The theory of tenancy fits nicely into a game model.
Institutional precondition: (Delimitation of property rights) The rights to use and earn income from certain property (or resource in other contexts) such as land, labor, and capital, along with the right to transfer these rights, shall be exclusively delimited by law to allow for market transaction.
Nature of contracts: Every transaction involves a contract, which is a transfer of property rights among individual contracting parties, so that property rights can be recombined to maximize property value.
Share contracting: multiple individual parties combine privately owned resources for the production of certain mutually agreed outputs, and the actual outputs will be shared by certain mutually accepted percentages as returns to the contracting parties for their productive resources forsaken.
Efficient use of resources: At societal level, efficient allocation of existing resources necessitates equal marginal products of each resource across all sectors it gets used, despite how the income is distributed, and is measured by the interest rate (i.e. marginal cost).
The tenancy game \( G = (M \cup N, S, \mathbf{u}) \) involves a group of tenants \( i \in M = \{ 1, \dots, m \} \) and a group of landlords \( j \in N = \{ 1, \dots, n \} \) in the same locality, where \( m \gg n > 1 \).
Landlords have land, which needs labor and non-land capital (such as seed, water, fertilizer, pesticide, and equipments) to produce. Regard the market value of annual crop yield \(Q\) as a function of land area (equivalent) \(a\), labor (equivalent) \(t\), and capital \(k\); land and labor are measured in homogenized equivalent amounts, as land differs in fertility and farmers differ in farming knowledge. Due to homogeneity of land, we can write the production function \( Q(a, t, k) = a q(\frac{t}{a}, \frac{k}{a}) \), where \(q\) is the annual yield value per unit land area equivalent. Regarding \(q\) as a univariate function, it should have properties: \( q'' < 0; q'(0) > 0, q'(\infty) < 0; q(0) = 0 \).
Consider the standard conditions of the tenancy game as when private property rights can be freely transferred and transaction cost is zero for all contractual arrangements.
Table: Strategy space and payoffs in three (abstract) types of contracts
| Contract | Wage | Fixed rent | Fixed share |
|---|---|---|---|
| \(S_i\) | \( t_i \in \{T_i\} \) | \( (t_i, a_i) \in \mathbb{R}^2_+ \) | \(\emptyset\) |
| \(S_j\) | \( t \in \mathbb{R}_+ \) | \( r \in \mathbb{R}_+ \) | \( (\frac{T_i}{a_i},s) \in \mathbb{R}_+ \times (0,1) \) |
| \(u_i\) | \( w t_i \) | \( a_i \left(q (\frac{t_i}{a_i}) - r \right) - p k_i \) | \( a_i q (\frac{t_i}{a_i}) (1-s) - p k_i\) |
| \(u_j\) | \( a q (\frac{t}{a}) - w t - p k\) | \(ar - p k_j\) | \( \sum_i a_i q (\frac{t_i}{a_i}) s - p k_j \) |
For tenancy farming, wage contract can be considered as the "exit" of tenant; and owner-farming as another "exit" of landlord. For each farmer, maximum labor equivalent \(T_i\) is determined by marginal cost of individual labor and market wage of labor. (Market value of labor is determined by the availability of specialized knowledge/skill in a population.) That is, individuals have no incentive to work beyond the point when their marginal costs equal labor wage. Also due to individual rationality, farmers will work no less than their maximum labor equivalent if they choose to farm at all. Thus for each individual, their labor strategy in wage farming can be reduced to a single-point set \( \{T_i\} \). On the other hand, landlords decide the amount of labor to hire and capital to invest; landlords preserve the right to earn income from land produce.
In fixed rent tenant contract, each landlord sets a fixed price of unit land, while each tenant chooses the size of land to work on; landlords earn a fixed amount of return (after harvest) and tenants keep the rest; capital investment can be freely distributed among the tenant and the landlord, which is stipulated in the contract.
In fixed share tenant contract, since landlords can stipulate labor with minimal transaction cost, e.g. by measuring production, while tenants will abide due to competition, share tenancy can be seen as an optimization of annual land rent (by the landlord) with two variables: share and nonland-to-land factor ratio, as formalized in {Cheung1969 2.B}. As with fixed rent tenant contract, capital investment can be freely distributed among the tenant and the landlord, which is stipulated in the contract.
With labor wage \(w\) and capital price \(p\) being market prices, landlords under wage contract will expect rent \( \hat{r} = \max q(\frac{t}{a}, \frac{k}{a}) - w \frac{t}{a} - p \frac{k}{a} \). Under fixed rent, tenants will maximize \( q (\frac{t_i}{a_i}) \) and farm as much land as they can at such intensity; while landlords will raise rent until the contract provides no more return to the tenant than in wage contract: \( w t_i^* = a_i \left( \max q (\frac{t_i}{a_i}) - r \right) - p k \). Under fixed share, landlords will maximize \( q (\frac{t_i}{a_i}, \frac{k_i}{a_i}) s - p \frac{k}{a_i} \), which is equivalent to the wage contract, and share will be set to compensate labor at the same level as its market wage. At equilibrium, all three types of contracts provide the same return per unit labor; the condition also holds for land.
Landlords and tenants will thus be indifferent about the choice of contracts, and equilibrium strategies under all types of contracts will lead to the same efficient resource allocation: \[\begin{aligned} w &= \frac{\partial Q}{\partial t}(a_i, T_i, k_i), &&\forall i \in M \\ \hat{r} &= \frac{\partial Q}{\partial a}(a_j, t_j, k_j), &&\forall j \in N \\ p &= \frac{\partial Q}{\partial k}(a, t, k) \end{aligned}\] Specifically, the most efficient crop (relative product price, labor) will be chosen for the land.
This is implied by Coase theorem {Coase1960}: With the initial delimitation of rights, the ultimate result (which maximises property value) is independent of the legal position if the pricing system is assumed to work without cost.
Transaction cost associated with contracts cut into resource value, which include negotiation costs and enforcement costs (implementation costs).
Natural risk refers to the variation in product value caused by nature or the state of the world, such as weather conditions and pests. The postulate of risk aversion states that given the same expected average income, an individual prefers one with lower variance.
Table: Transaction costs and risk dispersion in three (abstract) types of contracts
| Contract | Wage | Fixed rent | Fixed share |
|---|---|---|---|
| Negotiation | Landlords decide labor and crops given market price of labor. | Tenants decide land area and crops given market price of land rent. | The terms are mutually decided by the landlord and the tenant: crops, labor-to-land ratio, and share. |
| Enforcement | "shirking" of labor input. | Landlord must police the maintenance of soil and other assets owned by the landlord. | Landlord must ascertain the harvest yield; police the maintenance of soil and other assets owned by the landlord. Agent can collect payment from both the landlord and the tenant. |
| Risk | Landlord bears most of the risk. | Tenant bears most of the risk. | Risk is dispersed between the tenant and the landlord. |
{Cheung1969, Chap.4} laid out a qualitative analysis of choice of contractual arrangements (see table). Among the three types of contracts, fixed share tenant contract has least preferable transaction costs but most preferable risk dispersion. Depending on risk perception (which is unmeasurable) of the contracting parties, different contracts will be chosen.
Additionally, implementation costs may include the cost of measuring quantity of resource used, tenant moving costs at lease dismissal, etc. Negotiation cost may also cover the renegotiation of capital investment to adapt to change in price and technology, and the renegotiation of income distribution to adapt to changes in relative asset prices and unanticipated inflation (if paid in cash).
Compared with crop rent, cash rent has less risk when price raise partly compensates low production; but it has higher risk when unanticipated inflation happens.
Transaction costs also depend on the level of law enforcement effort, or corruption of courts.
Rent control is a form of legal restriction on property rights, in particular the right to earn income from property forsaken.
Under rent control alone, resource owners may rearrange the contracts to restore the original equilibrium. An offsetting contractual rearrangement is a contractual revision which, in the absence of transaction costs and risks, produces no effect on the initially contracted resource allocation and income distribution. Such rearrangements include:
If the law further prohibits compensating payments and the right to transfer, a different and inefficient equilibrium will be realized. With land rent capped and offsetting contractual rearrangement inhibitively costly, landlords will instead maximize production to partially recover their lost rent, and tenants will comply due to competition.
The discrepancies in the marginal products of resources (labor, capital; land) in the various sectors (agriculture/non-agriculture; tenant/non-tenant farm) imply economically inefficient use of the existing resources of the society.
Landowners' income will fall, so will the value of land used as tenant farms. The tenants' income from farming will rise while working longer hours. The increased tenant income will bid up labor wage, wage farmers will also work longer hours. The increase in tenant income will be largely invested in land: either by increasing the density of intramarginal crop planting such as better seeds, fertilizers, and pesticides, improving irrigation and fields, and planting the plants closer to each other; or by increasing the rate of crop rotation such as planting marginal crop in idle season, relaying, interplanting, or earlier harvesting. The total agricultural output on tenant farms will rise.
Individuals have different knowledge of the alternative uses of a resource, contractual arrangements and their transaction costs (See also {Hayek1945}). Without transaction cost, competition for and transferability of property rights ensures that the most valuable knowledge will be utilized and the maximum value of the resource realized.
Perfectly competitive market also has preferable transaction cost. Since competing parties will stand by to offer or accept similar terms, competition reduces the enforcement cost of a contract.
Production theorem of property right assignment {Cheung1969 4.D}:
For any production function requiring resource inputs h and t, if the right to a portion of the income from h, however small, is not appropriated or exclusively assigned, the ratio t/h will rise under competition, implying a fall in the marginal product of t and a rise in the marginal product of h; as the unassigned income from h increases, the marginal product of t will accordingly be lower, and may become negative when the entire income from h is not assigned to any individual party.
If income right is exclusively assigned, all joint owners shall have authority to make decisions concerning their share of the resource. As in Cheung's footnote: "The misallocation generated by a tax on resource use stems ... from the fact that no government official is thereby granted authority to make decisions on resource use".