Answer:
a) To calculate the marginal rate of substitution (MRS), we need to find the slope of the indifference curve. The utility function given is:
U(x1, x2) = (x1 + 2) (x2 + 10)
Taking the partial derivative of U with respect to x1 and x2 respectively, we get:
MUx1 = x2 + 10
MUx2 = x1 + 2
The MRS is the ratio of the marginal utilities, that is:
MRS = MUx1 / MUx2
Substituting the partial derivatives we found, we get:
MRS = (x2 + 10) / (x1 + 2)
b) The tangency condition is given by the equation:
MRS = P1 / P2
where P1 and P2 are the prices of goods x1 and x2 respectively. Substituting the MRS we found, we get:
(x2 + 10) / (x1 + 2) = P1 / P2
c) To find the demand functions for x1 and x2, we need to solve the tangency condition and the budget constraint:
P1 x1 + P2 x2 = m
where m is the consumer's income. Rearranging the tangency condition, we get:
x2 = (P2 / P1) (x1 + 2) - 10
Substituting this expression for x2 into the budget constraint, we get:
P1 x1 + P2 [(P2 / P1) (x1 + 2) - 10] = m
Simplifying and solving for x1, we get:
x1 = (m / P1) - (2 P2 / P1^2)
Substituting this expression for x1 into the expression we found for x2, we get:
x2 = (m / P2) - (2 P1 / P2^2) - 10
Therefore, the demand functions for x1 and x2 are:
x1(p1, p2, m) = (m / P1) - (2 P2 / P1^2)
x2(p1, p2, m) = (m / P2) - (2 P1 / P2^2) - 10
where p1, p2, and m are the prices of goods x1 and x2 and the consumer's income, respectively.
Step-by-step explanation:
hope this helps :))