This is on Public Finance, especially on cash transfers. Question is as follows:

Question

This is on Public Finance, especially on cash transfers.

Question is as follows:

Suppose government provides low-income individuals with cash transfers. The formula for benefits is as follows: B = max [294 - 3/10 y, 10] where B is the benefit and y is the earnings. There are two types of foods, food (F) and other consumption (X). The prices of both goods are unify so that the budget constraint is F + X = y + B. Finally, the utility function of individuals is U = 1/3 ln F + 2/3 ln X. Find the optimal consumption bundle for an individual with y = 300. Because of nutritional consideration, government decides to give a coupon for food instead of cash. However, the formula (1) continues to determine the value of the coupon. How does the optimal consumption change for an individual with y = 300? Does the new policy improve the welfare of the individual?

Explanation / Answer

To get optimal solution lagrangian expression is used as

Z = {(1/2)ln F + (2/3) ln X) + V (F + X - Y -B)

where V = lamda

Differentiating the expression with respect to F, X and V, we get First order condition as

X = (-2/3V) and F = (-1/2V) and F + X -Y-B = 0

by substituting we get,   (-1/2V) - (2/3V) - Y - B = 0..................(1)

when Y = 300, B = max (204 , 0)

putting Y = 300 and B = 204 in eq(1) we get V = -(1/432)

Putting V = (1/432) in F and X expression to obtain their values as

F =216, X = 288 which are optimal solution

b) Here we will get food coupon value as 204. So with y = 300, F cannot be obtained as 216 due to its restricted value to 204. Thus the solution changes as F = 204 and X = 288

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