(1) A farming family owns land of size a (acres), and farms it with labor l, using the production function Y 1001 1/2 a 1/2 = The farm has access to 4 total units of labor (you can think of 4 as the family size), which it divides as finely as it wishes (fractionally if needed) between working on the farm and off the farm. Off-farm employment yields a wage of 100 per unit. The farm can also hire in labor, again at the wage cost of 100 per unit. But unlike family labor, hired labor has to be supervised, and for this the farm has to hire a supervisor at a cost of 225. Once paid, the supervisor can costlessly supervise all hired labor. (a) Prove that if the family has less than 16 acres of land (a 16), it will devote family labor equal to a/4 to the farm, hire in no additional labor, and hire out the remainder 4 - (a/4) for off-farm employment. (b) Prove that if the family has between 16 and 49 acres of land, it will continue to operate as a full family farm, with all its family members working full time on it, but will not hire in any labor. Above 49 acres, it hires a supervisor and at this threshold, its hiring of outside labor jumps up from 0 to slightly over 8 units of hired labor, and then keeps climbing as a continues to rise. Now suppose that the family has the additional option of leasing out some or all of its land at a fixed rental rate of R per unit. But assume that it cannot lease in any land. (c) Calculate a threshold for R such that above this threshold, the family never farms any land, no matter how much or how little land it owns, and leases it all out. [Hint: work out the implicit return to land on the family farm after subtracting the imputed costs of family labor.] (d) Can you work out what would happen for lower values of the land rental? For instance, can you show values of R such that for small values of land, the family leases out nothing, then leases out some land, and then again goes back to leasing out no land as its holdings get large? (e) How would your answers to parts (b)-(d) change (if at all) if there were no fixed costs to supervision, and if hired labor costs 25 per unit to supervise instead?