Nadine sells user-friendly software. Her firm\'s production function is f(x_1, x
ID: 1193284 • Letter: N
Question
Nadine sells user-friendly software. Her firm's production function is f(x_1, x_2) = x_1 + 2x_2, where x_1 is the amount of unskilled labor and x_2 is the amount of skilled labor that she employs. In the graph below, draw a production isoquant representing input combinations that will produce 20 units of output. Draw another isoquant representing input combinations that will produce 40 units of output. Does this production function exhibit increasing, decreasing, or constant returns to scale? Constant. If Nadine uses only unskilled labor, how much unskilled labor would she need in order to produce y units of output? If Nadine uses only skilled labor to produce output, how much skilled y labor would she need in order to produce y units of output? If Nadine faces factor prices (1, 1), what is the cheapest way for her to produce 20 units of output? If Nadine faces factor prices (1, 3), what is the cheapest way for her to produce 20 units of output?Explanation / Answer
(a) Given production function is as follows -
f(x1, x2) = x1 + x2
Left-hand side of this function shows output produced and right-hand side of this function shows the combination of inputs needed to produce such output.
Case - I : Production of 20 units of output
In order to draw a production isoquant that represent input combination that will produce 20 units of output, we have to first calculate the maximum amount of x1 input or maximum amount of x2 input to produce 20 units, if we use them individually.
In other words, how many units of x1 is needed to produce 20 units of output, if we use only x1 and how many units of x2 is needed to produce 20 units of output, if we use only x2 .
This can be done as follows -
Calculaitng amount of x1 needed to produce 20 units, if only x1 is used -
f(x1, x2) = x1 + x2
20 = x1 + x2
Taking x2 = 0, as we are using only x1
20 = x1 + 2*0
x1 = 20
Thus, 20 units of input x1 can produce 20 units of output.
Calculaitng amount of x2 needed to produce 20 units, if only x2 is used -
f(x1, x2) = x1 + x2
20 = x1 + x2
Taking x1 = 0, as we are using only x2
20 =0 + 2x2
x2 = 10
Thus, 10 units of input x1 can produce 20 units of output.
The value of x1 is 20 units. This will act as X-intercept.
The value of x2 is 10 units. This will act as Y-intercept.
Joining these two intercepts, we get production isoquant representing input combination that will produce 20 units of output.
In the given graph, line drawn for 20 units of output is drawn in similar manner.
Similarly, one can draw the production isoquant for 40 units of output.
(b) In graph given in part (a), one can see that production isoquant is straight-line and not a curve.
Straight line production isoquant represent constant returns to scale as slope of such line remains same on all parts of line.
Instead of straight line, if we have downward sloping production isoquant then this shows decreasing returns to scale.
or,
Instead of straight line, if we have upward sloping production isoquant then this shows increasing returns to scale.
(c) Calculate amount of unskilled labor needed to produce y units of output, if only unskilled labor is used.
In the given question, x1 represent amount of unskilled labor and x2 represent skilled amount of labor.
As we are using only unskilled labor, this implies that there will be zero skilled labor used.
Given production function is as follows -
f(x1, x2) = x1 + x2
y = x1 + x2
Taking x2 = 0, as we are using only x1
y = x1 + 2*0
x1 = y
Thus, y units of input x1 can produce y units of output.
(d) Calculate amount of skilled labor needed to produce y units of output, if only skilled labor is used.
In the given question, x1 represent amount of unskilled labor and x2 represent skilled amount of labor.
As we are using only skilled labor, this implies that there will be zero unskilled labor used.
Given production function is as follows -
f(x1, x2) = x1 + x2
y = x1 + x2
Taking x1 = 0, as we are using only x2
y =0 + 2x2
x2 = y/2
Thus, y/2 units of input x2 can produce y units of output.
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