Assignment
(1) Added question (economies of scope). Let's look at a project that has a per firm technology of f(x)=-5+x for a unique input, x. We will assume if x is less than 5 output is equal to zero and not consider any range of inputs below 5.
(A) What type of economies of scale (remember there are three possible does this technology have for its entire range? Why? What is true of its average per unit cost? (Assume each unit of input x has cost one for this.) Can you graph that? What is the relationship between average costs and economies of scale?
(B) Let's look at a function that has different returns of scale defined by the average per unit of cost of AC(y)=y-14+51/y. [Hint this is a parabola with a definite minimum value.] Plot this. Where does this function have increasing, constant or decreasing returns? Note that returns to scale can imply to changes in inputs that are less than doubling.
(C) Suppose there are two firms with the same technology. Let's assume that we require any amount y to be produced by two firms who produce identical quantities y/2, let's call the resulting average costs to both firms combined as AC2(y)? It turns out that AC2(y) is the cheapest way to organize production between these two firms. Why?
(D) Plot AC(y) and AC2(y) on the same graph. Graph the minimum of the two functions. The region of the graph where AC(y) corresponds to the minimum is called the subadditive region. What do you think is the economic significance of this region? What is the relationship between subadditivity and returns to scale in this graph?
In general, can you think of an economic question where subadditivity might be a more useful concept than economies of scale?
(E) In models where firms produce multiple goods, the number of markets in which average costs are minimized by having positive production from the same firm is called economies of scope. Let's say there are two goods, y1 and y2. Can you give an average cost function with positive economies of scope? How about one with negative economies of scope?