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## Change in Income, Change in Price and Lagrangean Function

Change in Income and Change in Price:Example: M

_{0}= $800 , P_{x}= $20, P_{y}= $40. Depict the budget lines in each case.1) Income rises from $800 to $1,000

2) P

_{x}rises to $25 holding initial income and P_{y }constant.3) Now income increases from $800 to $1000 and P

_{x}rises to $25 and P_{y}rises to $50.:Optimal Choicemax x,y U (x, y)- ‘choose x and y to maximize utility’

Subject to- P

_{x}x + P_{y}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 P

_{x}x + P_{y}y = M .:Lagrangean Functionmax x,y U(x, y )

subject to- P

_{x}x + P_{y}y ≤ MWe define the Lagrangean (L) as L(x, y, λ) = U(x, y) +λ (M − P

_{x}x − P_{y}y) , where λ is a Lagrange multiplier. The first-order necessary condition (FOC) for an interior optimum (with x > 0 and y > 0) are:We can combine (1) and (2) to eliminate the Lagrange multiplier therefore FOCs reduce to:

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

Example: You are given U(x, y) = √x + √y = x

^{1/2}+ y^{1/2}and P_{x}x + P_{y}y = M. Now derive demand functions of x and y when the consumer is maximizing his utility.And

is:FOC of utility maximizationwhere x* and y* are utility-maximizing quantities demanded.

By squaring both sides of above equation and rearranging terms, we find that y * = x * (P

_{x}^{2}/P_{y}^{2}).Plugging the last expression into budget equation as well as simplifying we get:

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