1. 7. Concavity and convexity
Econ 494
Spring 2013

2. Why are we doing this? Desirable properties for a production function:
Positive marginal product 𝑓 𝑖 >0
Diminishing marginal product 𝑓 𝑖𝑖 <0
Isoquants should have
Negative rate of technical substitution 𝑑 𝑥 2 𝑑 𝑥 1 <0
Diminishing RTS 𝑑 2 𝑥 2 𝑑 𝑥 >0
We want to link these desired properties to the shape of the production function.
This will also apply to the utility function when we discuss consumer theory
Typo corrected

3. Where are we going with this? Why do we care?
Production function defines the transformation of inputs into outputs
Postulates of firm behavior
Profit maximization
Cost minimization
Results: shape of production fctn is key to FONC and SOSC
Especially in evaluating comparative statics.

4. Math review: Shape of functions
Concavity and convexity
Quasi-concavity and quasi-convexity
Determinant tests

5. Concavity Concavity is a weaker condition than strict concavity
f(x0)
f(x1)
qf(x0) + (1-q) f(x1)
Strict concavity
Concavity is a weaker condition than strict concavity
Concavity allows linear segments
Give an example on the board using y=ln(x)

7. Convexity Strict convexity and convexity
Reverse direction of inequality
qf(x0) + (1-q) f(x1)

8. Shape of production function
Is this production function concave? Convex? Both?
Concavity/convexity is usually defined for some region of f(x).

9. Quasi-concavity
Production functions are also quasi-concave

10. Quasi-concavity
A quasi-concave function cannot have a “U” shaped portion
f(x) is not quasi-concave over its whole domain.

11. Strict quasi-concavity
No linear segments (or “U” shaped portions)
Replace weak inequality with strict inequality

12. Quasi-convexity
For quasi-convex function, change direction of inequality and change “min” to “max”

13. Recap

14. Principal minors Best illustrated with an example:
This pattern applies for square matrices of any dimension

15. Remember Young’s theorem: fij = fji
Using Hessian matrix
Hessian matrix
Contains all 2nd partial derivatives
Remember Young’s theorem: fij = fji

16. Strict Concavity
The function 𝑦=𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 is strictly concave if its Hessian matrix is negative definite (ND).
A negative definite matrix has leading principal minors with determinants that alternate in sign, starting with negative:
𝐇 1 <0 𝐇 2 >0 𝐇 3 <0 etc.
Alternatively: −1 𝑛 ∙ 𝐇 𝑛 >0
Diagonal elements of H are all <0
This is a sufficient condition for ND

17. Recall SOSC for maximum (2 variables)
𝑓 11 <0 and 𝑓 11 𝑓 22 − 𝑓 >0
Note that 𝑓 22 <0 is implied by the above
𝑓 11 𝑓 22 − 𝑓 >0
𝑓 11 𝑓 22 > 𝑓 12 2
𝑓 22 < 𝑓 𝑓 Note sign reversal because 𝑓 11 <0.
𝑓 22 < 𝑓 𝑓 11 <0 Because 𝑓 >0 and 𝑓 11 <0
The above meet the conditions for a negative definite matrix:
−1 𝑛 ∙ 𝐇 𝑛 >0 (for all 𝑛)
Let 𝐇 1 = 𝑓 11 <0 and 𝐇 2 = 𝑓 11 𝑓 22 − 𝑓 >0

18. Concavity
The function 𝑦=𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 is concave if its Hessian matrix is negative semi-definite (NSD).
For NSD, replace strict inequality with weak inequality
Determinants still alternate in sign:
𝐇 1 ≤0 𝐇 2 ≥0 𝐇 3 ≤0 etc.
Alternatively: −1 𝑛 ∙ 𝐇 𝑛 ≥0
This is a necessary condition for NSD

19. Strict Convexity
The function 𝑦=𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 is strictly convex if its Hessian matrix is positive definite (PD).
A positive definite matrix has leading principal minors with determinants that are all strictly positive:
𝐇 1 >0 𝐇 2 >0 𝐇 3 >0 etc.
Alternatively: 𝑛 ∙ 𝐇 𝑛 >0
Diagonal elements of H are >0
This is a sufficient condition for PD

20. Convexity
The function 𝑦=𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 is convex if its Hessian matrix is positive semi-definite (PSD).
For PSD, replace strict inequality with weak inequality.
Determinants all non-negative:
𝐇 1 ≥0 𝐇 2 ≥0 𝐇 3 ≥0 etc.
Alternatively: 𝑛 ∙ 𝐇 𝑛 ≥0
This is a necessary condition for PSD

21. Quasi-concavity
The function 𝑦=𝑓 𝑥 1 , 𝑥 2 ,…, 𝑥 𝑛 is quasi-concave if its bordered Hessian matrix is negative definite (ND).
ND is sufficient for quasi-concavity
Strict quasi-concavity
No convenient determinant conditions for distinguishing quasi-concavity from strict quasi-concavity.
The bordered Hessian being NSD is a necessary condition for quasi-concavity.

22. Example 𝑦=𝑓 𝑥 1 , 𝑥 2

23. Concavity and quasi-concavity

24. Recap Concavity H is NSD (–1)n |Hn| ³ 0 necessary Strict concavity
H is ND
(–1)n |Hn| > 0
sufficient
Convexity
H is PSD
(+1)n |Hn| ³ 0
Strict convexity
H is PD
(+1)n |Hn| > 0
Quasi-concavity
BH is NSD
(–1)n |BHn| ³ 0
BH is ND
(–1)n |BHn| > 0