Negative Größen bei Diophant?

In this paper we champion Diophantus of Alexandria and Isabella Basmakova against Norbert Schappacher. In two publications ([46] and [47]) he puts forward inter alia two propositions: Questioning Diophantus' originality he considers affirmatively the possibility, that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls Basmakova's claim (in [5]), that Diophantus uses negative numbers, a "nonsense", reproaching her for her "thoughtlessness". First, we disprove Schappacher's Bourbaki thesis. Second, we investigate the semantic meaning and historical significance of Diophantus' keywords leipsis and mparxis. Next, we discuss Schappacher's epistemology of the history of mathematics and defend Basmakova's methods. Furthermore, we give 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a - b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis (pleyra) of a square number (tetragonos), which is afterwards computed by the formula (a - b)^2 = a^2 + b^2 - 2ab. Finally, we report how the topic "Diophantus and the negative numbers" has been dealt with by translators and commentators from Maximus Planudes onwards.

Individual Decision-Making and Inflation Persistence

The traditional task of a central bank is to preserve price stability and, in doing so, not to impair the real economy more than necessary. To meet this challenge, it is of great relevance whether inflation is only driven by inflation expectations and the current output gap or whether it is, in addition, influenced by past inflation. In the former case, as described by the New Keynesian Phillips curve, the central bank can immediately and simultaneously achieve price stability and equilibrium output, the so-called ‘divine coincidence’ (Blanchard and Galí 2007). In the latter case, the achievement of price stability is costly in terms of output and will be pursued over several periods. Similarly, it is important to distinguish this latter case, which describes ‘intrinsic’ inflation persistence, from that of ‘extrinsic’ inflation persistence, where the sluggishness of inflation is not a ‘structural’ feature of the economy but merely ‘inherited’ from the sluggishness of the other driving forces, inflation expectations and output. ‘Extrinsic’ inflation persistence is usually considered to be the less challenging case, as policy-makers are supposed to fight against the persistence in the driving forces, especially to reduce the stickiness of inflation expectations by a credible monetary policy, in order to reestablish the ‘divine coincidence’. The scope of this dissertation is to contribute to the vast literature and ongoing discussion on inflation persistence: Chapter 1 describes the policy consequences of inflation persistence and summarizes the empirical and theoretical literature. Chapter 2 compares two models of staggered price setting, one with a fixed two-period duration and the other with a stochastic duration of prices. I show that in an economy with a timeless optimizing central bank the model with the two-period alternating price-setting (for most parameter values) leads to more persistent inflation than the model with stochastic price duration. This result amends earlier work by Kiley (2002) who found that the model with stochastic price duration generates more persistent inflation in response to an exogenous monetary shock. Chapter 3 extends the two-period alternating price-setting model to the case of 3- and 4-period price durations. This results in a more complex Phillips curve with a negative impact of past inflation on current inflation. As simulations show, this multi-period Phillips curve generates a too low degree of autocorrelation and too early turnings points of inflation and is outperformed by a simple Hybrid Phillips curve. Chapter 4 starts from the critique of Driscoll and Holden (2003) on the relative real-wage model of Fuhrer and Moore (1995). While taking the critique seriously that Fuhrer and Moore’s model will collapse to a much simpler one without intrinsic inflation persistence if one takes their arguments literally, I extend the model by a term for inequality aversion. This model extension is not only in line with experimental evidence but results in a Hybrid Phillips curve with inflation persistence that is observably equivalent to that presented by Fuhrer and Moore (1995). In chapter 5, I present a model that especially allows to study the relationship between fairness attitudes and time preference (impatience). In the model, two individuals take decisions in two subsequent periods. In period 1, both individuals are endowed with resources and are able to donate a share of their resources to the other individual. In period 2, the two individuals might join in a common production after having bargained on the split of its output. The size of the production output depends on the relative share of resources at the end of period 1 as the human capital of the individuals, which is built by means of their resources, cannot fully be substituted one against each other. Therefore, it might be rational for a well-endowed individual in period 1 to act in a seemingly ‘fair’ manner and to donate own resources to its poorer counterpart. This decision also depends on the individuals’ impatience which is induced by the small but positive probability that production is not possible in period 2. As a general result, the individuals in the model economy are more likely to behave in a ‘fair’ manner, i.e., to donate resources to the other individual, the lower their own impatience and the higher the productivity of the other individual. As the (seemingly) ‘fair’ behavior is modelled as an endogenous outcome and as it is related to the aspect of time preference, the presented framework might help to further integrate behavioral economics and macroeconomics.