1. (6 points) Suppose a consumer has a utility functionU(x 1 ,x 2 )=x 1100 x 2100 and faces a budget constraintp 1 x 1 +p 2 x 2 =M. Derive her demand functions for good 1 and good 2 . Verify that both functions are homogeneous of degree 0 in(p 1 ,p 2 ,M). 2. (6 points) Suppose a consumer has a utility functionU(x 1 ,x 2 )=(x 1 +x 2 ) 2 and faces a budget constraintp 1 x 1 +p 2 x 2 =M. Derive her demand functions for good 1 and good 2 .
Question
- (6 points) Suppose a consumer has a utility functionU(x 1 ,x 2 )=x 1100 x 2100 and faces a budget constraintp 1 x 1 +p 2 x 2 =M. Derive her demand functions for good 1 and good 2 . Verify that both functions are homogeneous of degree 0 in(p 1 ,p 2 ,M). 2. (6 points) Suppose a consumer has a utility functionU(x 1 ,x 2 )=(x 1 +x 2 ) 2 and faces a budget constraintp 1 x 1 +p 2 x 2 =M. Derive her demand functions for good 1 and good 2 .
Solution
- The consumer's utility function is U(x1, x2) = x1^0.1 * x2^0.1 and the budget constraint is p1x1 + p2x2 = M.
To derive the demand functions, we need to maximize the utility function subject to the budget constraint. This is a standard optimization problem that can be solved using the method of Lagrange multipliers.
The Lagrangian is L = x1^0.1 * x2^0.1 + λ(M - p1x1 - p2x2).
Taking the partial derivatives and setting them equal to zero gives the following system of equations:
0.1 * x1^-0.9 * x2^0.1 - λp1 = 0, 0.1 * x1^0.1 * x2^-0.9 - λp2 = 0, M - p1x1 - p2x2 = 0.
Solving this system gives the demand functions:
x1 = M/(2p1), x2 = M/(2p2).
These functions are homogeneous of degree 0 in (p1, p2, M) because if we multiply all the arguments by a positive constant, the value of the function does not change.
- The consumer's utility function is U(x1, x2) = (x1 + x2)^2 and the budget constraint is p1x1 + p2x2 = M.
To derive the demand functions, we need to maximize the utility function subject to the budget constraint. This is a standard optimization problem that can be solved using the method of Lagrange multipliers.
The Lagrangian is L = (x1 + x2)^2 + λ(M - p1x1 - p2x2).
Taking the partial derivatives and setting them equal to zero gives the following system of equations:
2*(x1 + x2) - λp1 = 0, 2*(x1 + x2) - λp2 = 0, M - p1x1 - p2x2 = 0.
Solving this system gives the demand functions:
x1 = M/(p1 + p2), x2 = M/(p1 + p2).
These functions are homogeneous of degree 0 in (p1, p2, M) because if we multiply all the arguments by a positive constant, the value of the function does not change.
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