Consider an economy with two individuals, a poor individual whose income level is given by L>0 and a rich individual whose income level is given by H>L. Both consumers spend their income on a single consumption good (the price of which is normalized to unity) and share the same preferences given by u(c)=c1/2, where c denotes consumption. The government considers implementing a poverty alleviation policy which guarantees a minimum level of consumption of M, where L<M<H, for all individuals at the minimal cost.
1) Suppose first that the government can distinguish between the two individuals. Characterize the optimal individualized transfers set by the government (individualized transfers are denoted by {Note:Tj is T with lower j } Tj ; j=L, H,and are assumed to be nonnegative).
2) Suppose alternatively that the government is unable to distinguish between the two individuals and is hence offering both of them an identical transfer, denoted by T. Formulate the government optimization problem and calculate the optimal universal transfer level and the total government expenditure on the welfare program.
3) Now suppose that c is hot soup (measured in bowls). The government is considering opening a soup kitchen which will provide soup free of charge to all interested individuals. Eating in a soup kitchen entails some stigma costs. Suppose that the utility derived by an individual who eats in a soup kitchen and whose total consumption level is c is given by u(c,s)=c1/2-s (Note:C1/2 means Square root C), where s>0 denotes the stigma cost (entailed by eating in the soup kitchen). A senior economist interviewed for a talk show argued that the whole concept of opening a soup kitchen is flawed and will not save on government expenditure, as both types of agents incur identical stigma costs. Discuss the merits of this argument.
4) Denote by T>0, the number of bowls offered by the soup kitchen. Show that the benefit from eating in a soup kitchen is decreasing with respect to the individual’s level of income (Hint: consider an individual with an income level y, express the benefit, defined as the difference in utility b/w having or not having lunch in the soup kitchen, as a function of y, and show that this function is decreasing in y).
5) Formulate the government constrained optimization problem. You should properly define the objective function, the poverty alleviation constraint and two incentive compatibility constraints.
6) Suppose that H=100, L=36, M=80 and s=5/2. Show that the optimal number of soup bowls served by the soup kitchen would be given by T=44. Further show that the soup kitchen is welfare-superior to the universal transfer system (Hint: what would be the FB solution?)
7) Characterize a threshold level of stigma, above which opening a soup kitchen is welfare detrimental. Explain the underlying economic rationale.