Assignment:
HETERODOX MICROECONOMICS
The Simple Sraffa Model
Assume a two sector economy defined by the technology below:
a(11) = 0.2 a(12) = 0.1
a(21) = 0.3 a(22) = 0.4
L(1) = 0.2 L(2) = 0.3
1) What are the values of the two goods, v(1) and v(2)?
2) Setting p(2) = 1, what would the price of the first good and wage rate be if the rate of profit in the economy were 0%?
3) Setting p(2) = 1, what would the price of the first good and wage rate be if the rate of profit in the economy were 10%?
4) What seems to be the relationship between the rate of profit and the wage rate in this economy?
5) Under these conditions suppose capitalists in sector 1 discover the following new technique:
a(11)' = 0.2
a(21)' = 0.2
L(1)' = 0.26
Will capitalists in sector 1 replace their old technique with this new one?
6) What are the "values" of the two goods, v'(1) and v'(2) if capitalists in sector 1 do use the new technique?
7) Will capitalists in sector 1 serve the social interest when deciding whether or not to implement the new technology? Explain both your result and the intuition behind it.
The Frobenius-Perron and Okishio Theorems
The part of the Frobenius Perron theorem for A* non-negative and indecomposable that is critical for proving the Okishio theorem is:
For any p > 0 either: (1) min(j) {pA*(j)/p(j)} = a = max(j){pA*(j)/p(j)}
Or: (2) min(j) {pA*(j)/p(j)} < a < max(j){pA*(j)/p(j)}
(8) Provide an economic interpretation of pA*(j).
(9) If condition (1) holds for some particular p' what do we know about p' ?
(10) To prove condition (2) for p not an eigenvector of A* we assume pA* ³ ap and reason as follows: There is some y > 0 such that A*y = ay. How do we know this?
(11) Post multiplying pA* ³ ap by y gives: pA*y > apy. How do we know this?
(12) But pre multiplying A*y = ay by p gives: pA*y = apy. How do we know this?
(13) Therefore p*A ³ ap is false. How do we know this?
(14) Therefore there is some j for which pA*(j)/p(j) < a. How do we know this?
A Productive Economy and Fundamental Theorems
(15) Clearly, just because an economy is productive according to the weaker condition D2 does not imply that it is productive according to the stronger condition D1. However, if an economy is productive according to the weaker condition D4 it must also be productive according to the stronger condition D3. How do you explain this seeming contradiction?
(16) According to theorem 8 what does it mean to say that an economy is productive?
(17) It is very convenient to be able to define whether or not an economy is productive or not productive in terms of a single parameter, the value of its characteristic root. That is, it is convenient to be able to state: If and only if dom (A~) < 1 is an economy productive. Moreover, this allows us to say very precisely how much the productivity of the economy has grown in any year. Suppose a = 1.0 initially, and then technological changes implemented during the year change A~ so that a' = 0.9 in the following year. How much would you say the productivity of the economy has increased?
(18) (a) Why would nobody bother picking up a subsistence economy, A^ , off the beach if
A^ = A? (b) Why would capitalists never pick up A^ = A*? (c) Why might workers pick up
A^ = A*? (d) If A^ = A* and b is very large, is it consistent with common usage to call A^ a subsistence economy? In what sense might A^ still be considered unproductive?
(19) (a) Why might workers pick up a productive economy, A~ , off the beach if A~ = A? What would determine how happy they were with their find? (b) Why would capitalists pick up
A~ = A*? What would determine how happy they were with their find?
(20) The so-called Fundamental Marxist Theorem is: If and only if the rate of exploitation, e, is positive will the rate of profit, r, be positive - where e is defined as (1 - Vb)/Vb. First, explain what Vb represents, and therefore what e represents. Then compare what the Fundamental Marxist Theorem says as compared to what the Fundamental Sraffian Theorem, T 11, says.