(1) Suppose that Ted wants to make use of his East-LA backyard. He wants to grow up spinach. He got a bunch of baby spinach plants from his girlfriend so at this stage his only concern is whether to use fertilizer or organic manure. He has been told that 1 pound of fertilizer (F) can yield 20 spinach plants (S) and that 1 pound of manure (M) can yield 30 spinach plants, no matter how much of each has been used so far.

(a) What is Ted's production function? That is, what is Qspinach(M,F)?

(b) Does this function exhibit constant, increasing, or decreasing returns to scale?

(c) Suppose that a pound of fertilizer sells for $5 and a pound of manure sells for $10. If Ted's goal is to produce 300 spinach plants the cheapest way he can and sell these at the local farmer's market, how much fertilizer and manure should he get?

(d) What's the cost of doing this?

(e) What would be your answers to (c) and (d) if the price of fertilizer were $10 as well?

(f) What are the Average and Marginal products of fertilizer and manure? For each of these two inputs, when would average and marginal product coincide? When wouldn't they coincide?

(2) Suppose that Jill (Ted's sister), also gets excited about farming so she decides to plant something in the (huge) land she inherited. She decides to produce strawberries. There, she needs to use tractors (T) and farmers (F) who are willing to pick up the strawberries. The technology she has access to can be summarized by a Cobb-Douglas production function, where Qstrawberries(T,F)=10(TF)1/2
When she starts operations, there are 10 tractors for rent in the local market and she rented them all in advance. The rental rate is $1 per hour and she needs to keep them for at least 3 months. Agricultural wages are $7.50 per hour.

(a) Given that the number of tractors is fixed, would you say that she faces increasing, diminishing or constant returns to labor? This is, doubling the amount of labor more than doubles, less than doubles or exactly doubles the number of strawberries produced?

(b) How many workers is she going to hire if she wants to produce Q pounds of strawberries?

(c) What is Jill's cost function?

(d) What is her average cost function? And her marginal cost function?

(e) If she chooses to produce Q=50, what's her total cost, marginal cost and average cost? What if Q=100? Q=200?

(f) Graph the AC curve and the MC curve.

(g) Now assume that the number of tractors is still fixed, but you just know that it is fixed at some level T. What is the cost function now if you know that w=7.5, r=1 and we want to produce Q pounds of strawberries?

(h) From all the possible numbers of tractors that she could have faced (T), which one would have produced the lowest short run cost? Here you need to find what value of T minimizes the expression you found in (g).

(i) Substituting the value of T you just found into the cost function, what does it look like?

(j) Now assume that after the three months she can reallocate her budget to produce more efficiently. Show that she faces constant returns to scale.

(k) What is Jill's long run cost function?

(3) Mark sells small hand made chairs at a market. There are a lot of other furniture sellers along with him. His total cost function is the following: TC(q)=180+q+(1/20)q2

As you would expect, there's not only a large number of sellers, but also a large number of buyers every Sunday (say). Under these conditions, it is reasonable to assume that he cannot really set the price that he charges. He is a "price taker". Suppose he sells his output at p dollars.

(a) What's Mark's Marginal Cost function? What is his Average Cost Function?

(b) How much is his fixed cost? How much is his variable cost?

(c) At what level of production is his average cost minimum (check the example we saw during lecture)

(d) If the price per chair is $8, how many chairs will he make (and sell)? What will be his profit in this case?

(e) When he is just able to cover his total costs of production, we say that he is able to just "break even" This is, Total Revenue=Total Cost. If he stays producing the (optimal) number of chairs you found in (d), at what price would he be able to just break even?

(f) At what price would he be able to just cover his variable costs? This is called the shutdown price.