1.. Suppose that U(x; y) = min(x; y) with px = 1 and py = 1. Describe and
illustrate the income and substitution effects of an increase in the price of
good y. What does this imply about a tax imposed on good y..
2.. Let U(x; y) = 5x:8y:2 showing all derivation work, find:
(a) the Marshallian demand functions for x and y
(b) the Indirect Utility Function
(c) the compensated demand functions xc and yc
3….. Suppose that Timmy just graduated from college and has two job offers in two distinct
cities. Timmy gains utility from only the consumption of goods x and y and has a utility
function U(x; y) = . In City A, Timmy would earn $50 and the prices of x and
y are $42 and $12, respectively. In City B, Timmy would earn $40 and the prices of x
and y are $32 and $8, respectively. Will Timmy take the job in City A or City B and
how much utility would he gain in each city
none of the above
1. To describe and illustrate the income and substitution effects of an increase in the price of good y, we can use the concept of indifference curves.
- Income Effect: When the price of good y increases, the consumer's purchasing power decreases. This means that the consumer's real income is reduced, as they can now afford less of both goods x and y. The income effect depends on the income elasticity of each good. In this case, since the prices of x and y are the same (px = py = 1), the income effect is equal for both goods.
- Substitution Effect: The substitution effect occurs when the consumer substitutes the now relatively more expensive good (y) for the relatively cheaper good (x). In this case, since U(x, y) = min(x, y), the consumer will choose the minimum of x and y to maximize their utility. When the price of y increases, the consumer will substitute away from y and choose more of x.
Implication of a tax imposed on good y: A tax on good y would effectively increase its price. The income and substitution effects described above would still apply. The specific implications of the tax would depend on the amount of the tax and the relative magnitude of the income and substitution effects.
2. To find the required functions, we will need to calculate the derivatives of the utility function:
U(x, y) = 5x^0.8y^0.2
(a) Marshallian Demand Functions:
To find the Marshallian demand functions, we differentiate the utility function with respect to x and y:
∂U/∂x = 4x^(-0.2)y^0.2
∂U/∂y = x^0.8y^(-0.8)
To find the Marshallian demand functions, set these derivatives equal to the respective prices of x and y:
4x^(-0.2)y^0.2 = px
x^0.8y^(-0.8) = py
Solve these equations simultaneously for x and y to obtain the Marshallian demand functions.
(b) Indirect Utility Function:
The indirect utility function represents the maximum utility that can be obtained given the prices of x and y and a fixed income. It is found by plugging the optimal values of x and y from the Marshallian demand functions into the utility function:
V(p, I) = U(x(p,I), y(p,I))
(c) Compensated Demand Functions:
The compensated demand functions represent the demand for x and y when the consumer's income is adjusted to keep utility constant after a price change. You can find these functions by solving the utility maximization problem with a given utility level and prices. The resulting demand functions are the compensated demand functions.
3. To determine which job Timmy will choose and the utility he would gain in each city, we need to compare the utility levels in each city.
Utility in City A:
U(x, y) = x^0.5y^0.5
Given the prices in City A, the budget constraint is:
42x + 12y = 50
Solve this budget constraint along with the utility function to find the optimal x and y values. Then substitute these values back into the utility function to find the utility level in City A.
Utility in City B:
U(x, y) = x^0.5y^0.5
Given the prices in City B, the budget constraint is:
32x + 8y = 40
Solve this budget constraint along with the utility function to find the optimal x and y values. Then substitute these values back into the utility function to find the utility level in City B.
Compare the utility levels in each city to determine which job Timmy will choose.