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One of the fundamental topics taught in the microeconomics class is minimizing economic costs. It includes understanding the concept of derivatives and applying them. However, most of the first-year undergraduate students find calculus difficult to understand, which also results in poor knowledge of optimization. We use the method based on the weighted inequality between the arithmetic mean and the geometric mean to explore if such a non-calculus approach can help students solve selected problems and understand the underlying concepts. The method is transparent and intuitive and requires only algebraic manipulations. Although this approach is generally not superior to calculus, it is a good introductory method that can help students understand and adopt more complex terms and procedures required by calculus.

The aging of European societies is reshaping their population pyramids. The increase in life expectancy and the decrease in the fertility rate lead to an increasing share of the elderly population. This leads to rising age-related expenditures, especially public pension expenditures to GDP. Consequently, economies are reforming their pension systems to make them more sustainable. Next to the aging-related challenges, the new EU members, eleven post-socialist economies: Bulgaria, Croatia, the Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Romania, Slovakia, and Slovenia share a similar history of restructuring of their pension systems and establishment of a multi-pillar system. The objective of this article is to examine the similarities and differences between the pension systems of the selected post-transition economies of the European Union to establish the basis for further research, simulations, and assumptions on the impact of future pension reforms. For that purpose, we apply Ward’s clustering methodology on three variable groups in three selected years: 1996, 2006, and 2016. The idea of clustering economies in three years with a 10-year gap is relevant since it reveals how the cluster structure is changing over time. Additionally, three periods represent three different phases in the pension systems’ development. Three groups of variables were used for cluster analysis. First, pension systems’ characteristics include average effective retirement age, pension expenditure, and replacement rate. Second, demographics encompass fertility rate, life expectancy at the age of 65, net migration rate, and old-age dependency ratio. Third, the macroeconomics and labour market variables refer to the GDP growth rate, real labour productivity, labour force participation rate, and the unemployment rate. Results of cluster analysis show that the composition of the countries in the extracted clusters changes significantly, both throughout the observed period and when looking at different variable groups. Our results revealed that the pension systems of economies with later retirement proved to be more sustainable, during the three observed periods, due to lower pension expenditures that are in turn positively reflected in the favourable economic conditions and their labour market.

Model maksimizacije profita u uvjetima savršene konkurencije zauzima središnje mjesto u mikroekonomskoj analizi ponašanja poduzeća. U uvjetima proizvodnje jednog proizvoda u teorijskim i empirijskim istraživanjima tehnologiju poduzeća često opisuje Cobb-Douglasova funkcija proizvodnje. U literaturi je problem maksimizacije profita uz Cobb-Douglasovu tehnologiju gotovo uvijek riješen primjenom diferencijalnog računa. Stoga je cilj ovoga rada pokazati kako se primjenom težinske AG-nejednakosti problem maksimizacije profita uz Cobb-Douglasovu funkciju proizvodnje s dva faktora proizvodnje može riješiti na alternativni, novi način, bez derivacija. U usporedbi s diferencijalnim računom, primjena težinske AG-nejednakosti zaobilazi netrivijalnu provjeru nužnih i dovoljnih uvjeta za optimalno rješenje problema. Elegantnost novog načina za izračun maksimalnog profita proizlazi iz izravne primjene težinske AG-nejednakosti te same definicije strogog globalnog maksimuma, čime se otvara pogled na ovaj problem iz sasvim nove perspektive. Ipak, treba istaknuti da primjenu težinske AG-nejednakosti, koja spada u alate elementarne matematike, ne treba shvatiti kao superiorniji, već kao komplementaran način diferencijalnom računu u rješavanju i boljem razumijevanju ovog mikroekonomskog problema., Profit maximization in the case of the Cobb-Douglas production function is one of the fundamental problems in the microeconomic analysis of a company. In the literature, the problem of profit maximization with Cobb-Douglas technology is almost always solved by applying a differential calculus. Therefore, this paper aims to show how by applying the weighted arithmetic mean – geometric mean inequality (weighted AM-GM inequality) the problem of profit maximization with Cobb-Douglas production function with two input factors can be solved in an alternative, new way, without derivatives. Compared to the differential calculus, the application of the weighted AM-GM inequality bypasses a non-trivial check of the necessary and sufficient conditions for the optimal solution of the problem. The elegance of the new way of calculating the maximum profit originates from the direct application of the weighted AM-GM inequality and the very definition of a strict global maximum. However, it should be noted that the application of weighted AM-GM inequality, which belongs to the tools of elementary mathematics, should by no means be understood as superior, but as a complementary way to differential calculus in solving and better understanding this microeconomic problem.

In this article we analyse the short run profit maximization problem in a convex analysis framework. The goal is to apply the results of convex analysis due to unique structure of microeconomic phenomena on the known short run profit maximization problem where the results from convex analysis are deductively applied. In the primal optimization model the technology in the short run is represented by the short run production function and the normalized profit function, which expresses profit in the output units, is derived. In this approach the choice variable is the labour quantity. Alternatively, technology is represented by the real variable cost function, where costs are expressed in the labour units, and the normalized profit function is derived, this time expressing profit in the labour units. The choice variable in this approach is the quantity of production. The emphasis in these two perspectives of the primal approach is given to the first order necessary conditions of both models which are the consequence of enveloping the closed convex set describing technology with its tangents. The dual model includes starting from the normalized profit function and recovering the production function, and alternatively the real variable cost function. In the first perspective of the dual approach the choice variable is the real wage, and in the second it is the real product price expressed in the labour units. It is shown that the change of variables into parameters and parameters into variables leads to both optimization models which give the same system of labour demand and product supply functions and their inverses. By deductively applying the results of convex analysis the comparative statics results are derived describing the firm’s behaviour in the short run.

This paper examines a two-way relationship between convex analysis and microeconomic theory. Motivation for this paper are the observed similarities in the structure of the theory of consumer behavior and production theory. The fact that the behavior of variables is not determined by their nature but, rather, by their relationships is best illustrated and explained by using convex sets and convex analysis, which occupy central place in microeconomic theory. This paper is the result of efforts to make complex results of convex analysis and its application in microeconomic theory more transparent. Starting with the well-known economic phenomenon of profit maximization the authors derive in a novel way general results within the framework of convex analysis. From those results follow, directly and indirectly, the conclusions of the theory of consumer and producer behavior. The authors show that applying the Fundamental Theorems of Calculus opens up a new perspective in which the marginal cost curve can be interpreted as the marginal profit curve. This enables the derivation of Hotelling's lemma in a new way. Using the new interpretation of Hotelling's lemma, the authors reconstruct the cost function and confirm the Conjugate Duality Theorem of Legendre-Fenchel transformations. Relaxing the assumption of differentiability by describing the graph of the cost function as the envelope of its tangents, the authors rederive the properties of Legendre-Fenchel transformations and show that they hold in general. The path from the well- known economic facts to completely general conclusions of convex analysis is continued by applying the Conjugate Duality Theorem of Legendre-Fenchel transformations to the profit function. The essence of the dual characterization of technology by the profit function is illustrated by the graphical representation of linear homogeneity of the profit function. It results in the possibility to reconstruct the production function while using only the First Order Conditions to rederive Hotelling's lemma. It is this inductive-deductive approach used to examine the properties of Legendre-Fenchel trasformations and their application in the theory of consumer and producer behavior that establishes a two-way relationship between convex analysis and microeconomic theory.

Recent macroeconomic and demographic trends have resulted in new challenges for pension systems. One of these challenges is to create a sustainable pension system while simultaneously providing adequate pension benefits for current and future pensioners. This research explores how similar are pension systems of eleven European Union countries by using hierarchical cluster analysis for year 2016. Variables representing pension systems, as well as demographic, macroeconomic and labour market data were used to cluster these economies. Three clustering solutions were generated using hierarchical clustering approach, one for each variable group. Given the number of observed countries, only two cluster solutions were considered. According to the characteristics of the pension systems, countries that have greater problems of unsustainability are recognized. A similar group of countries also forms the cluster characterized by unfavourable demographic trends that make it more difficult to maintain sustainability. Romania stands out from other economies, based on macroeconomic indicators, as it recorded faster economic growth, greater labour productivity growth and lower unemployment rate in 2016. The findings of this study provide a guideline for future pension reforms, since they indicate which countries’ experience could be valuable in defining certain policy measures. This work is licensed under aCreative Commons Attribution-NonCommercial 4.0 International License.

Background: Duality in the microeconomic theory enables us to represent consumers’ preferences and production technology with various dual functions satisfying certain regularity conditions. Objectives: Since the basis for the application of duality in the microeconomic theory is the price taking behaviour, this paper takes the challenge of applying principles of duality to a monopolistic case where a single producer has an influence on the price which it charges for its product. Methods/Approach: The standard approach of deriving the profit function for the monopolist from the production function and the defined pseudoproduction function is accompanied by an alternative approach in which the starting point is the pseudocost function. Starting from the derived profit function, the pseudoproduction function and the pseudocost functions are recovered and a version of Hotelling’s lemma is given. Results: The structure of the profit maximization problem in a monopolistic case was made similar to the structure of the profit maximization problem in the perfectly competitive case and it is shown that all starting functions can be recovered back from derived functions. A version of Hotelling’s lemma is illustrated, which brings us indirectly from the profit function to the supply function. Conclusions: By introducing the pseudoproduction function in the profit maximization model of a monopolist, the structure of the problem becomes similar to the perfectly competitive case and duality results can be applied. The profit function is derived from the pseudoproduction and the pseudocost function, and all starting functions are recovered back from the derived profit function.

This paper analyzes the mark-up of the price of a product over marginal costs for a monopolist using Appelbaum’s theoretical model. The profit maximization model of an industry that uses the monopolist’s product as its input is formulated. Our goal is to express the monopolist’s mark-up as a function of the elasticity of substitution for the respective industry and to analyze how changes in the elasticity of substitution affect the mark-up ratio. Consequently, the CES production function along with its substitution parameter is chosen. An analytical description of changes in the elasticity of substitution and its influence on the monopolist's mark-up is given. All scenarios are supplemented by geometrical illustrations, economic interpretations and numerical examples.

This paper follows our theoretical research on duality in microeconomic theory and applies the duality principles, which rest on the price taking behaviour of economic subjects, on the monopolistic behaviour. The standard approach of deriving the profit function for the monopolist from the production function and defined pseudo production function is accompanied by the alternative approach in which the starting point is the pseudo cost function. Finally, starting from the derived profit function the pseudo production function is recovered and a version of Hotelling’s lemma is given. All results are illustrated with a numerical example.

In the Stackelberg leadership model the follower firm acts like in the Cournot model and for the given level of production of the leader firm determines its own production level which maximizes economic profit. The leader firm chooses quantity knowing that the follower firm will react to it according to its reaction function. Many economic phenomena have Stackelberg model structure and in this paper the relationship between the client and the producer of embedded sofware is modeled using this approach. The product quality affects the likelihood of product safety and the consumers' utility but depends on the invested effort. The goal is to find an equilibrium in the Stackelberg model where the client is the leader, and the producer is the follower. In the first step, the producer chooses the number of units of labor for the given price of a product in order to maximize the expected profit. In this way the producer indirectly affects the software quality. In the paper the effect of the product price change on producer's reaction is analysed. The comparative statics results confirm our intuition. The higher the product price, the greater the level of labor units employed by the producer and the better the product. Increase in labor costs has opposite effect on the producer's decision. In the second step the consumer determines the product price in order to maximize his or her expected utility. This decision is affected by the producer's reaction or the price effect on the product quality. The consumer's utility and the software quality are usually directly related in the existing literature. In this paper, this relationship is complemented by the probability of product safety. Theoretical optimization models which are solved by the client and the producer are illustrated numerically and the total welfare in the Stackelberg and the cooperative equilibrium is compared. The effect of the change in relationship between the employed labor units and the software quality on the given results is analysed. According to our numerical example, the more favorable this relationship is, the higher is the client's and the producer's welfare in the Stackelberg equilibrium. Even though fewer workers are employed, produced software is better. The total welfare loss is smaller due to more dominant effect of more favorable relationship between employees and the software quality in the Stackelberg equilibrium than in the cooperative equilibrium. Given the sensitivity of the results, future research should be devoted to examining the functional relationship between given variables in specific phenomena. The cost of software maintenance could also be taken into account.