Econ 604 Advanced Microeconomics
Econ 604 Advanced Microeconomics
Davis Spring 2006, April 6
Lecture 10
Note: No lecture next time (April 13)
Reading. Chap. 11. pp 268 - 274
Problems 11.1, 11.3, 11.5, 11.6, 11.7
Note: Problem 11.6 Part b.Typo.” RTS = (K/L)1-( (Inverted from text). Also, 11.7 part c is messy, I don’t expect a clean answer.
Next time: Chapter 12.
REVIEW
VIII. Comments on HW 9 that I return tonight.
In Chapter 7, after discussing market demand and problems of aggregation, we mentioned some relationships between elasticities. You should be certain that you know these. Specifically we know three.
(a) Engel’s Law: All income is spent, so the share weighted some of income elasticities equals one..
sxeX,I + sYeY,I = 1
(b) The Slutsky Equation in elasticities: Price elasticity of demand is the sum of compensated price elasticity of demand and a share weighted income elasticty.
eXP = eSXP - eXI sx
(c) Euler’s Therom Applied to Demand functions (that are homogeneous of degree zero)
The sum of income and price elasticties equal zero. Thus, a constant percentage increase in all prices and income will have no net effect.
eX,Px + e X,Py + e X,,I = 0
Your problem 7.9 was intended to review these concepts.
IX. Chapter 8 Expected Utility and Risk Aversion
A. Probability and Expected Value.
Probability is discussed as relative frequency
Expected value equals outcome weighted probabilities
B. Fair Games and the Expected Utility Hypothesis
Fair Games are those with equal expected values
Individuals often will not play fair games (e.g., the St. Petersburg Paradox). Risk aversion is a likely explanation. (problem 8.3)
C. The Von Neumann-Morgenstern Theorem
Under certain relatively innocuous assumptions, one can construct a utility index that ranks all uncertain outcomes. Further, individuals act as if they maximize the value of this expected utility index.
D. Risk Aversion
A preference for a less variable outcome relative to a more variable outcome is terms risk aversion. Risk aversion implies a diminishing marginal utility of income. Notice that to measure risk aversion we must estimate the curvature of the utility function. This entails making considerable stronger assumptions regarding the utlity function that we were forced to make in our previous analyses.
Risk Aversion and Insurance: Given a way to calculate the utility loss associated with a more variable outcome, we can calculate the amount individuals would pay to avoid the risk (problems 8.5 and 8.7)
E. Measuring Risk Aversion
We developed two measures of risk aversion:
A coefficient of absolute risk aversion
r(W) = - U”(W)/U’(W)
A coefficient of relative risk aversion
rr(W) = - WU”(W)/U’(W)
Finally, we observed that the risk posture of an individual may change as wealth changes. For a person exhibiting constant absolute risk aversion, risk posture will not change with wealth (this would be true, for example with an exponential utility function). A person exhibiting constant relative risk aversion, will be sensitive to wealth levels. However, the person’s attitude toward a given percentage gain and loss will remain constant (example a logarithmic utility function)
Note: We skip section F in chapter 8. This development can be useful (essentially, in it risk is analyzed in terms of an indifference curve analysis, where the two “goods” are wealth in the good and bad states. However, given the little time remaining in the semester, I prefer to move on to production theory.
PREVIEW
Chapter 11. Production Functions
A. Production Functions, and Marginal Productivity.
B. Isoquant Maps and the Rate of Technical Substitution
C. Returns to Scale
Constant Returns to Scale and the RTS
The n-Input Case
D. The Elasticity of Substitution
E. Some Common Production Functions
Linear
Fixed Proportions
Cobb-Douglas
CES
F. Technical Progress
Measuring Technical Progress
Growth Accounting
Lecture______________________________________________________
X Chapter 11. Production Functions. To this point, we’ve focused on the demand side of the output market. That is, we studied the optimizing decisions for the utility maximizing consumer. Now we turn our attention to the other half of standard economic analysis: the theory of the firm (who presumably maximizes profits in light of a cost constraint).
We approach the problem for the firm in a standard way. First (in this chapter) we consider production theory. This is a study of the activity of converting inputs into outputs. In this chapter we will identify an optimal input mix. Then in the following chapter, we will study costs conditions for the firm, under the condition that inputs are hired optimally. Finally, we will use the cost conditions to identify profit maximizing output levels, and ultimately market supply.
A. Marginal Productivity.
The production function. As a general matter, firms use a variety of inputs in order to produce outputs. A general expression for this relationship is
q = f(K, L, M, …)
For our purposes it is expedient to focus on a simple production function where only two inputs are available, L, labor and K, Capital. Thus we define the relation
q = f(K, L)
as a production function illustrating the maximum amount of q that can be produced from combinations of K and L.
Now to study optimal input use it will be useful to consider marginal contributions.
Marginal Physical Product: The additional output that can be produced by employing one more unit of that input while holding all other inputs constant. Mathematically
Marginal physical product of capital = MPK = (q/(K = fK
Marginal physical product of labor = MPL = (q/(L = fL
Diminishing Marginal Product. Importantly, these are partial derivatives, reflecting the fact that the marginal productivity of labor, for example, may be dependent on the quantity of capital employed. Parallel to utility, we would expect marginal productivity to fall as more of the input is used. This is the notion of “crowding” that underlies the standard assumption of a Law of Diminishing Returns. Mathematically
(MPK/(K = (2q/(K2 = fKK< 0
(MPL/(L = (2q/(L2 = fLL< 0
Often offsetting this diminishing marginal productivity is an increase in cross productivity. While labor may become marginally less productivity, holding the amount of capital fixed, increases in capital will shift the productivity of labor upward. Mathematically, we typically assume
(MPL/(K = (2q/(L(K = fLK> 0
Average Physical Product. Although marginal productivity is by far the most important term in determining optimal resource use, in many empirical studies average productivity is measured. Certainly “labor productivity” in the popular press refers to average product.
APL = output/Labor = q/L = f(K,L)/L
Example 11.1. A Two-Input Production Function: Consider a production function
Q= f(K,L) = 600K2L2 - K3L3
To construct marginal and average labor productivity (L), we must assume a value for K. Suppose K=10. Then
Q= 60,000L2 - 1000L3
Then the Marginal Product of Labor
MPL = (q/(L = 120,000L - 3000L2
Notice that here Marginal productivity diminishes, and eventually becomes negative. This implies q has a maximum value
120,000 = 3000L
L = 40
Thus, any units of labor beyond 40 actually reduce total output. For example, holding K =10, q = 32 million when L =40. When L = 50, q = 25 million
The Average Product of Labor is
APL = q/L = 60,000L – 1000L2
Again, this is an inverted parabola, and reaches a maximum when
(APL/(L= 60,000 - 2,000L = 0
This occurs when L=30.
Notice, that when L = 30
APL = 60,000(30) - 1000(30)2 = 900,000
Also
MPL = 120,000(30) - 120,000(30) - 3000(30)2
= 900,000
This is quite general. The marginal drives the average in the sense that when the marginal is above the average, it pulls the average up. When it is below the average, it pulls the average down.
In essence, the average conveys the same information as the marginal. However, the marginal is more volatile, because it carries with it the weight of inframarginal decisions.
B. Isoquant Maps and the Rate of Technical Substitution
To illustrate possible substitution of one input for another we use the production function’s isoquant map. The isoquant map illustrates all combinations of inputs that can be used to produce a given output. Mathematically, an isoquant records the set of K and L that satisfy
f(K,L) = qo
There are an infinite number of isoquants in the K, L plane, one for every level of output, as shown in the figure to the left. Notice the relationship between isoquants and the indifference map for utility functions. The most important difference is that isoquant are directly observable. Labor, Capital and levels of output are all measurable quantities, and economists devote considerable attention to measuring them.
The slope of the line tangent to an isoquant map reflects the tradeoff of one input for another that will keep total output constant. This relationship is termed the Marginal Rate of Technical Substitution (RTS).
Marginal Rate of Technical Substitution (RTS). The rate at which labor can be substituted for capital while holding output constant along an isoquant. Mathematically,
RTS (L for K) = -dK/dL| q=qo
RTS and Marginal Productivities. Total differentiation of the production function allows development of the RTS in terms of marginal productivities
dq = (f/(L dL + (f/(K dK =MPLdL + MPKdK
Along a given isoquant dq=0. Thus
-dK/dL|q=qo = MPL/MPK
It is obvious from the above relationship that isoquants must be negatively sloped, since both marginal productivities are positive. Nevertheless, parallel to the MRS in utility theory, we define the RTS as a positive number. We can construct examples with negative marginal productivities, but they don’t make sense economically.
Reasons for a Diminishing RTS
Not only are the isoquants in Figure 11.1 negative, but they are convex. Thus, the RTS is diminishing. One might generally assume that diminishing RTS follows from the law of diminishing returns. In fact, the relationship is a bit more complicated, since we increase one input as we decrease another (the law of diminishing returns holds other inputs fixed)
To see the problem, assume q = f(K,L), and that fK, fL>0. fKK, fLL 3K3L2
or KL200
However, this does not necessarily imply a continuously diminishing RTS.
fLK = 2400KL - 9K2L2
Observe that this function is positive only when
KL < 266.67 = 2400/9
Thus, in the range 200 ................
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