(1) When, if ever, will relative prices be proportional to labor "values?"
(3) When, if ever, will relative prices be equal to the left eigenvector for the matrix of produced inputs, A?
(2) Under what circumstances will the relationship between the uniform rate of profit, r, and the hourly wage rate, w, be not only negative but also linear?
(3) Suppose r > 0 and v(i) = v(j). But suppose more of the labor necessary to produce good j takes place farther in the past and more of the labor necessary to produce good i takes place closer to the present, what can you deduce about p(i) and p(j)?
(4) Suppose r = 0 and v(i) = v(j). But suppose more of the labor necessary to produce good j takes place farther in the past and more of the labor necessary to produce good i takes place closer to the present, what can you deduce about p(i) and p(j)?
(5) RAx* = x* - Ax* where x* is the right eigenvector for A. What can you deduce about
RAx and x - Ax for any x that is not proportional to x*?
(6) Is the matrix a(11) a(12) indecomposable?
0 0
(7) If a(11)L(2) - a(12)L(1) < 0 which industry has the higher capital/labor ratio?
(8) If A(n,n) is indecomposable what, if anything, can we conclude about A(n-1,n-1)? What, if anything, can we conclude about A(n-2,n-2)?
(9) In equation (22) how do we know that the numerator gets smaller and the denominator gets bigger when r increases?
(10) If the only non-basic good is good n, what must be true about the last row of A(n,n)?
(11) According to Theorem 14 - i the conditions of production in the basic industries only are sufficient to determine both the uniform rate of profit and the relative prices for all basics in the economy. This implies that even should there be technical changes which improve the efficiency of production in non-basic industries this can have no effect on the rate of profit in basic and non-basic industries. Explain how equation (23a) proves this surprising result.
(12) Suppose only good n is not a basic good, suppose w = 0, and suppose dom{A(n-1,n-1)} = 0.8. However, suppose a(nn) = 0.9 Why does this make it impossible for r(n) to be as high as the rate of profit in all the other sectors of the economy? Why can we not solve this problem by simply raising p(n) in order to raise r(n) until it is as high as the rate of profit in the other sectors? If this situation arose in the real world, what do you suppose would happen?
(13) What does it mean to set p(I-A)x* = 1? Why is this something we are free to do if we find it convenient?
(14) What does it mean to set Lx = 1? Why is this something we are free to do if we find it convenient?
(15) In a multisector model with heterogeneous labor and multiple non-labor primary inputs, if we produce gross outputs in the proportions given by the right eigenvector, x*, of the economy input output matrix, A, the relation between the distributive variables w(k) k=1...W, the different wage rates, u(q) q=1...U, the rents for different non-labor primary inputs; and r, the uniform rate of profit in the economy, is given by: wLx* + uTx* + (1/R)r = 1, where R is the maximum possible rate of profit in the economy, L is the matrix of direct labor inputs for the W different kinds of labor, and T is the matrix of primary inputs for the U different kinds of non-producible, primary inputs. What does this equation imply, or not imply? What are its political implications? How must one be careful when applying it to real world situations?