Microeconomics - Notes
                                             Simona Montagnana
                                              November 21, 2016
                                             Production and costs
         The main aim of this notes is to investigate the production and costs functions.
         The profit function π is defined to be the difference between total revenue TR and total cost TC,
         such that:
                                                  π = TR−TC
         Where the total revenue received from the sale of Q goods at price P is given by:
                                                    TR=PQ
         andwherethetotal cost relates the production costs to the level of output Q. As quantity produced
         rises, the cost also rises, so the TC is increasing.
         However in the short run, some of these costs are fixed (for instance the cost of land, equipment,
         rent, etc...) → FC=fixed costs
         ONthe other hand there are variable costs, that vary with output (cost of raw materials, compo-
         nents, energy, etc...) → VC=variable costs and VC(Q)=TVC=total variable cost
         The total cost is the sum of the contributions from fixed and variable costs:
                                               TC=FC+VC(Q)
         The average cost=AC is obtained by dividing TC by Q:
                                     AC=TC =FC+VC(Q)=FC+VC
                                            Q           Q          Q
         Let’s consider the following TR and TC curves:
         The two curves intersect at two points A and B, corresponding to output levels Q   and Q . At
                                                                                          A        B
         these points the costs and the revenue are equal and the firm breaks even. If Q < Q  or Q > Q ,
                                                                                           A           B
         then the TC curve lies above that of TR → cost exceeds revenue: for these levels of Qthe firm makes
         a loss.
                                    TC 
                                    TR 
                                                                                         Total Cost 
                                                  Total 
                                                  Revenue 
                                                               Maximum          B	
                                                               Profit 
                                                   A	
                                                 Q 	                          Q 	
                                                   A                            B                  Q	
           Weare interested also in the effect on TR of a change in the value of Q from some existing level →
           the marginal revenue MR of a good is defined by:
           MR=d(TR) →marginal revenue is the derivative of the total revenue with respect to demand
                      dQ
           The exact value of MR at point A is equal to the derivative: d(TR), and is given by the slope of the
                                                                                       dQ
           tangent at A. The point B also lies on the curve but corresponds to a a one-unit increase in Q. The
           vertical distance from A to B equals the change in TR when Q increases by one unit. The slope of
           the chord joining A and B is                            ∆(TR)
                                                                     ∆Q
           In other word, the slope of the chord is equal to the value of MR obtained from the non-calculus
           definition.
           Example: If the TR function of a good is given by:
                                                                              2
                                                               1000Q−4Q
           Write down an expression fro the marginal revenue. Then, if the current demand is 30, find the
           approximate change in tha value of TR due to a:
               1 three-unit increase in Q,
               2 two-unit decrease in Q.
                                                            MR=1000−8Q
           So when Q = 30, we get:
                                                        MR=100−8(30)=760
               1 ∆(TR)≈MR·∆Q=760·3=2280TRrisesbyabout2280
               2 ∆(TR)≈MR·∆Q=760·−2=−1520TRfallsbyabout1520
                                                                      2
                                     TC	
                                     TR	
                                                               B	
                                               Tangent                          TR curve 
                                                                Δ (TR) 
                                                     A	
                                                         Δ Q 
                                                       Q 	    Q 	
                                                        A      A+1                                   Q	
           The simple model of demand, assumed that price P and quantity Q, are linearly related according
           to an equation:
                                                                 P =aQ+b
           Where the slope is a, is negative and the intercept, b, is positive.
           Adownward-sloping demand curve such as this corresponds to the case of a monopolist. This is the
           case when the firm raises the price the demand falls. The associated revenue function is given by:
                                                            TR = PQ
                                                                   = (aQ+b)Q
                                                                           2
                                                                   = aQ +bQ
           The marginal revenue is obtained by differentiating TR with respect to Q:
                                                               MR=2aQ+b
           The marginal revenue curve slopes downhill exactly twice as fast as the demand curve. The average
           revenue AR, is defined by
                                     AR=TR→sinceTR=PQ,wehave: AR= PQ =P
                                               Q                                                 Q
           At the maximium point of the TR curve, the tangent is horizontal with zero slope and so MR is
           zero. At the other extreme from a monopoly there is the perfect competition. In this case the firm
           can sell only at the prevailing market price and because the firm is relatively small, it can sell any
           number of goods at this price. So if the fixed price is denoted by P, and the associated total revenue
           function is TR = PQ.
           In perfect competition the average and marginal revenue curves are the same, they are horizontal
           straight lines.
                                                                       3
                                                                 TC 
                  P                                              TR 
                   b 
                                                                                         Total 
                                                                                         Revenue 
                       MR 
                                        AC 
                             -b/2a          -b/a     Q                          -b/2a	     -b/a	       Q 
                                 P 
                                                      MR=AR 
                                  p  
                                   1
                                                                                    Q 
         Other important production function are:
         The marginal cost MC, by
              MC=d(TC) →marginal cost is the derivative of the total cost with respect to output
                        dQ
         The marginal product of labor MPL, by
           MP =d(TC) →marginal product of labor is the derivative of output with respect to labor
               L     dQ
                                                             4