Assume that an individual has the utility functio where X and Y represent units

Question

Assume that an individual has the utility functio

where X and Y represent units of goods X and Y, respectively, and U is some index of utility.MUx = Y+1 and MUy=X+2. This individual spends all of her money income, I = $72, on the two goods where the price of X, PX = $2, price of Y, PY = $4. The individual maximizes her utility by consuming 18 units of good X, 9 units of good Y and receives an amount of utility reflected by the index, U=200.

(a) Suppose the government decides to assist this individual by subsidizing her purchases of good X so that the effective price is now PX = $1 for this consumer. Compute the utility-maximizing levels of goods X and Y, along with the associated level of utility received by this individual. What is the total amount of the subsidy?

(b) Now assume that the government decides to simply pay the consumer the subsidy amount in the form of a cash transfer directly, thereby effectively increasing her money income, I, by this amount. Under these circumstances, compute the utility-maximizing levels of goods X and Y along with the level of utility received by this individual.

(c) Which government program (subsidy vs. cash transfer) would the individual prefer and why?

Explanation / Answer

MUx = 9+1 = 10 MUy = 18+2 = 20

a) Now if Px = 1 now budget constraint X + 4Y = 72

MUx/Px = MUy/Py

Y+1 = (X+2)/4

X+2 = 4Y+4

X-4Y = 2

solving budget constraint and above equation we get X = 36 Y =8.75 = 9

now associated level of utility for X and Y is 38 and 10 and total amount of subsidy is 36

b)Now let 2X+4Y = 72+36 = 108

MUx/Px = MUy/Py

(Y+1)/2 = (X+2)/4

X+2 = 2Y+2

X = 2Y

solving budget constraint and above equation we get X = 26 or 28 Y =14 or 13

associated utility of X and Y = 27 or 29 and 16 or 15

so total utility is maximum when X=28 and Y=13

c) Public will prefer cash transfer as total utility is more

U subsidy = 38*10 = 380

U transfer = 29*15 = 435

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