SPECIFIC Factors and Income Distribution – See ALSO HANDOUT OF GRAPHS

1.  To analyze the economy's production possibility frontier, consider how the output mix changes as labor is shifted between the two sectors.

  1. Graph the production functions for M and F: The production functions for goods M and F are standard plots with quantities on the vertical axis, labor on the horizontal axis, and QM= f(KM,LM) with slope equal to the MPLM, and on another graph, QF= f(KF,LF) with slope equal to the MPLF.
  1. Graph the production possibility frontier. Why is it curved? To graph the production possibilities frontier, combine the production function diagrams with the economy's allocation of labor in a four quadrant diagram. The economy's PPF is in the upper right hand corner, as is illustrated in the four quadrant diagram one the handout. The PPF is curved due to declining marginal product of labor in each good. The concave (to the origin) shape illustrates increasing opportunity cost as the country devotes more labor to the production of one good.
  1. Express the slope of the production possibility frontier as a ratio of the Marginal Product of Labor in the two goods. The slope of the PPF equals –MPLF/MPLM.
a.  Let PF/PM = 2 (M/F). Determine graphically the wage rate and the allocation of labor between the two sectors. To solve this problem, one can graph the demand curve for labor in M as
W= VMPLM = PM MPLM, and the demand curve for labor in F as
W = VMPLF = PF MPLF. Since the total supply of labor is given by the horizontal axis, the labor allocation between the sectors is approximately LM=29 and LF=71. The wage rate is about $0.98 or $1/hour.
b.  Using the graph drawn for problem 1, determine the output of each sector. Then confirm graphically that the slope of the PPF at the point equals the relative price. Use the graph drawn for problem 1b (or the table given in problem 1) to show that the M sector’s output is QM=48 and F sector’s is QF=88 (but the last digit is uncertain). (This involves combining the production function diagrams with the economy's allocation of labor in a four quadrant diagram; please see the handout.) Using the one we drew, it appears that the slope of the PPF = - MPLF / MPLM = -PM/PF = -0.5
First, note that PF/PM = 2 was given; it follows that PM/PF = 0.5. Since labor is mobile, in equilibrium, the wages rates in both industries will be the same. The wage is determined by the demand and supply of labor. Each industry is willing to pay a wage up to the Value of the Marginal Product of Labor (VMPL). Equilibrium occurs at the intersection of the two labor demand curves, where WF = WM and VMPLF = VMPLM. Next, PF MPLF = PM MPLM, by substitution. Now multiply both sides by -1/(PF MPLM). The result, -MPLF / MPLM = -PM/PF, shows that at the production point, the slope of the PPF equals the relative price of the x-axis good, M.
c.  Let PF/PM = 1 (M/F). Repeat parts a and b.
Use a graph of labor demands, as in part a, to show that the intersection of the demand curves for labor occurs at a wage rate close to $0.72. The decline in the price of good F caused labor to be reallocated; labor is drawn out of production of good F and enters production of good M (LM=63, LF=37). This also leads to an output adjustment, whereby production of good F falls to about 68 units and production of good M rises to about 76 units.
At the new equilibrium, the slope of the PPF = - MPLF / MPLM = -PM/PF = -1
Remember that PF/PM =1 was given; it follows that PM/PF = 1 also. Labor moves out of Food and into Manufacturing until WF = WM and VMPLF = VMPLM.
Next, PF MPLF = PM MPLM, by substitution. Given PF = PM =1, it follows that
MPLF = MPLM. Hence, the slope of the PPF = - MPLF / MPLM = -PM/PF = -1
  1. Calculate the effects of the price change on the incomes of the specific factors in M and F. With the relative price change from PF/PM=2 to PF/PM =1, the money price of good F has fallen by 50 percent (from $2 to $1/unit F), while the money price of good M has stayed the same ($1/unit M). Wages have fallen, but by less than the fall in PF (wages fell approximately 27 percent, from 0.98 to 0.72 $/hour). Thus, the real wage in terms of Food (w / PF) actually rises from (0.98/2 =) 0.49 to 0.72 units F/hour) while to real wage in terms of M (w / PM) falls (from 0.98 to 0.72 units M/hour; given PM = $1).
    Capital owners are better off due to the lower real wage they must pay and the lower relative price for the food they buy. Landowners are worse off due to the higher real wage they must pay and the lower relative price for the food they sell.
    One may calculate the real income of capitalists by subtracting what they pay the workers from their total production, QM – (real wage, w, times units of labor employed, LM). When PF/PM=2, QM – w(LM) = 48 - .98(29) = 19.6 units of M.
    When the relative price of F falls to PF/PM =1, then the real income of capitalists rises to QM – w(LM) = 76 - .72(63) = 30.6 units of M. By subtracting the old level of real income from the new, one can see that the capitalists gain (30.6 – 19.6 =) 11 units of M.
    Repeating this process, one may calculate the real income of landowners by subtracting what they pay the workers from their total production, QF – (real wage, w, times units of labor employed, LF). When PF/PM=2, QF – w(LF) = 88 - .49(71) = 53.2 units of F.
    When the relative price of F falls to PF/PM =1, then the real income of landowners falls to QF – w(LF) = 68 - .72(37) = 41.4 units of F.
    By subtracting the old level of real income from the new, one can see that the landowners lose (41.4 – 53.2 =) 11.8 units of F.
    Remember that the country’s income will equal the value of its production. With trade, it will increase production of its (more valuable) comparative advantage good and import some of its disadvantage good. As in the Classical and HO Models, trade raises the per capita income of participating countries. In the Specific Factors Model, owners of resources specific to the exporting industry gain. Owners of resources specific to the import-competing industry lose. In this example, the capitalists gained and the landowners lost real income. Mobile factors may gain or lose, as discussed in the answer to part e. You should be careful to contrast these results with the predictions of the HO Model (see the Factor-price Equalization and the Stolper-Samuelson Theorems).

e.  Discuss the effects of the price change on the income received by labor. To determine the welfare consequences for workers, we need to know the quantities of goods M and F that they consume. If workers consume only M, then they are worse off. If they consume only F, then they are better off.
(Extra) Given the above information, one can construct the budget constraint faced by labor and show that if workers initially consumed F and M in a ratio of 26/46, then their welfare would not change. If ratio of F to M was higher, then they would be better off as a result of the decrease in the price of food. However, the derivation of this ratio is unimportant for our purposes. We do not know the demands of workers for the two goods(DF, DM). Therefore we do not know whether they are helped or hurt by the decline in the price of food and the consequent decline in wages.

3.  What if the mobile factor, labor, increases in supply? The box diagram presented below is a useful tool for showing the effects of increasing the supply of the mobile factor of production, labor.

a.  Analyze the qualitative effects of an increase in the supply of labor in the specific factors model, holding the prices of both goods constant at PF/PM = 1 (M/F). For an economy producing two goods, X and Y, with labor demands reflected by their marginal revenue product curves, there is an initial wage of W1 and an initial labor allocation of Lx=OxA and LY=OYA. When the supply of labor increases, the right boundary of this diagram is pushed out to OY'. The demand for labor in sector Y is pulled rightward with the boundary. The new intersection of the labor demand curves shows that labor expands in both sectors, and therefore output of both X and Y also expand. The relative expansion of output is ambiguous; it depends on the shapes of the MPL curves. Wages paid to workers fall. The incomes of the capitalists and the landowners both increase. See answer to part b.

  1. Graph the effect on the equilibrium for the numerical example in problems 1 and 2, given PF/PM = 1 (M/F), when the labor force expands from 100 to 140.
    From the shape of the MPL curves, it is clear that labor will continue to exhibit diminishing returns. Using the numerical example (see graphs on handout), LM increases to 91 from 63 and LF increases to 49 from 37. Wages decline from $0.72 to $0.62. This new allocation of labor yields a new output mix of approximately QM=95 and QF=75. (Again, the last digit is approximate).
    Using a four quadrant diagram, you can demonstrate that the new production possibility frontier shifts out, becomes more concave and is steeper at the x-axis (flatter at the y-axis

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