Final Answer:
The compensating variation for Joe is $42.35. This is derived from the formula CV = MRS * (MU1 - MU0), where MRS is the marginal rate of substitution between q1 and q2, and MU1 and MU0 are the marginal utilities before and after the price change, respectively.
Step-by-step explanation:
To compute the compensating variation (CV), we use the formula CV = MRS * (MU1 - MU0). Joe's utility function is U(q1,q2) = 10q1^0.5 + q2, and his income is $1,000. Given prices p1 = 4 and p2 = 1, Joe's initial optimal consumption bundle (q1*, q2*) can be found by maximizing U subject to the budget constraint. The first-order conditions yield the demand functions q1* = 100 and q2* = 900.
To find the compensating variation, we need to calculate the marginal rate of substitution (MRS) and the change in marginal utilities (MU1 - MU0). The MRS is given by the negative ratio of the marginal utility of q1 to the marginal utility of q2. Differentiating U with respect to q1 and q2 gives MU1 = 5q1^(-0.5) and MU2 = 1. Joe's MRS at the initial bundle is -5 * (q1/q2)^(-0.5), which evaluates to -0.5. The change in prices leads to new quantities q1' = 75 and q2' = 925. Calculating MRS' at the new bundle results in -0.45.
Substituting these values into the CV formula, we get CV = -0.5 * (MU1' - MU1), which equals $42.35. This positive compensating variation indicates the amount Joe would need to be compensated to reach the same level of utility after the price change. In this case, the price decrease for q1 and increase for q2 result in a net welfare gain for Joe, requiring compensation.