[1].Insofar as there is more than one individual who wants the same property resource,competition is implied,and the number of competitors includes not only those who are actually using the resource but also potential owners（users）.The assumption of zero contracting cost substitutes for the sometimes dubious assumption of"pure"competition.Included in the general term"contracting cost"are the costs of negotiating and the costs of enforcing the stipulations of the contract.I shall discuss these and other problems of transaction cost in chapter 4.
[2].Depending on the production function,these pairs of curves may not be identical.We shall discuss this later.
B.The Solution in Mathematics
For simplicity of presentation,assume there are two homogeneous factors of production,h and t,where h is the amount of land per tenant farm and t is the amount of tenant labor per farm.Further assume that the production functions of each tenant farm are identical.With these assumptions,it follows that the land size,h,and the rental percentage,r,contracted for each tenant farm will be the same in equilibrium.
Let each tenant farm's production function be
q=q（h,t）
The amount of land per farm,h,is equal to the total landholding of the landowner,H,divided by the number of farms,m;that is,
The landowner's total rent,R,is then equal to the number of farms times the rent per farm;that is,
R=m·r·q（h,t）
Under competition,
Wt=（1-r）q（h,t）
where W is the market wage rate of the tenant labor,t.
The problem of the landowner is then to maximize R,through the choice of m,r,and t,subject to the constraint of competition;[1]that is,
max.R=m·r·q（h,t）
{m,r,t}
subject to Wt=（1-r）q{h,t）
Forming the Lagrangean expression,the problem is thus the maximization of
The necessary conditions are:
From equation（2）above,we have
λ=m
And noting that
,equation（1）becomes
that is,
This indicates that rent per acre of land equals the marginal product of land in equilibrium,a condition identical to that of a fixed-rent contract.
From equation（3）,we have
or the marginal product of tenant labor equals the wage rate,a condition identical to that of a wage contract.Finally,solving equations（1）and（4）for r,
That is,in equilibrium,the rental percentage must simultaneously satisfy the last two terms.In other words,in equilibrium,the elasticity of output with respect to land,
,equals the total yield net of tenant cost（rent）as a portion of the total product,
[1].Note that t and m need not be treated separately.Given t,an adjustment of m yields the same result as adjusting t while holding m constant.They are separated here for the purpose of conveniently deriving all the conditions in equilibrium.
C.The Geometric Solution and Further Exposition
The results derived in the preceding section can be demonstrated geometrically.In figure 3 we have the same dimensions as in figure 2.But in figure 3 we concentrate on only one of the tenants,which means that the total land space owned by the landlord may not be exhausted.The curve q/h represents the average product of land with one tenant family employed;that is,the average product with respect to the land size while holding all other farming inputs（of one tenant family）constant.The curve f/h,or the fixed total tenant farming cost divided by land area,reveals the cost of farming inputs（other than land）which yield the expected q/h.Assume for the moment that all nonland farming inputs are borne by the tenant;the f/h curve is the total cost other than land divided by the respective land space.It includes the cost of labor,seeds,fertilizers,and farming equipment associated with the period of time of the relevant production run.[1]
That is,f/h=（pt·t+pz·z+……）/h;where f is the constant total cost other than land,and pt,pz,···are the factor prices of tenant labor,t;fertilizers,z;……Since we hold the farming inputs constant,the f/h curve is a rectangular hyperbola.The vertical difference between q/h and f/h defines（q-f）/h,the rent per unit of land,taking into account the alternative cost of the tenant.[2]
The total amount of tenant inputs which define f/h are contrac-tually stipulated,which is essential because the tenant would commit less if only the rental percentage were prescribed.Given any rental percentage,only a portion of every unit of output produced will go to the tenant.If farming decisions were made entirely by the tenant,it would be to his interest that the cost of incremental tenant input be less than the associated marginal product.This would result in a condition inconsistent with equilibrium.A fuller discussion of this point will be presented in the next chapter.