Notes on 
          
                   Chapter 1: Monopoly I 
                      Microeconomic Theory IV 
                        3º - LE-: 2008-2009 
                               
                               
                         Iñaki Aguirre 
              Departamento de Fundamentos del Análisis Económico I 
                      Universidad del País Vasco 
          
          
                                    
          
                   Microeconomic Theory IV                                                                                                               Monopoly I 
                   Introduction 
                   1.1. Profit maximization by a monopolistic firm.  
                   1.2. Linear demand and constant elasticity demand. 
                   1.3. Comparative statics.  
                   1.4. Welfare and output. 
                    
                   Introduction 
                   We say that a firm is a monopoly if it is the only seller of a good (or goods) in a market.   
                   Problem: it is not easy to define good and market.  
                   A firm may become a monopoly by various reasons:  
                   - Control over raw materials. 
                   - Acquisition of the exclusive selling rights (by a patent, by a public auction etc.). 
                   - Better access to the capital market.  
                   - Increasing returns of scale etc. 
                    
                   In contrast with a perfectly competitive firm which faces a perfectly elastic demand (taking 
                   price as given), a monopolist faces the market demand. Therefore, a firm with monopolistic 
                   power in a market it is aware of the amount of output that it is be able to sell it is a 
                   continuous function of the price charged. Put differently, the monopolistic firm takes into 
                   account that a reduction in output will increase the price that can be charged. In 
                   consequence, a monopolist has the power to set the market price. While we can consider a 
                   competitive firm as a “price taker”, a monopolist is price decision-maker or price setter.    
                    
                    
                    
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                   Microeconomic Theory IV                                                                                                               Monopoly I 
                   1.1. Profit maximization 
                   (i) The problem of profit maximization in prices and in quantities. First order conditions. 
                   Second order conditions. A graphical interpretation of the profit maximization problem.  
                   (ii) Interpretation of marginal revenue. 
                   (iii) Marginal revenue equals marginal cost condition.  
                   (iv) Output and demand elasticity. 
                    (v) Lerner Index of monopolistic power.  
                   (vi) Graphical analysis. 
                   (vii) Second order conditions. 
                    
                    (i) The problem of profit maximization in prices and in quantities 
                   There are two types of constraint that restrict the behaviour of a monopolist: 
                   a) Technological constraints summarized in the cost function C(x). 
                   b) Demand constraints: x(p). 
                    
                   We can write the profit function of the monopolist in two alternative ways: 
                   -                                           by using the demand function.  
                      Π=()ppx(p)−C(x(p))
                   -                                     by using the inverse demand function.   
                      Π=()x        px()x−C()x
                   The demand, x (p), and the inverse demand, p(x), represent the same relationship between 
                   price and demanded quantity from different points of view. The demand function is a 
                   complete description of demanded quantity at each price whereas the inverse demand gives us 
                   the maximum price at which a given output x may be sold in the market.  
                                   maxΠ(p)                           maxΠ(x)
                                      p                                x≥0
                                            mm
                                           px
                                        ⇓≡                                ⇓
                                                                                        
                                    mm mm
                                       ==()                                    ()
                                  x        xp                       p px
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                   Microeconomic Theory IV                                                                                                               Monopoly I 
                   The problem of profit maximization as a function of price 
                    maxΠ≡(pp)           max x(p)−C(x(p))     
                                                                                 
                       pp
                       ''''
                    Π=()px()p+px()p−C(x(p))x()p=0      
                       ''             '             ''           ''              '       2       '             ''
                                                                             ⎡⎤                                                   
                    Π=()p 2x(p)+pxp()−C(x()p)x(p) −C(x(p))xp()<0     
                                                                             ⎣⎦
                    
                   The problem of profit maximization as a function of the output 
                                                                           
                    maxΠ≡(xp)           max (x)x−C(x)     
                      xx≥≥00
                       '''
                    Π=(0)        pC(0)−(0)>0⇒ (0p)>C(0) 
                       ''''m
                    Π=()xp()x+xp()x−C()x=0⇔ (Πx)=0  First order condition. 
                       ''             '             ''          ''
                    Π=()xp2 (x)+xp(x)−C(x)<0 Strictly concave profit function (regular case). 
                    
                                                              Π 
                     
                    
                                                                                             '     m
                                                                                        Π()x             =0
                     
                    
                                                                                                                              Π()x
                                                        '
                                                   Π>(0)             0
                                                                                                xm                                         x
                    
                     
                    
                    
                    
                    
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