Question

A firm has two variable factors and a production function f(x1; x2) = (2x1 + 4x2)^1/2. On a graph, plot three input combinations and draw production isoquants corresponding to an output of 3 and to an output of 4. Also, mention the technical rate of substitution(s) for the isoquants. Show all working.

Answer 1

production function f(x1; x2) = (2x1 + 4x2)^1/2

ISOQUANTS shows different combinations of two inputs which yield same level of output. the table below shows different combinations of two inputs which gives same level of output. Figure below shows isoquants. blue isoquants represent isoquants which yield output 3 and red isoquant yields output 4.

In contructing isoquant we consider two inputs are substitutes ( can be perfect, close or bad substitutes). Inorder to produce same level of output, when we increase employment of one intput other input has to be reduced. in isoquant (which produces output 3) we start by employing 1 unit of both inputs. now if we want to employ more of input 2 we have to reduce input one . hence we increased input two from 1 to 1.25 and reduced input one from 1 units to 0.5 units. And when we increase input one from 1 unit to 2 unit, we decreased input 2 from 1 unit to 0.5 unit.

Isoquant two also works in same way.

Technical rate of substitution of isoquant is commonly know as marginal rate of technical substitution of inputs (MRTS X1,x2).

MRTS X1,x2 shows the rate at which one input (say x2) is substituted for an additional unit of another input (say x1). hence it measures number of units X2 a producer is willing to giveup inorder to employ an additional unit of X1.

MRTS X1,x2=

thus MRTS X1,x2 measures the slope of isoquant.

for inputs which are close substitutes MRTS X1,x2 declines as emploment of x1 increases.

For perfect substitutes MRTS X1,x2 increases as emploment of x1 increases.

ISOQUANT 1 (OUTPUT 3)	ISOQUANT 2 (OUTPUT 4)
X1	X2	X1	X2
1	1	2	1
0.5	1.25	1	1.5
2	0.5	3	0.5