Assume that the endownment of capital and labor in a given society is 20 units of capital and 20 units of labor. The production function of good X can be expressed as X = K1/2L1/2, and the production of good Y can be expressed as Y = K + L. Which of the following describes society's production possibility frontier?
Solution:
A straight line from (X=20, Y=0) to (X=0, Y=40).
Explanation:
If society devotes all resources to making X, X = 20. If society devotes all resources to making Y, Y = 40. The production possibility frontier is a straight line because there are constant returns to scale for both goods.
I think it is line from (X=8.94, Y=0 to (X=0, Y=40). The line will be in a bow-shape.
(I presume by: X = K1/2L1/2 means K and L are raised to the 1/2 power.)
To see this, plot a graph. (Excel spreadsheets would be very helpful here).
To understand how to get the answer to this question, we need to first understand what a production possibility frontier is and how it is derived.
A production possibility frontier (PPF) represents the maximum combination of goods that a society can produce given its resources and technology. It shows the trade-off between producing one good and another, assuming all resources are fully utilized and used efficiently.
In this case, we have the endowment of 20 units of capital and 20 units of labor. The production function for good X is X = K^(1/2) * L^(1/2), where K represents capital and L represents labor. The production function for good Y is Y = K + L.
To find the PPF, we need to consider different combinations of X and Y that can be produced using the given resources. Let's analyze two extreme scenarios.
1. Scenario A: If society devotes all resources to making X, we allocate all 20 units of capital (K) and 20 units of labor (L) to the production of X. Using the production function for X, we substitute K = 20 and L = 20:
X = 20^(1/2) * 20^(1/2) = 20.
Hence, when society allocates all resources to making X, the maximum quantity of X that can be produced is 20.
2. Scenario B: If society devotes all resources to making Y, we allocate all 20 units of capital (K) and 20 units of labor (L) to the production of Y. Using the production function for Y, we substitute K = 20 and L = 20:
Y = 20 + 20 = 40.
Hence, when society allocates all resources to making Y, the maximum quantity of Y that can be produced is 40.
Now, we can draw the PPF based on these extreme scenarios. The PPF will be a straight line connecting the points (X = 20, Y = 0) and (X = 0, Y = 40), because the production functions for both goods have constant returns to scale. This means that for every unit of input (capital or labor) added, an equal unit of output will be produced.
Therefore, the correct description of society's PPF is a straight line from (X = 20, Y = 0) to (X = 0, Y = 40).