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

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