Cash Versus an Inframarginal Food Voucher
A household has a monthly income of 100 units. Food F costs 1 per unit and other goods O cost 1 per unit. Preferences are U = O + 10√F, so the marginal rate of substitution is 5/√F. The government offers either a food voucher of 60 units, redeemable only for food with no cash redemption, or a cash transfer of 60 units. Without any transfer the household chooses F = 25 and O = 75. An official argues the two programs are equivalent because both are worth 60 units. You are asked to draw the budget sets, find the optimal bundle under each program, calculate the household's cash-equivalent valuation of the voucher, and advise whether the voucher is justified if the objective is household welfare rather than food consumption.
1. Find the optimal bundle and utility under no transfer, under the cash transfer, and under the food voucher.
Read the worked answer
Without a transfer, the budget is F + O = 100. The household sets MRS = pF / pO, so 5/√F = 1, giving √F = 5 and F = 25. Then O = 75 and U = 75 + 10√25 = 75 + 50 = 125.
With a cash transfer of 60, income rises to 160. The same tangency condition gives F = 25 because the quasilinear utility has no income effect on food once income is above the threshold. Then O = 160 - 25 = 135 and U = 135 + 50 = 185.
With a food voucher of 60, the household must consume at least 60 units of food. Its unconstrained choice would be F = 25, so the voucher forces a corner solution at F = 60. The voucher pays for all 60 food units, leaving the entire cash income of 100 for O. Thus O = 100 and U = 100 + 10√60 ≈ 100 + 77.46 = 177.46.
Cash gives utility 185; the voucher gives 177.46. The cash transfer is preferred.
2. Calculate the household's cash-equivalent value of the voucher. Explain why it is below the voucher's face value.
Read the worked answer
The cash-equivalent transfer M* solves the indirect utility from cash equal to the voucher utility. With cash income M and no voucher, the household chooses F = 25 as long as M ≥ 25, so O = M - 25 and U = M - 25 + 50 = M + 25.
Set M* + 25 = 177.46, giving M* = 152.46. The voucher therefore has a cash-equivalent value of 152.46 - 100 = 52.46 units, not 60. The implicit welfare loss is 60 - 52.46 = 7.54 units.
The voucher is extramarginal: it forces food consumption of 60 when the household would voluntarily buy only 25. The household would rather take 25 food plus 135 other goods than 60 food plus 100 other goods. The face value of the voucher overstates its benefit because the household cannot resell the excess food or spend the voucher on other goods. If the voucher were no larger than 25 units, it would be inframarginal and equivalent to cash.
3. If the government's objective is household welfare, would you recommend the voucher? When might a voucher be justified?
Read the worked answer
If the objective is household welfare and the government can give cash, the cash transfer dominates. It gives the household the same budget value without distorting the food-other goods choice, and it raises utility by 185 - 125 = 60, compared with 177.46 - 125 = 52.46 for the voucher.
A voucher may still be justified if food consumption generates a positive externality or if the government has a paternalistic nutrition objective. In that case the relevant comparison is the externality benefit against the 7.54-unit welfare loss. A smaller voucher of up to 25 units would be inframarginal and would not distort the choice, but it would also do less to increase food consumption.
For poor households whose desired food spending exceeds the voucher, the voucher and cash are equivalent. The case for a voucher is strongest when food demand is high and the nutrition externality is well documented; otherwise cash is the cleaner instrument. The official's claim of equivalence is wrong for this household.