Questions

  1. Suppose that the following combinations of \(x_1\) and \(x_2\) all produce 100 bushels of corn.

    1. Calculate the \(MRS_{x_1x_2}\) and the \(MRS_{x_2x_1}\) at each midpoint.
remove(list=ls()) #clear all from memory

require(ggplot2)
require(data.table)

#define the variables
Combination <-  c("A","B","C","D","E")
x1 <- c(10,5,3,2,1.5)
x2 <- c(1:5)

#put them into a data.table
dat <- data.table(Combination,x1,x2)

#now we can use the shift fucntion to get the previous value
dat[, MRSx1x2 := (x2 - shift(x2))/(x1-shift(x1))]
dat[, MRSx2x1 := (x1 - shift(x1))/(x2-shift(x2))]

dat
##    Combination   x1 x2 MRSx1x2 MRSx2x1
## 1:           A 10.0  1      NA      NA
## 2:           B  5.0  2    -0.2    -5.0
## 3:           C  3.0  3    -0.5    -2.0
## 4:           D  2.0  4    -1.0    -1.0
## 5:           E  1.5  5    -2.0    -0.5

 

  1. Suppose that the price of \(x_1\) and \(x_2\) is each a dollar. What combination of \(x_1\) and \(x_2\) would be used to achieve the least-cost combination of inputs needed to produce 100 bushels of corn?  

    Since the price of each is one dollar, then the inverse price ration is \(-1\) for both, therefore costs will be minimised between combination C and D  

  2. Suppose that the price of \(x_2\) increased to $2. What combination of \(x_1\) and \(x_2\) would be used to produce 100 bushels of corn?

    Lets assume that \(x_2\) is on the vertical axis, therefore cost is minimised where \(MRS_{x_1x_2}\) is equal to the inverse price ratio as \(-\frac{Px_1}{Px_2}=-\frac{1}{2}=-0.5\) therefore cost will now be minimised between B and C.

  3. If the farmer was capable of producing 100 bushels of corn when the price of \(x_1\) and \(x_2\) were both $1, would he or she necessarily also be able to produce 100 bushels of corn when the price of \(x_2\) increases to $2? Explain.

    It is possible from a technical perspective but since every point on the isoquant yields the same output  

  1. For the production function \(y = 3x_1 + 2x_2\), find

    1. The \(MPP\) of \(x_1\).

    \[MPP_{x_1} = \frac{\partial y}{\partial x_1}=3\]

    1. The \(MPP\) of \(x_2\). \[MPP_{x_2} = \frac{\partial y}{\partial x_2}=2\]

    2. The marginal rate of substitution of \(x_1\) for \(x_2\) \[MRS_{x_1x_2} = -\frac{\partial y}{\partial x_1}/\frac{\partial y}{\partial x_2}=-\frac{3}{2}=-1.5\]  

  2. For the production function \(y = x_1^{0.5}x_2^{0.333}\), find

    1. The \(MPP\) of \(x_1\). \[MPP_{x_1} = \frac{\partial y}{\partial x_1}=-\frac{0.5x_2^{0.333}}{x^{0.5}}\]
    2. The \(MPP\) of \(x_2\). \[MPP_{x_2} = \frac{\partial y}{\partial x_2}=-\frac{0.667{x^{0.5}}}{x_2^{0.667}}\]
    3. The marginal rate of substitution of \(x_1\) for \(x_2\). \[ \begin{aligned} MRS_{x_1x_2} = -\frac{\partial y}{\partial x_1}/\frac{\partial y}{\partial x_2} = -\frac{-\frac{0.5x_2^{0.333}}{x^{0.5}}}{-\frac{0.667{x^{0.5}}}{x_2^{0.667}}} =-\frac{0.5x_2^{0.333}}{x^{0.5}}.\frac{x_2^{0.667}}{x^{0.5}{0.667}} &= -0.75\frac{x_2}{x_1} \end{aligned} \]  

  3. Assume that a farmer has budget of $200 to buy inputs. What is the slope of the isocost line when

    1. \(Px_1 = \$1; Px_2 = \$2.00\)?

    Lets assume that \(x_2\) is on the vertical axis, then the slope of the isocost line is \(-0.5\)

    1. \(Px_1 = \$1; Px_2 = \$1.75\)?

    Lets assume that \(x_2\) is on the vertical axis, then the slope of the isocost line is \(\frac{1}{1.75}=-0.57\)  

  4. Answer these two questions regarding input relationships:

    1. What is the difference between the marginal rate of substitution and the elasticity of substitution between two inputs?

    see notes

    1. What is the difference between Inputs which are substitutes or complements and the idea of substitution between Inputs?

    see notes

    1. Can two Inputs be both complements and substitutes at the same time?

    It is possible for two inputs such as capital and labour to be substitutes as one substitutes for the other along an isoquant. At the same time the two inputs can be complements. Adding capital to labour makes labour more productive, i.e. increases its MPP.