The theory of tenancy fits nicely into a game model.

## Preliminaries

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 proportions 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).

## Game

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: $q'' < 0; q'(0) > 0, q'(\infty) < 0; q(0) = 0$.

### Standard Conditions

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.

Game 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 the labor strategy of each individual 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

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.

Cost 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, Ch.4] laid out a qualitative analysis of choice of contractual arrangements (see table). Among the three types of contracts, fixed share tenant contract has the least preferable transaction costs but the 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.

### Regulation of Property Rights

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:

• Compensating payments
• Charge lump-sum annual payments offsetting the reduction in rent;
• Require the tenant to pay the full cost of the nonland capital;
• Forbid the use part of the land for the tenant's personal consumption;
• Tenure rearrangements
• Repossess the tenant's holding and adopt wage contract or owner-farming;
• Offer a fee to the tenant to buy back the lease right;
• Sell his holdings outright to his tenants;

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.

• Inefficient use of labor and capital: As more farming resources are directed to tenant farms, the marginal products of nonland inputs will be lower than those of similar resources employed elsewhere: $w > \frac{\partial Q}{\partial t}$, $p > \frac{\partial Q}{\partial k}$;
• Inefficient use of land: Under rent control, the marginal product of land is higher on tenant farms than elsewhere: $\hat{r} < \frac{\partial Q}{\partial a}$;

The discrepancies in the marginal products of resources (labor, capital; land) in the various sectors (tenant/non-tenant farm; agriculture/non-agriculture) imply economically inefficient use of the existing resources of the society.

Without controlled experiment, aggregate production data may be used to verify changes in marginal products. Since the output data for the prefectures in Taiwan did not separate tenant and owner farms, one may instead test that relative output responses are correlated with the degree to which the share restriction affected farming resources. Crops suitable for interplanting are counted in the interplanting margin only if they are clearly added for faster crop rotation: excluded are crops commonly grown and largely planted singly, crops regulated or mostly grown by the government. Major factors on yield were normal for the period under study (1948-51) in Taiwan: annual agricultural losses due to drought, flood and typhoon have no notable differences.

Implications and evidence on resource allocation and output:

• Landowners' income will fall, who compensates by maximizing production; so will the value of land used as tenant farms [Records, Reports].
• Farmer-land ratio increases in tenant farms and declines in owner farms [@Yearbook, Statistics; table 2];
• Among vegetables, those that require a shorter growing time or a higher cost of planting (associated with relatively high market value) tend to have higher percentage increases in crop area [@table 7].
• The tenants' income from farming will rise, while working longer hours and more days in a year [@Reports, Records] and investing more capital in land [@Implementation];
• Total yield per cultivated land in tenant farms rises [Yearbook]:
• intramarginal crops (rice) in tenant farms: crop area rises (single-crop to double-crop, mostly by adding early rice) [@Yearbook; table 3], crop hectare yield rises (better seeds/fertilizers/pesticides, improve irrigation and fields, plant closer) [@Yearbook];
• marginal crops added in tenant farms - internal land margin (citronella), interplanting margin (oriental pickling melons, cucumbers, watermelons, potatoes, and eggplants) and seasonal margin (vegetables): crop area significantly rises, crop hectare yield decreases [@Yearbook; table 5-7];
• Production of crops not included under the share restriction (horticulture, i.e. fruits) decreases: harvested area decreases [@table 3], plant yield decreases [@Yearbook; table 8];

Data Sources: Yearbook: Taiwan Agricultural Yearbook, by the Department of Agriculture and Forestry (DAF), 1948-1953, 1958; Statistics: Taiwan Agricultural Statistics 1901-1955, by Sino-American Joint Commission on Rural Reconstruction (JCRR), 1956; Reports: Annual Reports on Land Reform in the Republic of China, by JCRR, 1965; Implementation: Implementation of the 37.5 percent Farm Rent Limitation Program, by Taiwan Provincial Land Bureau, 1950; Records: Records of Taiwan Land Reform, by Cheng Chen, 1961;

Agricultural Terminology. area of cultivated land (耕地面积), crop area (作物面积); density of planting (种植密度), rate of crop rotation (轮作率); intramarginal crops (边际内作物), marginal crops (边际作物); internal land margin (内部边际土地), the pieces of land within a privately owned farm whose quality is so low that they are seldom utilized under a free market, such as hill slopes and badly eroded fields; interplanting margin (边际套种), another crop grows at the same time in the space between the rows of a planted crop; relaying (轮种), another crop is added in between and continues to grow alone after the initial crop is harvested; seasonal margin (季节性边际), planting marginal crops in idle season (winter); citronella (香茅草), horticulture (园艺);

## Conclusions

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".