A graded subring of an inverse limit of polynomial rings

We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R. Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in Gröbner Bases in R'. We show in Reverse lexicographic initial ideals of generic ideals are finitely generated that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order. If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in Generalized Hilbert Numerators that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I. In Gröbner bases for non-homogeneous ideals in R' we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated. The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In Topological properties of R' we show that with respect to this topology, locally finitely generated ideals in R'are closed.

Essays on Wage and Price Formation in Sweden

Study IReal Wage Determination in the Swedish Engineering Industry This study uses the monopoly union model to examine the determination of real wages and in particular the effects of active labour market programmes (ALMPs) on real wages in the engineering industry. Quarterly data for the period 1970:1 to 1996:4 are used in a cointegration framework, utilising the Johansen's maximum likelihood procedure. On a basis of the Johansen (trace) test results, vector error correction (VEC) models are created in order to model the determination of real wages in the engineering industry. The estimation results support the presence of a long-run wage-raising effect to rises in the labour productivity, in the tax wedge, in the alternative real consumer wage and in real UI benefits. The estimation results also support the presence of a long-run wage-raising effect due to positive changes in the participation rates regarding ALMPs, relief jobs and labour market training. This could be interpreted as meaning that the possibility of being a participant in an ALMP increases the utility for workers of not being employed in the industry, which in turn could increase real wages in the industry in the long run. Finally, the estimation results show evidence of a long-run wage-reducing effect due to positive changes in the unemployment rate. Study IIIntersectoral Wage Linkages in Sweden The purpose of this study is to investigate whether the wage-setting in certain sectors of the Swedish economy affects the wage-setting in other sectors. The theoretical background is the Scandinavian model of inflation, which states that the wage-setting in the sectors exposed to international competition affects the wage-setting in the sheltered sectors of the economy. The Johansen maximum likelihood cointegration approach is applied to quarterly data on Swedish sector wages for the period 1980:1–2002:2. Different vector error correction (VEC) models are created, based on assumptions as to which sectors are exposed to international competition and which are not. The adaptability of wages between sectors is then tested by imposing restrictions on the estimated VEC models. Finally, Granger causality tests are performed in the different restricted/unrestricted VEC models to test for sector wage leadership. The empirical results indicate considerable adaptability in wages as between manufacturing, construction, the wholesale and retail trade, the central government sector and the municipalities and county councils sector. This is consistent with the assumptions of the Scandinavian model. Further, the empirical results indicate a low level of adaptability in wages as between the financial sector and manufacturing, and between the financial sector and the two public sectors. The Granger causality tests provide strong evidence for the presence of intersectoral wage causality, but no evidence of a wage-leading role in line with the assumptions of the Scandinavian model for any of the sectors. Study IIIWage and Price Determination in the Private Sector in Sweden The purpose of this study is to analyse wage and price determination in the private sector in Sweden during the period 1980–2003. The theoretical background is a variant of the “Imperfect competition model of inflation”, which assumes imperfect competition in the labour and product markets. According to the model wages and prices are determined as a result of a “battle of mark-ups” between trade unions and firms. The Johansen maximum likelihood cointegration approach is applied to quarterly Swedish data on consumer prices, import prices, private-sector nominal wages, private-sector labour productivity and the total unemployment rate for the period 1980:1–2003:3. The chosen cointegration rank of the estimated vector error correction (VEC) model is two. Thus, two cointegration relations are assumed: one for private-sector nominal wage determination and one for consumer price determination. The estimation results indicate that an increase of consumer prices by one per cent lifts private-sector nominal wages by 0.8 per cent. Furthermore, an increase of private-sector nominal wages by one per cent increases consumer prices by one per cent. An increase of one percentage point in the total unemployment rate reduces private-sector nominal wages by about 4.5 per cent. The long-run effects of private-sector labour productivity and import prices on consumer prices are about –1.2 and 0.3 per cent, respectively. The Rehnberg agreement during 1991–92 and the monetary policy shift in 1993 affected the determination of private-sector nominal wages, private-sector labour productivity, import prices and the total unemployment rate. The “offensive” devaluation of the Swedish krona by 16 per cent in 1982:4, and the start of a floating Swedish krona and the substantial depreciation of the krona at this time affected the determination of import prices.