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
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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