I 1. The question is really if diminishing marginal return means another unit has no value. Look at the graphs below. Diminishing returns means we've reached point L1, thus the marginal product of additional labor was indeed decreasing. Most likely, the productivity of new labor had reached a point equivalent to L2, so even the average product was decreasing. But only when L3 is reached, so the marginal (net) product is negative, does additional labor mean there is no additional output.
I
2. To minimize average costs, banks should have between 100 million and 25 billion dollars in assets. Recent mergers have pushed banks beyond this range, so costs are higher than they should be.
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.
This inequality means the marginal product per last dollar spent on labor is bigger than the marginal product per last dollar spent on capital -- so dollars being spent on labor are giving a better return. The firm should substitute labor for capital, which will lower MPL and raise MPK.
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 = KaLb, 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.
MPL = bKaLb-1
MPK = a Ka-1Lb
MRTS = MPL/MPK=(bK/aL)
returns to scale: double both inputs to get Q* = (2K)a(2L)b, which equals 2(a+b) KaLb.
So Q*=2(a+b) Q. If a+b is less than 1, then 2(a+b) is less than 2, and Q* is less than 2Q, ie, decreasing returns to scale. If a+b exceeds1, then 2(a+b) exceeds 2, and Q* exceeds 2Q, ie, increasing returns to scale. If a+b=1, then 2(a+b) =2, and Q*=2Q, ie, constant returns to scale.
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=Q/1000).
If I=1000 then Q=1000R or R=Q/1000. Using the TC function, short run TC=100*Q/1000+1000=0.1Q+1000. Given this short run TC function, MC=0.1, ATC=0.1+1000/Q, AVC=0.1 and AFC=1000/Q.
When Q=10000, short run TC=2000, and when Q=12000 short run TC=2200.
b) With this function, the MPI=R and the MPR=I. Derive the optimal input combination, 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.
We need MPI/MPR=R/I=1/100 or 100R=I. And since Q=RI, Q/R=I, hence 100R=Q/R or 100R2=Q or equivalently, R=0.1Q½. Using these relationships in the TC function, we substitute for I to get TC=100R+I=100R+100R=200R. Now substitute for R to find that TC=200x 0.1Q½ =20 Q½ .
When Q=10000, TC=2000. When Q=12000, TC=2190.89
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?
Doubling both inputs shows that Q3=(2R)(2I)=4RI which exceeds 2RI=Q. Thus there are increasing returns to scale. Since TC=20Q½, ATC=20Q-½ which gets smaller as Q gets bigger, indicating economies of scale. They are consistent with one another.
d) How will the short run and long run cost functions change if the price of a reorder increases to 121 dollars?
Long run cost functions would increase. TC = 121R+I and from the ratio of MPI/MPR, 121R=I. Thus TC=24.2Q½