EconS 404, Professor Rosenman

HW4 Exercises

Analysis questions

1. During a crush to get a job done, a construction manager lamented that adding additional labor would not help because the staffing had reached a point of diminishing marginal returns.  Was the construction manager correct, and more labor would not help get the job done faster?

2.  An increasing tendency for the merger of commercial banks in the early 1990s raised the policy question of how much consolidation in the banking industry is good. The primary issue involved whether larger banks are more efficient than smaller banks just because they are bigger. While looking at this issue economists have to differentiate between long run and short run cost curves. The policy issue turned on long run cost curves, while most of the data available is about short run costs. Economists William C. Hunter and Stephen G. Timme addressed the short run versus long run question in an article in the Economic Review, published by the Federal Reserve Bank of Atlanta (May/June 1991). They focused on how the presence of fixed factors affected estimates of long run cost functions, but we can also use their analysis to see how long run and short run cost functions for the banking industry relate.

Small banks are those with total assets of $50 million to $100 million. Large banks are those with total assets exceeding $25 billion, while those in-between can be classified as moderate sized banks. In banking, as in other industries, fixed factors include physical capital. But other factors, termed quasi-fixed inputs, also cannot be changed quickly. These are factors related to transactions and information costs, including retail deposits that do not respond to interest rate changes, called "core deposits," and bank customer relationships. Core deposits are quasi-fixed because of set-up and specific transactions costs that firms incur when opening new accounts. Similarly, when lending money, banks incur costs of learning about borrowers behavior with respect to the reliability of the loan. These costs provide incentives for both sides to continue a relationship, setting them up as quasi-fixed inputs.

Hunter and Timme wanted somehow to differentiate between short run and long run costs. They did so by jointly estimating the total cost function and the variable cost function using data for banks of various sizes. Their reasoning relied on the fact that, when faced with fixed factors, banks would do the best they could in the short run by changing variable inputs. Thus, short run variable cost is a function of output, the prices of variable inputs, and the quantities of fixed and quasi-fixed factors. But in the long run, since all factors of production are variable, total cost would be a function of output and all input prices.

Of course, each different level of fixed factors gives a separate short run cost curve. Thus, by putting the quantity of fixed factors in the variable costs estimate, Hunter and Timme were able to trace out the short run cost curves for various levels of these factors. But when they used the price of fixed factors rather than the quantity, this implicitly allows them to vary, tracing out the long run cost curve. Graphically, their results are shown to the left.. Small banks and large banks both had short run average cost curves (SAC_s and SAC_l) that were higher than those of the mid-sized banks (SAC_1m and SAC_m2). But the long run average cost curve, LAC, had a large flat portion, indicating that for the range of mid-sized banks, the short run average cost curves at various levels of fixed factors were similar.

Question:  With a Long Run average cost as shown, how big should banks be?  Have recent mergers been a good idea?

TECHNICAL QUESTIONS

1. If a firm which uses capital (K) and labor (L) to produce some good finds that MPL/w>MPK/r, where w is the cost per unit of L and r is the cost per unit of K, should it substitute capital for labor? Explain.

2.  The CEO of a major Washington State utility company once argued that when faced with changes in the demand for service, electric companies cannot always respond by producing the quantity at the lowest possible cost. It was clear that he was talking about the different ability a company has to respond to changes in demand in the short run and the long run. Explain why in the short run a firm facing a change in demand may be forced to an economically inefficient method of production, and how it would adjust in the long run. Also explain why this doesn't mean the firm will be technically inefficient. Use an isocost/isoquant graph.

3.  Consider the production function Q = K^aL^b, where Q is output, L is labor inputs and K is capital inputs. Examine the marginal products, the marginal rate of (technical) substitution, and the returns to scale demonstrated by this production function.  (Calculus practice.  Find the partial derivative of Q with respect to K for the MPK, and the partial derivative of Q with respect to L for the MPL).  

4.  Inventory problems offer a simple example relating production to cost. Let Q be the annual flow of goods through storage or service. This could be, for example, light bulbs for a university. We will assume that the flow is met by inventory (I) and reorder (R). Thus, the production function is simply Q=RI. Inventory is the amount kept on hand, and R is the number of times the storage must be replenished. Let the cost of maintaining a unit of storage for a year be $1.00 per unit, and let the cost of each reorder equal $100.00. 

a) Suppose I is fixed in the short run at 1000. Derive the short run cost function, and the average total cost, average fixed cost, average variable cost and marginal cost of output. Find the short run total cost for Q=10000 and for Q=12000. (HINT: TC=100R+I, with I fixed at 1000. Also, since I=1000, R=Q/I). 

b) With this function, the MPI=R and the MPR=I. Derive the optimal input combination (hint:  remember you need MPI/MPR=price of I/price of R), and use this ratio with the production function and the total cost function to find the long run total cost function. Find the long run total cost for Q=10000 and Q=12000. Why are these different from the short run costs. 

c) Using the production function find the returns to scale, and the long run cost function find the economies of scale (look at the function for long run ATC). Are they consistent? 

d) How will the short run and long run cost functions change if the price of a reorder increases to 121 dollars?