Econ 604 Advanced Microeconomics



Econ 604 Advanced Microeconomics

Davis Spring 2006, April 20

Lecture 11

Reading. Chapter 12

Next time: Finish Ch. 12 (if necessary) Ch. 13

Problems: 12.3; 12.5; 12.7, 12.9

REVIEW

X. Chapter 11. Production Functions

A. Production Functions, and Marginal Productivity.

Relationship between Marginals, Averages and Totals (Problem 11.1)

B. Isoquant Maps and the Rate of Technical Substitution

RTS (L for K) = -dK/dL| q=qo

RTS and Marginal Productivities

Different types of isoquant maps

Reasons for a Diminishing RTS. (quasi-concavity conditions)

(Totally differentiate RTS=fL/ fK

dRTS = (fLK (fK) - fLfKK) /fK2 dK + (fLLfK - fLfKL)/fK2 dL

Then divide through by dL, using the fact that dK/dL = -fL/fK along an isoquant, and Young’s theorem

dRTS/dL = (f2LfKK + f2KfLL-2 fKfLfLK ) /(fK)3

Observe that the above condition is negative if the products are complements fLK>0, or if substitutes, if the own effects dominate the cross effects.)

C. Returns to Scale

Constant Returns to Scale and the RTS

q = f(X1, X2, .. , Xn)

Consider

mkq = f(mX1, mX2, .. , mXn)

k=1 we have constant returns to scale

k>1, increasing returns

k 0

and

(TC/(w = (L/(w = L > 0

While input substitution effects may damp this effect, costs would certainly not fall. Were costs to fall, the firm could not have been optimizing in the first place. For exactly the same reason average costs should also increase.

Marginal costs will also generally increase as well. A possibility exists that an input may be inferior, which may, perhaps surprisingly, cause marginal costs to fall. (That is, the cost of a input increases, and you use so much less of that input that marginal costs fall.)

MC = (TC/(q = ( L/(q = (

Consider the effects on MC of a change in the rental price of capital

(MC/(v = (2 L/((q(v) = (2 L/((v(q) = (K/((q)

This latter term is positive or negative depending on whether Capital is inferior or not.

b. Input Substitution. Consider now more formally, the effects of a change in relative input prices on the optimal K/L. We examine the derivative

((K/L) / ((w/v)

along a given isoquant. Expressed in proportional terms yields s,

s = [((K/L)(w/v)]/ [((w/v)(K/L)]

which, of course, is the elasticity of substitution (where the input price ratio substitutes for the RTS).

In the two input case, s must be nonnegative, and an increase in w/v must increase the ratio K/L, or (in the case of perfect complements) leave it unchanged.

c. Partial Elasticity of Substitution. In the more general case (with multiple inputs) we write, for inputs Xi and Xj

sij = [((Xi/ Xj)(wj/ wi)]/ [((wj/ wi) (Xi/ Xj)]

where output and other input prices are held constant. We term above sij as a partial elasticity of substitution. Non-negativity may not hold in this case, as the increase in, say, wj may cause increased use of a third factor, which may result in decreased used of Xj as well as Xi.

To be concrete, consider a production process consisting of capital, labor and, say, energy. An increase in the price of energy may result in decreased use of capital and increased use of labor. The partial elasticity of substitution is useful for studying the derived demand for inputs, and can be calculated if the production function is known. However, we won’t pursue it in detail here.

d. Quantitative Size of Shifts in Cost Curves. From the preceding discussion we know that increases in an input price will increase total, average and (in most instances) marginal costs. Consider briefly the quantitative impact of increasing input prices. We confine comments to some intuitive observations.

1) Costs will increase a lot if the input is an important part of production. (e.g., a labor cost increase will affect operations at a call center very considerably, since labor is such a large percentage of the total costs for the firm)

2) Costs will increase less if the input has good substitutes. For example, an increase in copper prices did not affect importantly electricity distribution costs, since suppliers could easily switch from copper to aluminum. On the other hand, gold jewelry costs move very closely with the price of gold, since no substitutes for this critical input exist.

3. Technical Progress. Costs will also decrease with technical improvements, since the technical improvements allow the same output to be produced with fewer inputs. This is easy to illustrate in the case of constant returns to scale. With CRS

TCo = C0(q,v,w) = qCo(v,w)

Now consider a production function that allows for technical progress, as we did at the end of chapter 11.

q = A(t)f(K,L)

Where at time t0, A(t0) = 1. Thus, unit costs at time t (in terms of output qo) become

Ct(v, w) = [C0(v, w)qo]/ [A(t)qo] = Co(v, w)/ A(t)

Total Costs are

TCt (q, v, w) = Ct(v, w)qo = TCo/A(t)

(Notice, in the above example, I’m abusing the notation a bit by defining q0 = f(K,L). We do not consider inter-temporal output effects, so q need not be subscripted. However, even holding output fixed, I need some way to indicate that fewer inputs were required to produce that output. Thus, we define the production function in terms of initial period outputs.

Example 12.2 Cobb-Douglas Cost Function. Consider again the Cobb-Douglas production function qo= 10K ½L ½. Although generating costs from a production function can be tedious, the process is rather straightforward here.

Given v and w, sellers minimize the cost of producing q0 when

w/v = K/L

Given

q= 10K ½ L ½

q/K= 10(L/K) ½

Substituting

q/K= 10(w/v) ½

Solving for vK renders

vK = (q/10)(wv) ½

Reasoning similarly

wL = (q/10)(wv) ½

Thus

TC = vK + wL

= (q/10)(wv) ½ + (q/10)(wv) ½

= (2q/10)(wv) ½

which is the cost function for our Cobb-Douglas production relation. Thus, for example, suppose that w= v = $4.

Average costs are

TC/q = .2(wv) ½

so, with w= v = $4,

AC = .2(4) = .8

Marginal costs are

(TC/(q = .2(wv) ½ Identical to average costs.

Input Price Changes. Now consider the effects of an input price change. Suppose v = $9 and w = $4. Then

TC’ = (2q/10)(4*9) ½

= 1.2q

Hence AC and MC increase from $0.80 to $1.20. Observe in this case that it was unnecessary to go back and recalculate the cost minimizing input choices here. That is because we have an expression for the cost function. Input changes are accounted for in this expression automatically.

Technical Progress Suppose we can write the production function in terms of a temporal element, as we did in Example 11.4.

qt = A(t)f(K,L) = e.05tf(K,L) = e.05tqo

Then total costs at any time t are

TCt = TC0/A(t) = e.-05t[.2qo(wv) ½]

Thus, after, say, 10 years of progress, costs are

TC10 = .607TC0. = .121qo(wv) ½

With w = v = 4,

TC10 = .48q0

So average and marginal costs have fallen by 40% (from 80 to 48. 48/80 = 6/10)

E. Short Run, Long Run Distinction. It is conventional in economics to distinguish between a long run, when all inputs are variable, from a short run, where at least one input is fixed. The former is termed the “planning horizon,” where all inputs are variable. The latter, the “operating horizon” assesses the optimal level of short run output. Here we assess the effects of having some factors in fixed supply by holding capital constant, at a level K1.

Thus q = f(K1, L)

1. Short Run Total Costs Short run total costs become

STC = vK1 + wL

2. Fixed and Variable Costs. Costs for the fixed capital stock K1 are termed Short Run Fixed Costs (SFC). Labor costs, wL are short run variable costs (SVC).

3. Nonoptimality of Short-Run Costs. Importantly, short run costs are not the minimal costs for producing various output levels, because the firm cannot adjust optimally the capital stock

The figure to the left illustrates. Although the firm optimally uses labor and capital for production level q1, either decreasing or increasing output from L1, K1 requires deviation from the efficient set of input combinations, because K is fixed at K1.

4. Short-Run Marginal and Average Costs. We will often refer to short run costs on a per unit basis. Thus, we develop definitions that parallel the long run case

SATC = STC/q

SMC = (STC/(q

5. Short-Run Average Fixed and Variable Costs. Similarly, it is instructive to break total costs into fixed costs and variable cost components.

SAFC = SFC/q

SAVC = SVC/q

Where SFC and SVC denote short run fixed and short run variable costs, respectively.

6. Relationship Between Short-Run and Long-Run Cost Curves. Consider the function

STC(q, K) = total cost.

Now increasing the allowed capital stock generates a new STC curve. The family of such curves (generated by application the envelope theorem) illustrate the minimum cost combinations output at each production level. Taking the minimum points for each individual curve generates the long run total cost curve, as illustrated below in total cost space (on the left) and in unit cost space (on the right)

i [pic]

7. Per-Unit Cost Curves. An important aspect of the unit cost curves, shown on the right above is that the optimal point of operation for a firm in the short run typically will not be at the point of minimum short run costs. Only if the firm is at the optimal scale of operation (e.g., at the base of the LRATC schedule) will the firm operate at the point of minimum costs. Otherwise the firm will use relatively too much or too little of the inputs available in fixed supply. The Long Run Equilibrium condition for efficient operation is as follows

AC = MC = SATC = SMC.

Example 12.3 Short Run Cobb-Douglas Cosst. Consider the effects of holding capital fixed at a level K1 in terms of the Cobb Douglas production function we used in the previous example.

q = 10K1 ½ L ½

Thus the total productivity of Labor is

L = q2/[100K1]

Short run costs are

STC = vK1 + wL

= vK1 + wq2/[100K1]

Thus, for example, if K1 = 4.

STC = 4v + wq2/[400]

To calculate total costs, we need v and w. Let w = v = $4 and let K = 1, 4, and 9. This yields the following costs

| q | STC(K=1) | STC(K=4) | STC(K=9) | TC |

|0 |4 |16 |36.00 |0 |

|10 |8 |17 |36.44 |8 |

|20 |20 |20 |37.78 |16 |

|30 |40 |25 |40.00 |24 |

|40 |68 |32 |43.11 |32 |

|50 |104 |41 |47.11 |40 |

|60 |148 |52 |52.00 |48 |

|70 |200 |65 |57.78 |56 |

|80 |260 |80 |64.44 |64 |

|90 |328 |97 |72.00 |72 |

|100 |404 |116 |80.44 |80 |

(Recall, that the TC figures are given by TC = .2q(wv)1/2)

Notice in the table that there exists only a single point of overlap between any short run total costs curve, and the long run cost curve. Further, except for that single point of tangency, short run costs exceed TC.

An Envelope Derivation. In fact, the long run total cost curve can be derived from the short run curves, by application of the envelope theorem. Consider again the STC expression, but where v = w = $4.

STC = 4K + 4q2/[100K]

But now let K vary. Differentiating w.r.t. K

(STC/(K = 4 - 4q2/[100K2]

Setting this derivative to zero (since we wish to minimize STC at each output level), yields

4 = 4q2/[100K2]

Solving

K = q/10.

Substituting back into the STC function yields the TC function (or the envelope of minimum STC points)

TC = 4q/10 + 4q2/[100(q/10)]

= .4q + .4q

= .8q

Per Unit Cost Function. Recall that long run marginal and average costs were constant at $0.80.

SATC =

vK1/q + wq/[100K1]

SMC = 2wq/[100K1]

Again setting v = w = 4

SATC =

4K1/q + q/[25K1]

SMC = .08q/K1

The chart to the left illustrates these curves for the cases where K = 1, 4, and 9. Notice particularly in the chart that each short run curve is tangent to the MC=AC curve only once.

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