Suppose the government implements an income support program with the intention of making sure residents are able to purchase sufficient food. The government pays a cash benefit to all individuals with incomes less than $1000 according the following formula:

cash benefit (CB) = $200 - 0.2*(earned income(I))

Households spend all of their income on food (F) and other goods (X). The price of food and other goods are normalized to 1. A households budget constraint is

F + X =  CB + I

Households have the following preferences:

U = 0.25*ln(F) + 0.75*ln(X)

1-Which program does the household with $300 earned income prefer and why?

a- The food stamp (voucher) program. The household prefers to maximize expenditures on food.

b- The cash benefit program. The household prefers to spend some of the benefit on goods other than food.

c- The household is indifferent between programs. It spends the same on food and other goods in both cases.

d- The cash benefit program. The household prefers to spend all of its income on goods other than food.

 

2-Repeat the exercise for a household with earned income of $800. What program is preferred and why?

a- The cash benefit program. The household prefers to spend some of the benefit on goods other than food.

b-The cash benefit program. The household prefers to spend all of its income on goods other than food.

c- The food stamp (voucher) program. The household prefers to maximize expenditures on food.

d- The household is indifferent between programs. It spends the same on food and other goods in both cases.

 

Answer:

1-Option b is correct.

2-Option a is correct.

Step-by-step explanation

The maximizing condition where the individual maximize the utility with respect to budget constraint,

Maximize 0.25*ln(F) + 0.75*ln(X) with respect to F + X =  CB + I = TI (total income)

Marginal utility of food = dU/dF = 0.25/F

Marginal utility to other good = dU/dX = 0.75/X

The slope of budget line F+X = TI is 1.

At optimum, the ratio of MU of F and MU of X should be equal to the slope of budget line i.e. price ratio.

(0.25/F) / (0.75/X) = 1

X/F = 3

X = 3F

Put this condition in budget equation,

F+3F = TI

4F = TI

F* = TI/4

X* = 3TI/4

1-If the earned income is $300, the cash benefit received by the individual will be,

CB = 200 - 0.2($300) = 200 - 60 = $140

The optimal consumption of food and other goods under cash benefit will be, F = (CB+I)/4 = ($140+$300)/4 = 110 and X = 3(CB+I)/4 = 3($140+$300)/4 = 330.

In case of food vouchers, the vouchers will only be used for the purchase of food. Thus, the whole earned income will be spent on X. The optimal amount of X = 3(0+300)/4 = 225.

The difference between the change in consumption of X is 105, which is substantial. Also, the CB received is a large proportion of the individual income. Thus, the individual will prefer cash benefit as the spending on X can be increased in this case keeping the food consumption fixed in both the cases. Option b is correct.

2-The cash benefit if the earned income is $800 will be,

CB = 200 - 0.2($800) = 200 - 160 = $40

The optimal amount of consumption of F under CB = ($40+$800)/4 = 210

The optimal amount of consumption of X under CB = 3($40+$800)/4 = 630

In case of food vouchers, the food consumption is fixed. All of the income will be spent on X. Thus, the optimal amount of X = 3($0+$800)/4 = 600.

The net change in the consumption of X is only 30. Also, the cash benefit is also small. However, it still gives higher amount of goods and services. Thus, option a is correct.