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Let say the utility function as U(X,Y) = lnX + Y. Show that the marginal rate...

Let say the utility function as U(X,Y) = lnX + Y. Show that the marginal rate of substitution (MRS) is the same on all of the indifference curves at a given X. Explain and include graph. (5 Marks)

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The MRS would be as or or . For different indifference curve with different U's, for a given X=c, the MRS would be . The reason being that the MRS does not include Y, due to the functional form. If that was the case, we would have for a given X, ie Y would change directly proportional with U. Hence, the MRS is same for all indifference curve at given X. The graph is as below. As can be seen, all the slopes are same since the tangents are parallel to each other.

As can be seen, even for different indifference curves, the MRS is same for X=10.

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