I've been reading a book that discusses, among other things, the price equations of Piero Sraffa. Most of the math has not been too difficult (I don't have much mathematical knowledge beyond basic college algebra), but today I came across an example that I don't understand.
The author describes how Sraffa determines the relative prices of goods produced by two industries that use one another's products as inputs. (This makes the simplifying assumption that there are only two industries, and two types of products, in the entire economy.) The price of the product manufactured by Industry 1 is designated as p1, the price of the product manufactured by Industry 2 as p2. Each price is the sum of the production coefficients of both types of product used in the production process (i.e. xp1 + yp2 = p1, ap1 + bp2 = p2, with each coefficient <1),multiplied by one plus the rate of profit (1 + r). Note thar you typically also account for wage costs, but in the example I'm talking about, the author has already aggregated them into the production coefficients for p1 and p2 in both industries.
The problem that's been giving me...problems is as follows:
(0.28p1 + 0.48p2)(1 + r) = p1
(0.62p1 + 0.22p2)(1 + r) = p2
with the author stating that the relation between the solutions is p1/p2 = 0.93.
I've tried working out the equations while assuming that the profit rate is zero in order to simplify things, but I can never come up with the proportion of .93 between p1 and p2. When I plugged .93 and 1 into the p1 and p2 positions, respectively, the solutions on the right side did come out to a .p1/p2 = 0.93 proportion, but in both cases, p1 and p2 represented a different value on the left side of the equation than they did on the right side! I assume that there is some method of working this out that I am not yet familiar with, since I am only acquainted with basic solving-for-X procedures.
Any help would be greatly appreciated, as this has been driving me up the wall all day. If you need more context, please let me know.