Chapter 5, Problem 5.16P

To determine

(a)

To show:

At interior optimum, the demand schedule for good x is:

  $x=\frac{I}{2Px}-5$

Explanation of Solution

In Marginal utility (MU), the extraadvantageresulting from the consumption of one more unit of a good or service.

Budget constraint/line shows all probable combinations of two products that can be bought.

given the level of earning of the consumer and the market values of both products.

Indifference curve (IC) displays all probable combination of two goods that gives the same level of satisfaction to the consumer.

The consumption bundle is optimalat the point of tangency of the budget line and the indifference curve. The slope of the IC and the budget line are equal at this point.

Consumer's marginal utility functions

  $MUx=yandMUy=x+10$

The Consumer has income I.

Let the prices of x and ybe denoted as P_x and P_y.

The interior optimum basket is specified by the solution:

  $\frac{MUx}{MUy}=\frac{Px}{Py}$ and $Pxx+Pyy=I$

Usingabove equation to find the demand for x as follows:

  $\frac{MUx}{MUy}=\frac{Px}{Py}$

  $\begin{array}{l}\frac{MUx}{MUy}=\frac{Px}{Py}\\ \Rightarrow \frac{y}{x+10}=\frac{Px}{Py}\\ \Rightarrow Px\left(x+10\right)=ypy\\ \Rightarrow \left(x+10\right)=\frac{ypy}{Px}\\ \Rightarrow x=\frac{ypy}{Px}-10\\ \Rightarrow \frac{x}{2}=\frac{Pxx+Pyy}{2Px}-5\\ \text{since,}\\ Pxx+Pyy=I\\ x=\frac{I}{2Px}-5\end{array}$

To determine

(b)

To calculate:

The value of P_x for which consumer would ever purchase any X.

Answer to Problem 5.16P

  $Px=10$

Explanation of Solution

Utility function refers to measure preferences over set of product and services. it is majorly used in economics. A function is to satisfy consumers for all grouping of products consumed.

Marginal utility (MU) can be defined as the extra benefit added from the consumption of one more unit of a product or service.

The demand curve is the graph display the demand for products or service an alteration with a change in price.

Given

I =100, and

Since

  $Pxx+Pyy=I$

  $x=\frac{100}{2Px}-5$ .........Equ (1) (The value of x can't be negative)

The maximum value for which the consumer would buy can be calculated as −

From Equ (1), x = 0 (as it can't ne negative)

  $0=\frac{100}{2Px}-5$

  $5=\frac{100}{2Px}$

  $Px=\frac{100}{2\times 5}$

  $\begin{array}{l}Px=\frac{50}{5}=10\\ \text{Hence},Px=10\end{array}$

To determine

(c)

If $Px=20\text{ and }Py=20$, then the graph for optimal consumption bundle of x and y is to be drawn.

Explanation of Solution

Pattern of consumption is the expenses pattern of income groups in the categories of products like food, clothing and optional items.

The budget constraint traces the highest indifference curve accessible; the consumer is defined to be optimizing consumption.

Suppose that $Px=20\text{ and }Py=20$.

Find out the optimal consumption basket of x and y.

  $Px>10$ by graphing the consumer's budget line $20x+20y=100$ and an indifference curve is $U\left(x,y\right)=xy+10y$.

  

The utility function $U\left(x,y\right)=xy+10y=50$ and it intersect the vertical axis.

This implies that the consumer could have a positive level of utility with purchases of y and no units of x.

The Utility is maximized in the graph $x=0,y=5$.

Since this corresponds to a corner point optimum where, $Px>10\left(Px\text{ exceed }10\right)$

To determine

(d)

The marginal rate of substitution ofx for y with the ratio $\frac{Px}{Py}$ at the optimum in part c is to be compared and whether consumer decreases utility if she purchased a positive amount of X is to be described.

Explanation of Solution

Marginal rate of substitution is used in economics. In this,the consumer gets some amount of one product in return for another product. It keeps the same level of utility.

The corner point has the price ratio $\frac{Px}{Py}=\frac{20}{20}=1$

To compare this ratio to the marginal rate of substitution of x and y, use the equation for the marginal rate of substitution.

  $\begin{array}{l}MRSx,y=\frac{MUx}{MUy}\\ MUx=y,MUx=x+10\\ MRSx,y=\frac{MUx}{MUy}\\ =\frac{y}{x+10}\end{array}$

As the marginal rate of substitution of x for y is a smaller amount than the price ratio of 1 and the marginal rate of substitution of x for y is diminishing as xrises.

This confirms that the consumer would decrease utility if a positive amount of x were purchased.

To determine

(e)

If income remains at 100 then the demand schedule for x for all values P_x is to be determinedand location dependsupon thevalue of P_y is to be checked.

Explanation of Solution

Demand curve is the graph showing the demand for goods or service changes with change in price.

Let us draw the demand schedule for x for all values of Px when I =100 which is given by the equation is $x=\frac{100}{2Px}-5$

  

The demand curve for x is downward sloping and bowed towards the origin.

As the demand schedule equation $x=\frac{100}{2Px}-5$ does not contain any terms related to the price of y. P_y, the location of the demand schedule does not depend on the value of P_y.