Answers

for Production Possibilities Questions

1. If Lx = 40, Qx = 3 (40) = 120
2. Labor productivity coefficient = 3; each worker produces 3 units of output in a unit of time.
3. Activity requirement = 1/3; it takes 1/3 of a unit of time to produce one unit of output.
4. If w = $9, the minimum possible price is (1/3) w = (1/3) $9 = $3 or else the selling price will not cover the labor costs.

2. Qx = 5 Lx

1. If Ly = 40, Qy = 5 (40) = 200
2. Labor productiity= 5
3. Activity requirement = 1/5 or 0.2; it takes 1/5 of a unit of time to produce one unit of good Y.
4. If w = $9, the minimum price which would cover labor costs is
   (1/5) w = $9/5 = $1.80

5. Labor is more productive in good Y -- one unit of labor produces 5 units of output in good Y and only 3 units of output in good X. (Be careful about concluding that it is therefore a better idea to devote labor to good Y, since nothing has been said about the value of the two goods. What if good X is houses and good Y is jellybeans ?)
6. The activity coefficient is higher in good X -- 1/3 = .33 > .2 = 1/5 This is simply saying that it takes more work to make a unit of good X.
7. We found that with a wage of $9, Px = $3 and Py = $ 1.80. The relative price of the two goods is simply the ratio of their prices
   Px/Py = $3/$1.80 = 1/0.6 = 10/6 = 5/3

   Note that although it may be convenient to think in terms of money prices, there is no real need to do so, and the money wage did not have to be given. Different money wages would not change the relative price, since

   Px/Py = (1/3) w / (1/5) w = (1/3) / (1/5) = 5/3

   Whatever the money wage w is, it will cancel out when you compute relative prices. This means that the dollar sign "cancels out" as well; relative prices are not being expressed in terms of money, but as a pure number. This is very convenient in thinking about international issues, since some money prices would be in dollars, some in francs, marks, rubles, yen and so on.

   3. The opportunity cost of good X is the number of units of good Y we given up to produce one more unit of good X. To calculate it, we must

   1. calculate the extra labor we would need to produce one more unit of good X. This is given by the activity requirement, namely 1/3 more hour of labor.
   2. take that amount of labor away from the production of good Y.
   3. calculate the amount of good Y given up. In our case, since Qy = 5 Ly, and we must take away 1/3 of a unit of labor,
      Qy = 5 Ly

      or in "differenced" form, with dQy = difference or change in Qy and dLy = difference or change in Ly,

      dQy = 5 dLy

      or

      dQy = 5 (-1/3) = -5/3

      To produce one more unit of good X we must give up 5/3 unit of good Y. The opportunity cost of good X is therefore 5/3. It is not accidental that opportunity cost is the same as the relative price of the two goods.

      4. The table can be filled in by using the production functions or by transforming the equation into activity coefficent form. If given, for example, the fact that 90 units of good X are produced, you can tranform the production function
      Qx = 3 Lx to Lx = (1/3) Qx = (1/3) 90 = 30
      to find that 30 workers would be required in the X industry.

      Note that to fill in the entire row given only one entry, one would want to make an efficient allocation of labor. But given two entries, the production plan may be either impossible, as in the case of entry H, or inefficient, as in the case of entries F and G.

      Labor allocations and Production Possibilities

      Case	Lx	Ly	Qx	Qy
      A	0	-120-	-0-	-600-
      B	-60-	60	-180-	-300-
      C	-30-	-90-	90	-450-
      D	-100-	-20-	-300-	100
      E	40	-80-	-120-	-400-
      F	-30-	-20-	90	100
      G	20	20	-60-	-100-
      H	-120-	-16-	360	80

      Note that entry F requires the employment of only 50 of the labor force of 120; entry G employs only 40 of the 120 workers. Plotting the output points here would take us well within the PPF.

      Entry H sets out a production plan of 360 units of X and 80 units of Y. However, the 360 units of X would by themselves require the employment of 120 workers, we would not have any workers left over to produce good Y.

      5. The graph of the PPF would run in a straight line from point (0, 600) on the Y axis to point (360, 0) on the X-axis. It would have a slope of

      -600/360 = -100/60 = -5/3

      You should note that the slope is the same in absolute value as the opportunity cost of good X, for the very simple reason that the slope of the PPF shows the tradeoff between the two goods.

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      Back to UPJ Economics Department Last modified: Wed Sep 11 08:34:48 1996