An individual utility function is given by U(x,y) = x·y. Let I = $800, px = $20 and py = $40. Suppose that a tax t of $10 per unit is imposed on good x. Assume that the full burden of this excise tax is borne by this consumer, i.e. the new price this individual faces for good x is p’x = px + t = $20+ $10 = $30. With the tax, this person chooses X* = 13.33 (you may want to use the demand/optimal purchase equation for x you derived earlier to check on this result). The total amount of tax collected is thus T = t·x*= ($10)13.33 = $133.33. Therefore, an income tax that would generate the same tax revenue would reduce this individual net income to I’ = $800 - $133.33 = $666.67. Use the indirect utility function you derived earlier to compute this person new utility level with the income tax. It is equal to ??? how many utils (this should create a improved 133.33 utils that was derived from the excise tax but still less than 200 utils).
1) An individual utility function is given by U(x,y) = x·y. Derive this individual indirect utility function. Using this individual indirect utility function, compute her level of utility when I = $800, px = $20 and py = $40. It is equal to 200 utils.
2) An individual utility function is given by U(x,y) = x·y. Let I = $800, px = $20 and py = $40. Suppose that a tax t of $10 per unit is imposed on good x. Assume that the full burden of this excise tax is borne by this consumer, i.e. the new price this individual faces for good x is p’x = px + t = $20+ $10 = $30. Use the indirect utility function you derived earlier to compute this person new utility level with this excise tax. It is equal to 133.33 utils.
Question: An individual utility function is given by U(x,y) = x·y. Let I = $800, px = $20 and py = $40. Suppose that a tax t of $10 per unit is imposed on good x. Assume that the full burden of this excise tax is borne by this consumer, i.e. the new price this individual faces for good x is p’x = px + t = $20+ $10 = $30. With the tax, this person chooses X* = 13.33 (you may want to use the demand/optimal purchase equation for x you derived earlier to check on this result). The total amount of tax collected is thus T = t·x*= ($10)13.33 = $133.33. Therefore, an income tax that would generate the same tax revenue would reduce this individual net income to I’ = $800 - $133.33 = $666.67. Use the indirect utility function you derived earlier to compute this person new utility level with the income tax. It is equal to ??? how many utils (this should create a improved 133.33 utils that was derived from the excise tax but still less than 200 utils).
To solve the given problem, determine the demand function and indirectly utility function.
Lagrange function,
L = U(x,y) +λ (I – px*x - py*y)
L = x y + λ (I – px*x - py*y)
Determined the first-order condition and put equal to zero.
dL/dx = y – λ px = 0
λ = y/px ..... (1)
dL/dy = x – λ py = 0
λ= x/px ..... (2)
dL/d λ = I – px*x - py*y
I = px*x + py*y ...... (3)
Solving 1 and 2,
y/px = x/py
y = px*x/py ...... (4)
Put this value of y in equation 3.
I = px*x + py* (px*x/py )
I = px*x + px*x
I = 2px*x
x*= I/2px
Put the value of x in 4.
y =(px/py )* ( I/2px)
y* = I/2py
The indirect utility function,
U(x,y) = x y
U(x*,y*)= (I/2px )* (I/2py)
U* = I^2/ 4px*py
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