Problem Set 6

Due 4/14/97


1. Consider the principal-agent model studied in class. Suppose that the cost of effort for the agent (in certainty-equivalent units) is given by c(e) = g e + (1/2) d e^2.

(a) Solve for the agent's choice of effort, given the principal's contract (a and b).

(b) Solve for the principal's choice of commission.

(c) For each of the following parameters, specify how the choice of b changes with the parameter, and further write two-three sentences providing intuition about the role of the parameter in determining the commission rates. [Hint: think about how the parameters affect the agent's willingness to supply an additional unit of effort as well as how responsive the agent is to changes in b].

Consider each of these parameters: r, s^2, g, d.

2. Consider a profit-maximizing firm that makes two varieties of a single product and whose costs are a convex quadratic function of the two output levels x and y. A price premium of D is earned on one of the varieties. The firm's profits are given by

px + (p+D) y - C(x,y)

where C(x,y) = (ax^2 + by^2)/2 + gxy

Assume that a>0, b>0, and ab-g^2>0.

(a) Suppose that y is fixed. The firm chooses x to maximize profits. Apply the "increasing differences comparative statics theorem" to determine whether the optimal choice of x is increasing or decreasing in the price p. Solve for the optimal choice of x as a function of p and y, and check your answer.

(b) Suppose that x is fixed. The firm chooses y to maximize profits. Apply the "increasing differences comparative statics theorem" to determine whether the optimal choice of y is increasing or decreasing in the price p. Solve for the optimal choice of y as a function of p and x, and check your answer.

(c) Now, suppose that the firm chooses both x and y. Under what conditions does the profit function satisfy increasing differences in (x,y)? In (x,p) and (y,p) [which you checked above]?

(d) Solve for the optimal choices of x and y as a function of p when both x and y are choice variables. Calculate the derivatives of x* and y* with respect to p. If a=2, b=1, and g=1.1, how does each choice change with p? What if g =.5?

(e) In general, what do your answers to this question lead you to conclude about the following possibilities: (i) If increasing differences between two choice variables fails, an exogenous increase in the returns to both choice variables might lead to a situation where one choice variable falls. (ii) If increasing differences between two choice variables fails, an exogenous increase in the returns to both chioce variables never causes both choice variables to rise.

3. The following model is a simplified version of one developed by Milgrom and Roberts (1990). We focus attention on three decisions that a firm makes in organizing its production operations: the number of varieties of its product (n), the average run size (x), and the flexibility of its equipment and methods (y). We omit from our analysis any effect that changing costs has on the firm's market share and scale of operation, which are affected by competitors' adjustments as well, focusing our modeling attention only on the choices outlined above. Thus we fix the firm's output level at q and assume that the mean price the firm receives for its products is P(n), which depends on the number of varieties produced. The firm's revenues are therefore qP(n). The cost it incurs in our model is the sum of several terms:

C(q) + h n x /2 + s(y)q/x + K(y,t)

The C(q) term is the direct cost of production in each period. The next two terms are derived from the "economic order quantity" inventory model. A firm that sells a total volume of q units andproduces lots (each of a single variety) of size x maintains an average inventory (per variety) of x/2 and has q/x set-ups, so its inventory holding costs are proportional to nx/2 and its set-up costs are proportional to q/x. In the cost formula, h and s(y) are the holding costs per unit and the costs per set-up, respectively. The costs of set-up depend on the "flexibility" of equipment and methods y. Changes in this firm's decisions arise because we assume that the marginal cost of flexibility is falling, that is, Ky is decreasing in t, where t is increasing over time as technology improves.

(a) Show that you can "reorder" one of the choices (i.e. consider the firm's choice over the negative of a variable) so that the firm's profits satisfy increasing differences between each pair of (appropriately ordered) choices.

(b) Show that you can apply the comparative statics theorem to make predictions about how the firm's choices change over time.

(c) Interpret your results from part (b), discussing each interrelationship between variables as well as the comparative statics conclusion.


1. According to Milgrom and Roberts "Does Organization Matter," why did Toyota choose JIT inventories and outside suppliers, while GM chose to keep large inventories and make its own inputs?

2. A common complaint of university students is that professors seem to remote and uninterested in teaching them. Suppose that professors can engage in three activities: teaching, research, and consulting.

(a) What incentives do universities provide for these activities through their compensation, promotion, and tenure policies? [Universities often place restrictions on outside activities, while tenure is based mainly on research and a little bit on teaching. Professors turned down for tenure are generally evaluated almost exclusively based on research when they look for new jobs.]

(b) Evaluate these incentives in the context of the multitasking model discussed in class. Why not provide higher-powered incentives for teaching? What are the incentives for junior professors to teach well? How do you expect behavior by tenured and nontenured faculty to differ?

(c) Discuss the pairwise relationships between each of the incentive instruments from the perspective of the firm: are incentives for teaching, research, and consulting substitutes or complements? That is, does a higher incentive on one dimension increase or decrease the returns to using a higher incentive on another dimension? Based on this, how should universities think about designing their incentives as a system?

3. Benetton, the Italian clothing manufacturer, has an information system in place which allows the stores to transmit daily sales data from retail stores to its manufacturing facilities. Can you predict what policies Benetton is likely to follow in the following areas: (i) inventory holdings in back rooms at stores, (ii) inventory holdings in warehouses, (iii) automation of distribution and the use of electronic barcodes in shipping, (iv) the use of scanners and barcodes in stores, (v) the ability of the manufaturing facilities to produce multiple products on the same line, (vi) the length of production runs, (vii) the relationship of Benetton to its suppliers of materials. Argue answers for each as resulting from direct or indirect interactions with the information system. Give an explanation for each interaction.

4. A management team is debating the merits of a switch from mass production to "modern manufacturing." One member of the team advocates the switch. Another counters, "I have done internal studies in our firm, and if any of the components of our system are changed individually, our profits will go down. Thus, the switch must be a bad idea." What is wrong with this logic?