2.Properties of PS.
• Additive (free entrance) :
y Y,and y Y, then y y Y • Convexity: y Y,and y Y, then y (1 )y Y, here [0,1]
See the fig.
• Proposition1: if Y is convex, so is V(q). • Proposition2: if V(q) is convex, f(x) is quasiconcave.
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y Y, ay Y, a 0
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4.Returns to scale
• Proposition 3: Y is constant returns to
scale if Y is both “additive” and “convexity”. • Proposition 4: single production, if and only if f(.) is homogenous of degree 1, Y is constant returns to scale.
the array y y, and y Y means y Y • No free lunch: Y n {0}
n n
See the fig.
• Free disposal:
See the fig.
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y n Y
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xj xi
• Question2: calculate the “TRS” and the
“elasticity of substitution” of CobbDouglas technology and CES technology.
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– Homothetic function:
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Assignment
• Textbook: ex.1.1, ex.1.3, ex.1.7. ex.1.9
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– Some of yi in y are restricted on z . – Short-run production set.
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1. Production set
• Input requirement set:
– All yi in y are negative, let them be –x (then x is positive), and the rest yj to be q . I O n – So, x and q , and y (q,-x) – the input requirement set is :
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4.Returns to scale
• Homogeneous and homothetic tech.
– Homogeneous of degree k:
f (tx) t k f (x) t 0
•
f (x) g (h(x)) h(x) is HD1, g (.) is a possitive monotonic function. Elasticity of scale: d ln q(t ) e( x ) q(t ) f (tx) d ln t t 1
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– If O = 1,and I = n - 1, then:
2.Properties of PS.
• Y is nonempty: we have something to do. • Y is close: Y contain it’s boundary,
y2
T ( y )
{ y : T ( y) 0}
y1
MRT12
{ y : T ( y) 0}
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1. Production set
• Production plan (production vector, or
input-output vector): y ( y1 , y2 , , yn ) • Production set Y: all technological feasible y. Y {y n : y are technologically feasible} • Restricted production set Y (z):
Choice
(purchase)
Choice
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Technology
• Contain:
– “production (possibilities) set” (PS) and “production function”; – Properties of the “PS”; – Technical rate of substitution; – Returns to scale.
4.Returns to scale
• Nonincreasing returns to scale:
y Y, ay Y, a [0,1] y Y, ay Y, a 1
• Nondecreasing returns to scale:
• Constant return to scale:
Advanced Microeconomics
(lecture 1: production theory I)
Summary
• textbook: • assignments:
– Varian, Hal R., 1992, Microeconomics Analysis, 3rd ed. – Mas-Colell, A., M. Whinston, and J. Green, 1995, Microeconomics Theory. – twice a week; – Team work; – Deliver on the class. – Mid-term: by the assignments; – Final-term: 80% of the questions coming from the assignments.
MRTS ji (x )

f (x) / x j f (x) / xi
x=x
• As x changes , we got technical rate of
substitution
TRS ji
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f (x) / x j f (x) / xi
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3.Technical rate of substitution
• The elasticity of substitution: the
curvature of the isoquant.
ji
( ) /
xj xi
xj xi
TRS ji / TRS ji
0
d ln( ) d ln(TRS ji )
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4.Returns to scale
• Why we assume that Y is non-increasing
(or decreasing for usual) returns to scale? • Suppose Y is decreasing returns to scale, and it’s PF f(x), now we introduce new input z, and difine a new PF: F ( z , x) zf (x / z ) F(.) is homogenous of degree 1.
{y : T (y ) 0}
y2
Y {y n : T (y) 0}
y1
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1.Production set
• Production function:
q f (x) when T (q , x) 0 – means: f (x) ={q q : (q, x) Y} • Isoquant: Q(q) {f (x) q : (q, x) Y} • Question1: calculate the PS, RS, TF, PF, Isoquant of Cobb-Douglas technology and Leontief technology.
I V (q) {x : (q, x) Y}
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