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Ex 2.3: A PPF with Capacity Constraints Some PPF's can have kinks in them: that...
Apr 9, 2024
Ex 2.3: A PPF with Capacity Constraints Some PPF's can have kinks in them: that is, points at which the MRT is discontinuous as you move along the PPF. These kinks can arise for a number of different reasons; this question analyzes just one of those. Suppose masks (good 1) and ventilators (good 2) can be produced according to the production functions x1=f1(L1,K1)=L1K1x2=f2(L2,K2)=min⁡{L2,K2} \begin{array}{l} x_{1}=f_{1}\left(L_{1}, K_{1}\right)=\sqrt{L_{1} K_{1}} \\ x_{2}=f_{2}\left(L_{2}, K_{2}\right)=\min \left\{L_{2}, K_{2}\right\} \end{array} Suppose the economy has Lˉ=400\bar{L}=400 units of labor and Kˉ=400\bar{K}=400 units of capital. In the short run, though, capital has already been allocated, as follows: K1=100K_{1}=100 units of capital have been allocated to producing masks, and K2=300K_{2}=300 have been allocated to producing ventilators. Labor can be allocated to either good. (a) Plot the (short-run) PPF for masks and ventilators. (b) If the economy is producing x1=50x_{1}=50 masks, what is the MRT? What about if it's producing x1=150x_{1}=150 masks ??
Answer
The short-run PPF for masks and ventilators will be a curve that starts at the maximum number of ventilators and ends at the maximum number of masks, with a kink at the point where labor shifts from being the limiting factor to capital being the limiting factor for ventilator production. The MRT when producing 50 masks is 1, and when producing 150 masks, it is undefined due to the kink in the PPF.
Solution
a
To plot the short-run PPF, we need to find the maximum number of masks and ventilators that can be produced with the given capital constraints and the total labor available. For masks, using all labor, L1=Lˉ=400 L_1 = \bar{L} = 400 , the production function is x1=L1K1=400×100=200 x_1 = \sqrt{L_1 K_1} = \sqrt{400 \times 100} = 200 . For ventilators, using all labor, L2=Lˉ=400 L_2 = \bar{L} = 400 , the production function is x2=min⁡{L2,K2}=min⁡{400,300}=300 x_2 = \min\{L_2, K_2\} = \min\{400, 300\} = 300 . The PPF will start at (0,300) and end at (200,0), with a kink where the production of ventilators is limited by K2 K_2 rather than L2 L_2
b
The MRT is the rate at which one good must be sacrificed to produce an additional unit of another good. When producing 50 masks, x1=50 x_1 = 50 , we can find the corresponding L1 L_1 using the production function for masks: 50=L1×100 50 = \sqrt{L_1 \times 100} , so L1=2500/100=25 L_1 = 2500/100 = 25 . The remaining labor for ventilators is L2=400−25=375 L_2 = 400 - 25 = 375 , and x2=min⁡{375,300}=300 x_2 = \min\{375, 300\} = 300 . If we produce one more mask, x1=51 x_1 = 51 , then L1=2601/100≈26 L_1 = 2601/100 \approx 26 , and L2=400−26=374 L_2 = 400 - 26 = 374 , x2=min⁡{374,300}=300 x_2 = \min\{374, 300\} = 300 . The MRT is Δx2/Δx1=0/1=0 \Delta x_2 / \Delta x_1 = 0/1 = 0 . However, this is not the typical MRT calculation since x2 x_2 is not decreasing. The MRT is effectively the slope of the PPF, and at this point, the slope is 1 because the PPF is linear before the kink. When producing 150 masks, the MRT is undefined because we are at the kink of the PPF where the slope changes abruptly
Key Concept
The Production Possibilities Frontier (PPF) with capacity constraints and the Marginal Rate of Transformation (MRT) at different points on the PPF.
Explanation
The PPF illustrates the trade-offs between two goods when resources are limited, and the MRT tells us how much of one good we must give up to gain an additional unit of another good. In the presence of capacity constraints, the PPF can have kinks, leading to discontinuities in the MRT.
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