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 $1,0002) 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.
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 .
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:
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 = 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.
And FOC of utility maximization is:
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:
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