Question

For the two preferences described below, please answer the following questions.

(i) Write down a utility function that is consistent with the description of the preference.

(ii) Draw indifference curve maps for the individual. Label both axis. Indicate marginal rate of substitution on your graph and whether it is diminishing.

(iii) Obtain the demand function for both good x and y.

(iv) Describe how an increase in income would affect John and Mary’s consumption bundles. John and Mary’s preferences are described below.

(a) John gets utility from coffee (x) and cream (y). Coffee and cream must always be consumed with a fixed proportion of 1:5.

(b) Mary consumes ice creams (x) and pizzas (y). Mary has diminishing marginal utility for pizzas, but not for ice creams. His marginal utility for ice creams is constant and equals 1.

Answer 1

1. Write down a utility function that is consistent with the description of the preference.

This is normal to ponder about the preferences, that are mostly more convenient to associate different numbers to different goods and the agent have a choice to choose the good with the highest number, generally called as utilities. Utility function denotes the utility that are associated with each x ∈ X usually by u(x) ∈ Ŗ

A utility function u(x) denotes an agent’s preferences as

u(x) ≥ u (y), only when and if x < y

This refers that an agent will do the same choices he uses her preference in relation to < or his utility function u(x).

(ii) Draw indifference curve maps for the individual. Label both axis. Indicate marginal rate of substitution on your graph and whether it is diminishing.

(Depiction of diminishing marginal rate of substitution in the above mentioned figure)

As the consumer has more of good X, he is prepared to less of good Y. The above mentioned fig1 state that when the consumer comes down from A to B on the indifference curve he will give up for AY₁ of good Y as in compensating form of ∆X of good X. The MRS at a point on the indifference curve is equal to the slope of indifference curve that is tangent on the X axis. Fig 2 have three tangents as GH, MN KL on points P & Q & R. Following which Marginal rate of substitution of X and Y diminishes as the consumer line comes down on the indifference curve.

iii) Obtain the demand function for both good x and y.

Suppose x is a normal good and for normal goods, the income and substitution effect go in the same direction means an increase in the price of good x makes good y more desirable and demandable.

iv) Describe how an increase in income would affect John and Mary’s consumption bundles. John and Mary’s preferences are described below.

(a) John gets utility from coffee (x) and cream (y). Coffee and cream must always be consumed with a fixed proportion of 1:5.

(b) Mary consumes ice creams (x) and pizzas (y). Mary has diminishing marginal utility for pizzas, but not for ice creams. His marginal utility for ice creams is constant and equals 1.

Answer (a). As the income of John rises his consumption of Coffee (x) and Cream (y) with a fixed proportion remains unchanged as the consumption of normal goods will increase with an increase in an income.

Answer (b). As the Mary has a diminishing marginal utility for pizza. Mary would purchase less of pizza and more of ice cream until Mary reaches for Marginal rate of diminishing for ice cream. Ultimately, items purchased the marginal utility per as per increase in the income spent on the two goods such as pizza and ice cream is/are the same and no other combination of pizza and ice cream will give greater utility. At last... marginal utility of what anything it can buy declines.

Therefore, MU_pizza/P_{pizza =} MU_{ice cream}/MU_{ice cream}

Answer 2

Solution:

a) For John, two goods are coffee (x) and cream (y). Since, they are always consumed together, in a fixed proportion or ratio, this implies the two goods are perfect complements for John. Thus, we will have kinked indifference curves.

i) Utility function: Uj = min{5x, y}

ii) Following is the indifference curve map as required/asked in the question. Clearly, MRS is diminishing (at extremes) that is, as John increases the consumption of coffee, MRS falls straight from infinity to 0 along an indifference curve.

iii) Finding the demand functions:

From the utility function, we know that John consumes at the kink. Thus, at kink 5x = y

Assuming income with John = M, prices of two goods are Px and Py, our budget line becomes:

x*Px + y*Py = M

With the condition at kink, x*Px + 5x*Py = M

x(Px + 5*Py) = M

So, x = M/(Px + 5*Py)

And so, y = 5*x that is y = 5*M/(Px + 5*Py)

iv) With what we obtain in (iii), it is easy to notice that income, M increases, John's consumption bundle goes up, with consumption of both goods increasing. So, with this new increased income, say M', consumption bundles become:

new x = M'/(Px + 5*Py) and new y = 5*M'/(Px + 5*Py)

b) Similarly, as above for Mary:

i) The way the preferences of Mary are described, it seems that she has a quasi-linear utility function. So, from ice-creams , since the marginal utility constant for 1, this is the good with linear form to

Um = x + ln (y) (or Um = x + y^1/2)

ii) Following is the indifference curves map for Mary. Clearly, MRS is diminishing that is, as Mary increases the consumption of coffee, MRS falls along an indifference curve (arrows show direction of increasing utility).

iii) Obtaining the demand function of the two goods:

Optimal consumption takes place at the tangency of the budget line and indifference curve.

So, MRS = MUx/MUy

MUx = = 1 (as already given).

MUy = = 1/y

So, MRS = MUx/MUy = 1/(1/y) or y

So, at tangency, when MRS = Px/Py

y = Px/Py.

Using the budget line, x*Px + y* Py = I, where I is Mary's income

x*Px + (Px/Py)*Py = I, so, x = (I - Px)/Px

iv) It's clear from the demand function of good y, the pizzas are independent of income, I, so income increase will have no impact on the demand for that good. While the one with declining MRS, the good x demand is a function of income, and is related to it positively, so an increase in income will increase good x consumption, leaving good y consumption unchanged.