In the rent extraction-efficiency trade-off analyzed so far, the principal wants to minimize the information rent left to the agent for a given level of output. The principal has no redistributive concerns vis-à-vis the agent. In the optimal tax-ation literature, starting with the seminal paper of Mirrlees (1971), the principal (generally a government or a tax authority) wants to redistribute wealth among members of society according to a particular social objective function G(·), which we will assume is increasing and strictly concave (G > 0 and G < 0). Of course, for the redistribution problem to be nontrivial, agents have to be heterogeneous. We will thus assume that with probability v(resp. 1 − v) an agent is a high- (resp. low- ) productivity one having a cost of production θ (resp. θ¯). An agent’s utility function is thus written as usual as U = t θq. The principal’s objective is instead .

This redistributive objective of the government is limited by the government’s budget constraint. Typically, if the return from production of each type is S(q), the budget constraint requires that the government cannot redistribute more that what is actually produced, i.e., Using the definition of the information rents U  and U¯, the budget constraint can be rewritten as Under complete information, the principal can distinguish between high and low productivity agents. The optimal redistribution scheme must solve the follow- ing problem: The problem is concave and the first-order conditions are necessary and suf- ficient  for  optimality.  Optimizing  with  respect  to  U  and  U¯ respectively  yields where µ is the positive multiplier of (3.112).

When  G(·) is  strictly  concave,  the  full  information  policy  calls  for  complete redistribution,  so  that  U = U¯*  = U.

Optimizing with respect to outputs yields the usual first-best productions qand  q¯*.  Hence,  any  agent,  whatever  his  type,  gets Under complete information, the government chooses to maximize the “size of the cake” before redistributing equal shares of it to everybody. There is no conflict between efficiency and equality.

Let us now turn to the more realistic case where the agent’s productivity is nonobservable. An incentive-feasible redistribution policy must now satisfy not only the budget constraint (3.112) but also the following incentive constraints: First,  note  that  the  optimal  first-best  policy  is  such  that  ,  i.e.,  the  high-productivity  agent’s  incentive  constraint  is  violated.  Hence, we suspect (3.115) to be binding under asymmetric information, and we look for an optimal second-best policy as a solution to the following program: Denoting the respective multipliers of (3.112) and (3.115) by µ and λ, the first-order  conditions  for  optimality  with  respect  to  U  and  U¯ yield,  respectively, Summing those last two equations, we obtain and the budget constraint is again binding. Also, we compute Because  G(·) is  concave  and is  necessary  to  satisfy  the  incen-tive constraint (3.115) if q¯ > 0, we have λ > 0, and the incentive compatibility constraint is also binding.

Optimizing with respect to outputs immediately yields qSB = q, i.e., no dis- tortion of the high-productivity agent’s output and a downward distortion of the low-productivity  agent’s  output.  Indeed,  we  have ,  where Using the definitions of λ and µ given above, we finally obtain We summarize all those results in proposition 3.7.

Proposition 3.7: Under asymmetric information, the optimal redistribu- tive policy calls for a downward distortion of the low-productivity agent’s output, ,   and   a   positive   wedge   between   the   low-   and   the high-productivity  agents’  utilities, .

To induce information revelation by the high-productivity type, the princi- pal raises his after-tax utility level and reduces that of the low-productivity type. Introducing this unequal distribution of utilities is costly for the principal, who maximizes a strictly concave social objective. To reduce this cost, and thereby to reduce inequality, the principal decreases the low-productivity agent’s output. Under asymmetric information, there exists a true trade-off between equity and efficiency.

Remark: It is interesting to give an approximation of the distortion described in (3.122) when Δθ is small enough. Using simple Taylor expansions,   we   get  Hence, we finally obtain As  the  degree  of  the  government’s  inequality  aversion increases, the principal becomes more averse to inequality, and he must more significantly reduce the low-productivity agent’s output.

The taxation literature has been mostly developed, following Mirrlees (1971), in the case of a continuum of productivity shocks. As ourtwo-type model has already shown, the technical difficulties of such models come from the impossibility to proceed in two steps as usual, i.e., first, find the distribution of utilities and, second, optimize with respect to output. Those two steps must be performed simultaneously by relying on complex optimiza- tion techniques (calculus of variations or Pontryagin Principle). This makes the analysis quite difficult, and explicit solutions are generally not available (see Atkinson and Stiglitz 1980, Stiglitz 1987, and Myles 1997 for the tech- niques needed to solve this problem in the case of a continuum of types). A second peculiarity of the optimal solution with a continuum of types is that both the lowest and the highest productivity agents produce the first- best output, provided that second-order conditions are satisfied (Lollivier and Rochet 1983). Otherwise, it may sometimes be optimal to have the least pro- ductive agents producing zero output. For all other types, the production is downward distorted as in our two-type example. The clear advantage of the continuum model is that it gives realistic predictions on the taxation sched- ule. This allows discussion of the progressivity or regressivity of this schedule. The fact that high-productivity agents produce efficiently also implies that the marginal tax rate faced by the highest productivity agents should be zero in the optimal taxation literature. This seems nevertheless to contradict most empirical observations.

Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.