University of Hong Kong
                                      The shape of Production Possibility Frontier
                           A question students frequently asked regarding the PPF is what
                        shape it should take on. The answer is it depends! Let me look at
                        the three main possibilities. Consider the following scenario. There are
                        100 units of labor, which is the only factor of production. There are two
                        products, food and shelter.
                           a) Linear PPF. Suppose one unit of food requires one unit of labor
                        to produce, and one unit of shelter requires two units of labor. (There
                        are constant returns to scale.) Then x units of good and y =(100−x)/2
                        units of shelter are produced. The PPF is given by y =(100− x)/2,
                        which is clearly a straight line.
                           b) Concave (towards origin) PPF. Suppose x, the quantity of food
                        produced, is equal to q.5 and y, the quantity of shelter produced, is equal
                                               x
                        to 0.5q.5 where q and q istheamountoflaborusedtoproducefood
                               y          x       y
                        andshelter, respectively. Note that in this case, doubling the input leads
                        to an output that is less than doubling. For instance, while 4 units of
                                                                                             √
                        labor can produce 2 units of food, 8 units of labor can produce 2 2 or
                        2.818 units of food, less than 4 units of food. What this means is that
                        earlier labor have a greater productivity than those hired later. (We
                        call this diseconomies of scale, or decreasing returns to scale.) Under
                        full utilization of resource, q  and q mustaddupto100. Fromthe
                                                       x       y
                                                                  2             2
                        production technologies we know q = x and q =4y . Therefore 100 =
                                                       √ x               y
                                    2     2           1          2
                        q +q =x +4y . Or, y =            100−x . When you plot this curve, you
                         x    y                       2
                        will see the following:
                                        y5
                                          4
                                          3
                                          2
                                          1
                                          0
                                            0   1  2   3   4   5   6  7   8   9  10
                                                                                 x
                                         PPFbetween shelter (y) and food (x)
                           This is the typical PPF we see from the textbook.
                                                             1
                     Remark 1 The absolute value of the slope of the PPF is −dy/dx =
                                    2 −0.5            √        2
                     −(1/4)(100 − x )    (−2x)=x/(2 100−x ), which is increasing in
                     x. (To see this it suffices to note that the numerator is increasing in x
                     and the denominator is decreasing in x,hencethewholetermmustbe
                     increasing in x).
                        c) Convex (towards origin) PPF. Suppose x, the quantity of food
                     produced, is equal to q2 and y, the quantity of shelter produced, is equal
                                          x
                     to 0.5q2. Note that in this case, doubling the input leads to an output
                           y
                     that is more than doubling. For instance, while 1 units of labor can
                     produce 1 unit of food, 2 units of labor can produce 4 units of food.
                     What this means is that earlier labor have a smaller productivity than
                     those hired later. (We call this economies of scale, or increasing returns
                     to scale.) Under full utilization of resource, q and q must add up to
                                                                x      y           √
                                                                       √
                     100. From the production technologies we know q =   xandq = 2y.
                                               √     √             x        √ y
                                                                    1           2
                     Therefore 100 = q + q =     x+ 2y.Or,y = (100− x) . When
                                      x    y                        2
                     you plot this curve, you will see the following:
                                    y5000
                                     4000
                                     3000
                                     2000
                                     1000
                                       0
                                         0    2000  4000  6000  8000 10000
                                                                      x
                     Remark 2 The absolute value of the slope of the PPF is −dy/dx =
                             √          √              √      √
                     −(100− x)(−1/(2 x)) = (100− x)/(2 x), which is decreasing in
                     x.(Toseethisitsuffices to note that the numerator is decreasing in x
                     and the denominator is increasing in x, hence the whole term must be
                     decreasing in x).
                        Besides the three possibilities introduced above, in principle we can
                     have more complicated PPFs. Consider the following, which is a mar-
                     riage between (b) and (c).
                        d)Mixed Shaped PPF.Amoreinterestingpossibilityisthatoneprod-
                     uct is produced under diseconomies of scale while the other economies
                                                      2
                     of scale. Let’s combine the food technology in case (b) with the shether
                     technologyincase(c). Thatis, supposex,thequantityoffoodproduced,
                     is equal to q.5 and y, the quantity of shelter produced, is equal to 0.5q2.
                                 x                √                                      y
                                               2                                       2 2
                     Then we have q +q = x + 2y = 100. Hence, y =(1/2)(100−x ) .
                                     x   y
                     Plotting it, we have:
                                    5000
                                    4000
                                    3000
                                    2000
                                    1000
                                      0       246810
                                                       x
                     Remark 3 TheabsolutevalueoftheslopeofthePPFequals−dy/dx =
                               2                    2
                     −(100−x )(−2x)=2x(100−x ). Thatis,itisincreasingandthende-
                     creasing. To calculate the point of inflexion, we can simply differentiate
                     the marginal cost, set the resultant term constant, and have it solved.
                              d                      2                       2
                             dx(−dy/dx)=2(100−x )+2x(−2x)=200−6x =0
                              ∗   p             √
                     Hence, x =     200/6=10/ 3=5.7735. Alternatively, you can simply
                                2     2
                     plot the −d y/dx against x :
                                     600
                                     400
                                     200
                                      0       246810
                                                       x
                                            The marginal cost of x
                                                       3