emotional reward, then this will evidently raise p. See p.361-362. Furthermore, note that individuals socialize in their environment, and in this manner also appropriate certain norms. Those who have internalized a certain p, will also use it in laboratory experiments. See p.264 in*Behavioral economics*(2014, Springer Gabler) by H. Beck. (back) - See p.268 in
*Behavioral economics*, or p.88 and further in*Économie comportementale*(2016, Economica) by D. Serra. According to Serra, in such situations typically 50% of the test subjects prefer cooperation. Your columnist believes that in*economic*transactions this fraction will be lower, due to the careful reflections and the inherent urge to compete. The famous sociologist J. Coleman applies in chapter 9 of*Foundations of social theory*(1990, Harvard University Press) the game of the prisoners to the stock market. See p.215 and further. (back) - See p.532-533 in
*Sozialpsychologie*(2008, Springer-Verlag) by L. Werth and J. Mayer. (back) - The model is copied from paragraph 3.2.3.2 in
*The economics of business enterprise*. Apparently the model originates from G.J. Stigler. (back) - For, g(n, p
_{0}) = n × (1 − p_{0}/p_{max})^{n-1}/ p_{max}. Therefore one has E_{n}(p_{0}) = ∫_{0}^{pmax}p_{0}× n × (1 − p_{0}/p_{max})^{n-1}/ p_{max}dp_{0}. This can be rewritten as E_{n}(p_{0}) = ∫_{0}^{pmax}n × (1 − p_{0}/p_{max})^{n-1}dp_{0}− ∫_{0}^{pmax}n × (1 − p_{0}/p_{max})^{n}dp_{0}. Integration yields E_{n}(p_{0}) = [-p_{max}× (1 − p_{0}/p_{max})^{n}]_{0}^{pmax}+ [p_{max}× (1 − p_{0}/p_{max})^{n+1}× n/(n+1)]_{0}^{pmax}= p_{max}× (1 − n/(n+1)) = p_{max}/ (n+1). Which had to be proved. (back) - See p.85 in
*The economics of business enterprise*. Ricketts does not elaborate on the case of bad luck. (back) - See paragraph 7.1.1 in
*Rational-Choice-Theorie*. An interesting curiosity is that on p.287 of*An introduction to behavioral economics*this table is called the servers game. In a game of tennis the server decides whether he will aim at the forehand or backhand. The receiver tries to anticipate on a service on the forehand or backhand. One may also consider penalties, where both the football player and the goal keeper select a corner. It is obvious that in sports the mixed strategy is morally acceptable. Besides, it is indispensable for winning. (back) - See paragraph 5.2 in
*Rational-Choice-Theorie*. The model has often been applied in order to analyze the effectiveness of criminal law. (back) - See for this kind of arguments p.442 in
*Sozialpsychologie*. (back) - See p.237-238 in
*Sozialpsychologie*. Thus the childern reduce the mental discrepancy between their behaviour and their needs. (back) - See p.162 in
*Group dynamics*(1983, Brooks/Cole Publishing Company) by D.R. Forsyth. (back) - See for instance p.141 and further in
*De kern van de sociale psychologie*(1990, Van Loghum Slaterus) by P. Veen and H.A.M. Wilke. When a member decides to violate the group norms, then he probably thinks that his outcomes in that group are not particularly satisfying. In the Netherlands Veen and Wilke were leading in this scientific field. Their book indeed excels by originality. Your columnist read it 22 years ago for the first time. In paragraph 3.1.2 of*Économie comportementale*it is mentioned, that the membership is sometimes temporary. Someone will enter the group on the basis of his insigths. He is willing to adapt (socialization). This phase must result in a stable relation. In case of conflicts between the member and his group a correction will be necessary (re-socialization). The outcomes of the member can become so unfavourable, that he leaves the group (at least when the exit option is available). This whole trajectory is a learning process for the member and the group. The same phase model can be found on p.95-99 of*Group dynamics*. (back) - Here notably paragraph 5.3 of
*The economics of business enterprise*is consulted. (back) - During the postwar decades, a strong movement formed in the west for supporting participation of the wage earners in management, at least for social matters, but possibly also for commericial ones. It could become so strong, because the hallmark of this period is a continuous growth and the economic risks seemed insignificant (
*Trentes glorieuses*,*Wirtschaftswunder*,*Great society*). At the time of*New Left*this movement degenerated into demands, which approached worker's co-management. In a down-to-earth perspective it is fair that only the supplier of capital decides about commercial matters, because he will inevitably lose his source of income, when the enterprise fails. This fact also gives him a strong incentive to take decisions, which guarantee the existence of the enterprise. (back) - The phenomenon of touching indifference curves is characteristic for the optimum in the Edgeworth box. Also elsewhere in the Gazette, it has often been used, for instance in the column about the hedonistic wage-theory, in the column about wage bargaining and in the column about the contract of the direction. (back)
- In the previous column about the principal-agent problem with observable effort the expected profit is also maximized. The mathematical approach allows to model a continuum of q-values, instead of merely q
_{H}and q_{L}. The maximalization of E(π) is limited by the participation condition, which determines the power of the workers, via their reservation utility U. The reservation utility could for instance originate from the wage in the collective agreement, which has been concluded by the trade union. The minimum in the agreement again depends on the unemployment in the concerned branch and in the economy as a whole. (back) - Perhaps this statement suggests more unfavourable morals of the workers than is actually the case. In any case the socialist class struggle has the effect, that even nowadays many workers feel exploited and a victim of management. The trade union as an intermediary can bargain about the labour contracts in a matter-of-fact manner. Unfortunately this is not always in the interest of the union, and at the time of the
*New Left*it often stirred up the emotions.

According to social psychology, supervision has two effects. On the one hand, the mutual competition has the result, that the worker will perform better than without supervision. On the other hand, the worker can experience it as a pressure, which discourages him. See the paragraph 8.1 of*Sozialpsychologie*for respectively the*social facilitation*and*inhibition*. Therefore the supervision must be organized in such a manner, that it does not hurt the job motivation. In paragraph 9.2 the attention is drawn to*social loafing*, that is to say, the inclination to reduce the personal effort in a collective task. It turns out that this "laziness" is a universal phenomenon. When the effort is not observable, then idleness is called the*Gimpel*effect (p.359). The invisibility undermines the mutual competition. See also p.269 and further in*Group dynamics*for the discussion of social loafing. Besides, the separate worker usually values his effort higher than it truly is (p.360 in*Sozialpsychologie*). The over-estimation of the personal performance is also observed in the so-called*dictator games*with preceding competition. This is called a*self-serving bias*. See p.345 in*An introduction to behavioral economics*, and p.60 in*Behavioral economics*. (back) - This statement is also found in the previous column about the principal-agent problem with unobservable effort. There it is derived mathematically. (back)
- In the previous column about the principal-agent problem with unobservable effort this dotted curve is called the
*incentive*-constraint. (back) - it is worth mentioning that in paragraph 5.5 in
*The economics of business enterprise*an extension of the model is discussed, where the entrepreneur maintains the performance agreement in the labour contract with a probability of p_{2}. Moreover Ricketts adds the probability ψ, that the entrepreneur errs in his observation of e, and then unjustly imposes the punishment s. This form of supervision and maintenance confronts the worker with a lottery, when he must decide about the extra effort Δe. Thus the effects of information and of norms are integrated in a single model! (back)