Answer to Question #182058 in Microeconomics for Abiy

Question

6. Find the demanded bundle for a consumer whose utility function is. u( x y) = x3/2 y and her budget constraint is 3x+4=100.

7. The demand function for a particular good is x = a + bp. What are the associated direct and indirect utility functions?

8. The demand function for a particular good is x = a+ bp+ cm.What are the associated direct and indirect utility functions? 

9. Explain how the downward sloping demand curve can be derived using indifference curves and budget lines. Using the same technique, show that it is theoretically possible to have an upward sloping demand curve.

Answer 1

(6) Optimal bundle is where

$\frac{MUx }{ MUy} = \frac{Px}{ Py}$

U = X^3/2Y

Optimal bundle is where MUx / MUy = Px / Py

Where, MUx = Marginal utility from good x

MUy = Marginal utility from good x

Px = Price of good x = 3

Py = Price of good y = 4

U = X^1.5 Y

Differentiating the above utility function with respect to x, to get MUx

dU/Dx = MUx

MUx = 1.5 X^0.5 Y

 U = X^1.5 Y

Differentiating the above utility function with respect to y, to get MUy

dU/Dy = MUy

MUy = X^1.5

MUx / MUy = Px / Py

1.5 X^0.5 Y / X^1.5 = 3 / 4

1.5 * Y / X = 0.75

Y / X = 0.75 / 1.5

Y / X = 0.5

Y = 0.5 X

 

3X + 4Y = 100

Substituting Y = 0.5 X in 3X + 4Y = 100

3X + 4 (0.5 X) = 100

3X + 2X = 100

5X = 100

X = 100/5

X = 20

 

Y = 0.5 X

Y = 0.5 * 20

Y = 10

The optimal consumption bundle is 20 units of good X and 10 units of good Y.

(7)To know how much a consumer can obtain at the most, with the given alternatives, the utility of "indirect" nature is used. 

For the given function of demand  $x=a+bp$

 The function for "indirect utility" can be written as 

$V(x,p,m)=max u(x,z)$

where, 

z=other goods

m=income/wealth

Indirect utility function$=−\frac{\frac{∂V}{∂p}}{\frac{∂V}{∂m}}$

Indirect utility function$=x(p,m)$

Thus, the consumer can only get as much utility as limited to the price and also to their income/wealth. 

Now, for the utility function of "direct" nature, 

$U(x,z)=Min V(x,p,m)$

(8)Utility function: In economic aspects, utility capacity is a significant idea that actions inclinations over a bunch of labour and products. Utility addresses the fulfilment that shoppers get for picking and burning through an item or administration

Direct utility is something that the consumer gets from the consumption of goods and services. The consumer has the choice of direct utility by consuming the good a and b.

The p is the price of the good which is decided by the market. The c will be the consumer income and m will be considered as money. The direct utility will get by consuming the good a and b by the consumer.

The consumer will get indirect utility by spending the money for the goods purchased by the consumer. The income does not provide the utility directly but it helps in generating utility indirectly by paying for the goods.

Hence, the goods and services help in providing satisfaction to the consumer.

(9)To derive the demand curve from the indifference curve and the budget line is the indifference curve shows the various combinations on the budget line.

It shows all the points which are demanded on the budget line and the demand is the downward-sloping curve.If we violate the principle that more is preferable to less, a series of indifference curves may be upward sloping. When a series of indifference curves are upward sloping, it means one of the products is a "negative" in the sense that the buyer would rather have less of it than more of it.