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Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 15 Outline of the course
I. Introduction II. Individual choice III. Competitive markets IV. Market failure Outline of the course
I. Introduction II. Individual choice III. Competitive markets Partial equilibrium (Ch. 16) Exchange economies (Ch. 31) Production (Ch. 1823) IV. Market failure Production
Technological constraints of a firm represented by a production set in inputoutput space. Consider the one input one output case. Boundary of production set is y D f .x/;
where the function f is called the production function. Production
Now consider the two input one output case. Production function: y D f .x1; x2/;
where y is the output level, and x1 and x2 are the input levels. Inputs 1 and 2 are sometimes called production factors. Production
N Isoquants: Set of input combinations .x1; x2/ with f .x1; x2/ D y , for given y. N Three leading examples
a b CobbDouglas: f .x1; x2 / D Ax1 x2 . Perfect complements: f .x1; x2/ D minfx1; x2g. Perfect substitutes: f .x1; x2/ D ax1 C bx2. Standard assumptions on technology
Monotonicity: production function f .x1; x2/ is increasing in x1 and x2 Convexity: isoquants are convex Marginal product. Technical Rate of Substitution.
Consider two input model. Marginal product of factor 1: MP1.x1 ; x2 / D
and analogously for factor 2. Technical rate of substitution: @f .x1 ; x2 /; @x1 TRS.x1; x2/ D slope of the isoquant that passes through .x1; x2/ MP1.x1 ; x2/ MP2.x1 ; x2/ Diminishing MP and TRS
Two standard properties of technologies: Law of Diminishing MP: MP1 decreases in x1 for fixed x2, and likewise for MP2. Diminishing TRS: jTRSj decreases in x1 along isoquant. Longrun vs. shortrun
Within a certain fixed period of time it might be infeasible to change some production factors. y D f .x1; x2/ N
with x2 fixed. This is the short run. N Eventually, all factors can change. This is the long run. Returns to scale
When we talked about the law of diminishing MP, we consider the change in MP1 due to a change in x1 holding x2 fixed. Now we are going to consider joint changes in x1 and x2. If we double all inputs, does output double, less than double or more than double? Returns to scale
Constant returns to scale: for all t > 1, f .tx1; tx2 / D tf .x1; x2/
Increasing returns to scale: for all t > 1, f .tx1; tx2 / > tf .x1; x2/
Decreasing returns to scale: for all t > 1, f .tx1; tx2 / < tf .x1; x2/ Profit maximization
One output y , two inputs x1 and x2. A competitive firm takes prices p , w1 and w2 as given and solves: .y;x1 ;x2 / max py w1x1 w2x2 subj. to y D f .x1; x2/
or, equivalently,
.x1 ;x2 / max pf .x1; x2/ w1x1 w2x2: Profit maximization
Firstorder conditions: MP1.x1; x2/ D w1=p MP2.x1; x2/ D w2=p;
which is TRS.x1; x2/ D
(Note the analogy with consumer theory.) w1 : w2 This characterization is valid whenever isoquants are smooth and convex and solution is interior (i.e., both x1 and x2 are positive). Constant returns implies zero profits
If profits were not zero, by doubling input one could double the profit. If initial profit were positive, firm could not be choosing an input combination that maximizes its profits! Let us see this more formally.... At the optimum, profit is ... D pf .x1 ; x2 / w1x1 w2x2 ; * where .x1 ; x2 / is the profitmaximizing combination of inputs. Using the constant returns to scale assumption, doubling the inputs yields a profit of pf .2x1 ; 2x2 / 2w1x1 2w2x2 D 2...; which is strictly greater than ..., unless ... D 0. Hence, ... must be zero, otherwise .x1 ; x2 / would not be profitmaximizing. Cobbdouglas example
a b Technology: y D f .x1; x2/ D x1 x2 . FOC:
a b ax1 1x2 a b bx1 x2 1 w1 D MP1.x1; x2/ D p w2 D MP2.x1; x2/ D p look up the algebra in the textbook A bit of algebra (see appendix of Chapter 19) yields
aCb 1 a b yDp a w1 a 1 a b b w2 b 1 a b Notice that this is not well defined for a C b D 1 (constant returns). Cost minimization
For each output level y , a competitive firm must choose a combination of inputs to minimize cost subject to the constraint that the combination of inputs yields output y . Cost minimization: .x1 ;x2 / min w1x1 C w2x2 subj. to f .x1; x2/ D y The value of w1x1 C w2x2 at the minimum is denoted c.y/ and it is called the cost function. Cost minimization
Profit maximization then becomes: max py
y 0 c.y/ with firstorder condition: p D c 0.y/
for interior solutions y > 0. M C.y/; Cost minimization
Now let c.y/ D cv .y/ C F , where F > 0 is the fixed cost and cv is the variable cost, i.e., cv .0/ D 0. The average cost is defined as AC.y/ D c.y/=y , and the average variable cost is AV C.y/ D cv .y/=y . The marginal cost curve intersects the average cost and the average variable cost curves at their minima. In fact, if y minimizes AC then 0 D AC 0.y / D M C.y /y c.y / y 2 ) M C.y / D c.y /=y D AC.y /; and analogously for AV C . Supply curve
The (inverse) supply curve is obtained from the firstorder condition p D M C.y/;
with two exceptions. First, the firm would never choose an output level y with p D M C.y/ when M C is decreasing around y . Otherwise, by increasing y a little bit it would increase profits. Second, the firm will choose a positive output level y with p D M C.y/ only if this yields higher profits than producing zero: py cv .y/ F F ) p AV C.y/: In sum, the inverse supply curve is the increasing portion of the marginal cost curve that lies above the average cost curve. inverse supply curve ...
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 '09
 SAMUELSON
 Microeconomics

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