Change in Income, Change in Price and Lagrangean Function

Change in Income and Change in Price:

Example: M0 = $800 , Px = $20, Py = $40. Depict the budget lines in each case.
1) Income rises from $800 to $1,000
2) Px rises to $25 holding initial income and Py constant.
3) Now income increases from $800 to $1000 and Px rises to $25 and Py rises to $50.

Optimal Choice:

max x,y  U (x, y)- ‘choose x and y to maximize utility’

Subject to- Px x + Py y ≤ M - ‘expenditures on x and y must not surpass the consumer’s income’

If the consumer likes more of both goods the marginal utilities of good x as well as y are both positive. At an optimal basket every income will be spent. Therefore the consumer will choose a basket on the budget line Px x + Py y = M .

974_optimal choice.jpg

Lagrangean Function:

max x,y U(x, y )
subject to- Px x + Py y ≤ M

We define the Lagrangean (L) as L(x, y, λ) = U(x, y) +λ (M − Px x − Py y) , where λ is a Lagrange multiplier. The first-order necessary condition (FOC) for an interior optimum (with x > 0 and y > 0) are:

1524_langrangean function.jpg

We can combine (1) and (2) to eliminate the Lagrange multiplier therefore FOCs reduce to:

855_langrange multiplier.jpg

From the above equations we are able to derive demand function of x and y.

Example: You are given U(x, y) = √x + √y = x1/2 + y1/2 and Px x + Py y = M. Now derive demand functions of x and y when the consumer is maximizing his utility.

286_maximizing langrange utility.jpg

And FOC of utility maximization is:

1299_FOC of utility maximization.jpg

where x* and y* are utility-maximizing quantities demanded.

By squaring both sides of above equation and rearranging terms, we find that y * = x * (Px2/Py2).
Plugging the last expression into budget equation as well as simplifying we get:

2012_langrangean function.jpg

Latest technology based Microeconomics Online Tutoring Assistance

Tutors, at the www.tutorsglobe.com, take pledge to provide full satisfaction and assurance in Microeconomics help via online tutoring. Students are getting 100% satisfaction by online tutors across the globe. Here you can get homework help for Microeconomics, project ideas and tutorials. We provide email based Microeconomics help. You can join us to ask queries 24x7 with live, experienced and qualified online tutors specialized in Microeconomics. Through Online Tutoring, you would be able to complete your homework or assignments at your home. Tutors at the TutorsGlobe are committed to provide the best quality online tutoring assistance for Microeconomics Homework help and assignment help services. They use their experience, as they have solved thousands of the Microeconomics assignments, which may help you to solve your complex issues of Microeconomics. TutorsGlobe assure for the best quality compliance to your homework. Compromise with quality is not in our dictionary. If we feel that we are not able to provide the homework help as per the deadline or given instruction by the student, we refund the money of the student without any delay.

©TutorsGlobe All rights reserved 2022-2023.