Question

Consider an individual with utility of the form: U(x,y) = x^ay^b where (a+b)=1. The price of good x is p_x and the price of good y is ­p_y. The individual faces a budget constraint of I (or income).

A. Find the demand functions for the individual in question.

B. Suppose the price of each good increases by a factor of T (therefore the price of good x is (1+T)p_x and the price of good y is (1+T)p_y). Prove that the increase in price does not change relative amount of each good the individual purchases, however, the individual purchases less of each good. How much additional income would the individual need to obtain the same utility as in part A)?

C. Suppose a policymaker is worried about consumption of good x (suppose production of good x pollutes and creates a negative externality on the environment, for example) and decides to subsidize the production of good y, thereby decreasing its price. The hope of said policymaker is that people will substitute away from the consumption of good x towards good y because it now costs less. If all individuals in the society in question have utility that is representative of parts A) and B), will this policy work? Why or why not?

Answer 1

Given: There are two goods x and y. the prices of the goods are P_x and P_y respectivily. A consumer's utility function is given by U(x,y) = x^a y^b. And the individuals income is given by I.

Solution:

A) Demand function:

Utility function is given by U(x,y) = x^a y^b

And the individuals budget constrint can be given by P_x * x^a + P_y * y^b = I.

And the marginal rate of substitution of the utility function is given by , MRS = ay / bx.

To find the demand function of the for these two goods we want to find the solution for the goods x and y.

Since we know the price of the two goods the price ratio is given by p_x / p_y.

Now we can use the rule of utility maximization such as,

MRS = p_x / p_y

ay / bx = p_x / p_y

y = bp_x / ap_y x

Now we got the y value now we can subsitute y value in the budget constraint equation

  P_x * x + P_y * y = I

  P_x * x + P_y * bp_x / ap_y x = I

  P_x * x + b/a*p_xx = I

(a + b/a) p_xx = I

x* = (a/ a+b) *I/p_x

Similarly, we can find the value of y in the same method from which we get

y* = (b/a+b) * I/p_y

Thus the solutions for the x and y are called as the consumers demand function.

B) suppose the price of the both goods increases by a factor of T.then the price of the good x changes to (1+T)p_x and the price of the good y changes to (1+T)p_y. the the butget constraint of the consumer is given by

(1+T)p_x *x + (1+T)p_y *y= I. The price ratio before price change is p_x/p_y and the price ratio after the price change is (1+T)p_x/ (1+T)p_y.

p_x/p_y = (1+T)p_x/ (1+T)p_y

p_x/py = p_x/p_y

Therefore the increase in the price does not change the relative consumption of the consumer.

c) YES , this policy works only if the the goods x and y are substitutes to each other. if it is true then the people will buy more of good y because the price of the good y is less than the good x. if the goods x and y are not substitutes then decreasing the price of good y will not impact the externality caused by good x.