How should we allocate capital and share risks?

January 25, 2024 Agriculture

Without information frictions, capital should be allocated according to individual productivity and risks should be diversified. But what happens when individual output is only observed privately? TSE Sustainable Finance Center’s Jean-Charles Rochet and Stéphane Villeneuve teamed up with former colleague Bruno Biais (HEC), as well as Hans Gersbach (ETHZ) and Ernst Ludwig von Thadden (Mannheim), to study how incentive compatibility constraints affect capital accumulation and risk sharing between a principal and many agents.

How do you explore the dynamics of capital allocation?

We consider an economy with many risk-averse agents and a single good that can be consumed or invested as capital. Each agent operates a project whose output is proportional to the amount of capital under their management and is subject to idiosyncratic shocks. Individual unit outputs are randomly distributed. We assume agents privately observe their individual output and can secretly consume some of it. In contrast to output, capital is observable.

From a mathematical viewpoint, finding the optimal mechanism in the context of a continuum of agents is challenging. But assuming logarithmic utility enables us to fully characterize the optimal dynamics of capital and consumption as well as the distribution of continuation utilities across agents.

What does this approach reveal about the optimal mechanism?

Our paper helps to clarify the consequences of informational frictions. For example, existing research emphasizes how wealth affects the ability to invest in capital, and the willingness to hold risky capital. Our approach reconciles these effects, demonstrating the role of incentive compatibility in constraining how much capital agents are allocated and how much of the corresponding idiosyncratic risk they must bear. Consequently, frictions unambiguously lower capital accumulation. 
The optimal contract allocates more consumption and capital for agents whose output is larger than expected. Lucky agents that perform better get more capital to manage in the next period. This is not because they are more skilled – performance is randomly distributed across agents and across periods – but because it provides agents with incentives to truthfully report good performance instead of diverting output. In contrast with the symmetric information case, perfect insurance is prevented by incentive compatibility. While there is no aggregate risk, the optimal mechanism exposes agents to a fraction of their particular risk.

The optimal mechanism is remarkably simple: consumption and capital are allocated among agents in proportion to each agent’s lifetime stream of consumption. The consumption of each agent grows at a constant rate in expectation, but is impacted by the agent’s performance. The change in the growth rate of an agent’s consumption or capital is proportional to their own output shock. By raising the agent’s exposure to this risk, the principal can relax the incentive compatibility condition and extract more rents. This reduces insurance, creating a tradeoff between rent and efficiency.

Because agents are exposed to their own shocks, incentive constraints generate increasing inequality over time and agents become more and more heterogenous. Moreover, while aggregate capital and output grow over time, growth is lower than under symmetric information. This is because incentive compatibility constrains how much new capital can be delegated to agents.

How can governments use money and taxes to implement the optimal mechanism?

We show that a more decentralized implementation is possible, when agents trade goods for money in a competitive market and the government intervenes only by issuing money and taxing wealth. This money has value because the agents are required to use it to pay the taxes. When trading, agents choose how much to invest in capital and money, bearing in mind that the former has higher expected returns but is riskier than the latter. 

To implement the optimal mechanism, the government influences the agent’s portfolio choice by controlling money supply and thus inflation. This affects the attractiveness of risky capital over safe money, so that agents’ risk exposure is the same in the market equilibrium as in the optimal mechanism. In general, the government uses both taxation and seigniorage to regulate inequality, raise revenue and extract rents from the agent.

To illustrate our intuition for using money and taxes as a policy tool, consider the simple two-period version of our model: at time 1, agents with high output will sell some of it, increasing their money holdings. This allows them to hold more capital because they can afford to pay higher taxes, leading to larger consumption at time 2. Unsuccessful agents will buy goods to smooth consumption, but cannot afford high taxes so have low capital and low time 2 consumption. This is consistent with theories of money as a “memory” of good performance and a safe store of value.


Money and Taxes Implement Optimal Dynamic Mechanisms’ and other publications by Jean-Charles, Bruno and Stéphane are available to view on the TSE website.

Article published in TSE Reflect, February 2024