Complementary good

<h2 id="examples">Examples</h2>

<p>An example of this would be the demand for <a href="/facts/Car/cnhHaaHQ">cars</a> and <a href="/facts/Petrol/lcnxq5BS">petrol</a>. The <a href="/facts/Supply_and_demand/KprL1fvc">supply and demand</a> for cars is represented by the figure, with the initial demand 
  
    
      
        
          D
          
            1
          
        
      
    
    {\displaystyle D_{1}}
  
. Suppose that the initial price of cars is represented by 
  
    
      
        
          P
          
            1
          
        
      
    
    {\displaystyle P_{1}}
  
 with a quantity demanded of 
  
    
      
        
          Q
          
            1
          
        
      
    
    {\displaystyle Q_{1}}
  
. If the price of petrol were to decrease by some amount, this would result in a higher quantity of cars demanded. This higher quantity demanded would cause the demand curve to <a href="/facts/Demand_curve/lALNI8BX">shift rightward</a> to a new position 
  
    
      
        
          D
          
            2
          
        
      
    
    {\displaystyle D_{2}}
  
. Assuming a constant supply curve 
  
    
      
        S
      
    
    {\displaystyle S}
  
 of cars, the new increased quantity demanded will be at 
  
    
      
        
          Q
          
            2
          
        
      
    
    {\displaystyle Q_{2}}
  
 with a new increased price 
  
    
      
        
          P
          
            2
          
        
      
    
    {\displaystyle P_{2}}
  
. Other examples include automobiles and fuel, mobile phones and cellular service, printer and cartridge, among others.
</p>
<h2 id="perfect-complement">Perfect complement</h2>

<p>A <i>perfect complement</i> is a good that <i>must</i> be consumed with another good. The <a href="/facts/Indifference_curve/BhgfCINQ">indifference curve</a> of a perfect complement exhibits a right angle, as illustrated by the figure.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> Such preferences can be represented by a <a href="/facts/Leontief_utility/sPP9ydq0">Leontief utility</a> function.
</p><p>Few goods behave as perfect complements.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> One example is a left shoe and a right; shoes are naturally sold in pairs, and the ratio between sales of left and right shoes will never shift noticeably from 1:1.
</p><p>The degree of complementarity, however, does not have to be mutual; it can be measured by the <a href="/facts/Cross_elasticity_of_demand/zHKo0jUD">cross price elasticity of demand</a>. In the case of video games, a specific video game (the complement good) has to be consumed with a video game console (the base good). It does not work the other way: a video game console does not have to be consumed with that game.
</p>
<h3>Example</h3>
<p>In <a href="/facts/Marketing/LttVQ66t">marketing</a>, complementary goods give additional <a href="/facts/Market_power/oyYb7b3L">market power</a> to the producer. It allows <a href="/facts/Vendor_lock-in/BLndHDVl">vendor lock-in</a> by increasing <a href="/facts/Switching_cost/SqKdLlI6">switching costs</a>. A few types of <a href="/facts/Pricing_strategies/GafT35PV">pricing strategy</a> exist for a complementary good and its base good:
</p>
<ul><li>Pricing the base good at a relatively low price - this approach allows easy entry by consumers (e.g. low-price consumer printer vs. high-price cartridge)</li>
<li>Pricing the base good at a relatively high price to the complementary good - this approach creates a barrier to entry and exit (e.g., a costly car vs inexpensive gas)</li></ul>
<h2 id="gross-complements">Gross complements</h2>
<p>Sometimes the complement-relationship between two goods is not intuitive and must be verified by inspecting the cross-elasticity of demand using market data.
</p><p>Mosak's definition states "a good 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is a gross complement of 
  
    
      
        y
      
    
    {\displaystyle y}
  
 if 
  
    
      
        
          
            
              ∂
              
                f
                
                  x
                
              
              (
              p
              ,
              ω
              )
            
            
              ∂
              
                p
                
                  y
                
              
            
          
        
      
    
    {\displaystyle {\frac {\partial f_{x}(p,\omega )}{\partial p_{y}}}}
  
 is negative, where 
  
    
      
        
          f
          
            i
          
        
        (
        p
        ,
        ω
        )
      
    
    {\displaystyle f_{i}(p,\omega )}
  
 for 
  
    
      
        i
        =
        1
        ,
        2
        ,
        …
        ,
        n
      
    
    {\displaystyle i=1,2,\ldots ,n}
  
 denotes the ordinary individual demand for a certain good." In fact, in Mosak's case, 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is not a gross complement of 
  
    
      
        y
      
    
    {\displaystyle y}
  
 but 
  
    
      
        y
      
    
    {\displaystyle y}
  
 is a gross complement of 
  
    
      
        x
      
    
    {\displaystyle x}
  
. The elasticity does not need to be symmetrical. Thus, 
  
    
      
        y
      
    
    {\displaystyle y}
  
 is a gross complement of 
  
    
      
        x
      
    
    {\displaystyle x}
  
 while 
  
    
      
        x
      
    
    {\displaystyle x}
  
 can simultaneously be a gross substitutes for 
  
    
      
        y
      
    
    {\displaystyle y}
  
.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p>
<h3>Proof</h3>
<p>The standard <a href="/facts/John_Hicks/8AgpsS4F">Hicks</a> decomposition of the effect on the ordinary demand for a good 
  
    
      
        x
      
    
    {\displaystyle x}
  
 of a simple price change in a good 
  
    
      
        y
      
    
    {\displaystyle y}
  
, utility level 
  
    
      
        
          τ
          
            ∗
          
        
      
    
    {\displaystyle \tau ^{*}}
  
 and chosen bundle 
  
    
      
        
          z
          
            ∗
          
        
        =
        (
        
          x
          
            ∗
          
        
        ,
        
          y
          
            ∗
          
        
        ,
        …
        )
      
    
    {\displaystyle z^{*}=(x^{*},y^{*},\dots )}
  
 is
</p><p>
  
    
      
        
          
            
              ∂
              
                f
                
                  x
                
              
              (
              p
              ,
              ω
              )
            
            
              ∂
              
                p
                
                  y
                
              
            
          
        
        =
        
          
            
              ∂
              
                h
                
                  x
                
              
              (
              p
              ,
              
                τ
                
                  ∗
                
              
              )
            
            
              ∂
              
                p
                
                  y
                
              
            
          
        
        −
        
          y
          
            ∗
          
        
        
          
            
              ∂
              
                f
                
                  x
                
              
              (
              p
              ,
              ω
              )
            
            
              ∂
              ω
            
          
        
      
    
    {\displaystyle {\frac {\partial f_{x}(p,\omega )}{\partial p_{y}}}={\frac {\partial h_{x}(p,\tau ^{*})}{\partial p_{y}}}-y^{*}{\frac {\partial f_{x}(p,\omega )}{\partial \omega }}}

</p><p>If 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is a gross substitute for 
  
    
      
        y
      
    
    {\displaystyle y}
  
, the left-hand side of the equation and the first term of right-hand side are positive. By the symmetry of Mosak's perspective, evaluating the equation with respect to 
  
    
      
        
          x
          
            ∗
          
        
      
    
    {\displaystyle x^{*}}
  
, the first term of right-hand side stays the same while some extreme cases exist where 
  
    
      
        
          x
          
            ∗
          
        
      
    
    {\displaystyle x^{*}}
  
 is large enough to make the whole right-hand-side negative. In this case, 
  
    
      
        y
      
    
    {\displaystyle y}
  
 is a gross complement of 
  
    
      
        x
      
    
    {\displaystyle x}
  
. Overall, 
  
    
      
        x
      
    
    {\displaystyle x}
  
 and 
  
    
      
        y
      
    
    {\displaystyle y}
  
 are not symmetrical.
</p>
<h2 id="effect-of-price-change-of-complementary-goods">Effect of price change of complementary goods</h2>
<ul><li><a href="/facts/Substitute_good/oK7UJNjw">Substitute good</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Carbaugh, Robert (2006). Contemporary Economics: An Applications Approach. Cengage Learning. p. 35. ISBN 978-0-324-31461-8. <a href="978-0-324-31461-8" target="_blank">978-0-324-31461-8</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>O'Sullivan, Arthur; Sheffrin, Steven M. (2003). Economics: Principles in Action. Upper Saddle River, New Jersey: Pearson Prentice Hall. p. 88. ISBN 0-13-063085-3. <a href="0-13-063085-3" target="_blank">0-13-063085-3</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Customer in Marketing by David Mercer". Future Observatory. Archived from the original on 2013-04-04. <a href="https://web.archive.org/web/20130404042855/http://futureobservatory.dyndns.org/9432.htm" target="_blank">https://web.archive.org/web/20130404042855/http://futureobservatory.dyndns.org/9432.htm</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Newman, Peter (2016-11-30) [1987]. "Substitutes and Complements". The New Palgrave: A Dictionary of Economics: 1–7. doi:10.1057/978-1-349-95121-5_1821-1. ISBN 978-1-349-95121-5. Retrieved 2022-05-26. <a href="978-1-349-95121-5" target="_blank">978-1-349-95121-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Huh, Young Eun; Vosgerau, Joachim; Morewedge, Carey K. (2016-03-14). "Selective Sensitization: Consuming a Food Activates a Goal to Consume its Complements". Journal of Marketing Research. 53 (6): 1034–1049. doi:10.1509/jmr.12.0240. ISSN 0022-2437. S2CID 4800997. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Mankiw, Gregory (2008). Principle of Economics. Cengage Learning. pp. 463–464. ISBN 978-0-324-58997-9. <a href="978-0-324-58997-9" target="_blank">978-0-324-58997-9</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Mankiw, Gregory (2008). Principle of Economics. Cengage Learning. pp. 463–464. ISBN 978-0-324-58997-9. <a href="978-0-324-58997-9" target="_blank">978-0-324-58997-9</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Mosak, Jacob L. (1944). "General equilibrium theory in international trade" (PDF). Cowles Commission for Research in Economics, Monograph No. 7. Principia Press: 33. <a href="https://dspace.gipe.ac.in/xmlui/bitstream/handle/10973/38888/GIPE-014030.pdf?sequence=3" target="_blank">https://dspace.gipe.ac.in/xmlui/bitstream/handle/10973/38888/GIPE-014030.pdf?sequence=3</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Complementary good open-in-new

Complementary good