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Mariah consumes music downloads (*M*) and concert tickets (*C*). Her utility function is given by *U = 0.5M ^{2} + 2C^{2}*, where

**a. Write an equation for MRS _{MC} = **
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The marginal rate of substitution between two goods is the ratio of the two goods’ marginal utilities. So, *MRS*_{MC} = MU_{M}/MU_{C}. The problem gives the marginal utility of music downloads as *M*; the marginal utility of concert tickets is given as *4C*. So *MU*_{M}/MU_{C} = M/4C. For further review see section “Indifference Curves”.

Incorrect. Any two bundles that give a person the same level of utility must lie on the same indifference curve. When *M = 4* and *C = 1*, Mariah’s utility is *U = 0.5(4)*^{2} + 2(1)^{2}, or *10*. When *M = 2* and *C = 2*, Mariah’s utility is *U = 0.5(2)*^{2} + 2(2)^{2}, or *10*. Because both bundles give Mariah *10* units of utility, they must lie on the same indifference curve. For further review see section “Indifference Curves”.

Correct! Both bundles give Mariah *10* units of utility, so both bundles must lie on the same indifference curve. For further review see section “Indifference Curves”.

**c. Calculate MRS _{MC} when M = 4 and C = 1 and when M = 2 and C = 2.**

**When M = 4 and C = 1, MRS _{MC} = **
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**When M = 2 and C = 2, MRS _{MC} = **
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In part a, you found that *MRS*_{MC} = M/4C. To find the *MRS* for any particular bundle, simply plug in the number of units of *M* and *C* the bundle contains into the formula. When *M = 4* and *C = 1*, *MRS*_{MC} = (4)/4(1), or 1. When *M = 2* and *C = 2*, *MRS*_{MC} = (2)/4(2), or *0.25*. For further review see section “Indifference Curves”.

Incorrect. In part b, you found that the bundles (*M = 2* and *C = 2*) and (*M = 4* and *C = 1*) both lie on the same indifference curve. In part c, you found that when *M = 2* and *C = 2, MRS*_{MC} = 0.25; when *M = 4* and *C = 1, MRS*_{MC} = 1. Note that as *M* increases from *2* units to *4* units, *MRS*_{MC} increases. This means that as *M* increases, the indifference curve gets steeper instead of flatter. Therefore, this indifference curve is concave to the origin, not convex. For further review see section “Indifference Curves”.

Correct! Comparing the two bundles in part b, as *M* increases from *2* units to *4* units, *MRS*_{MC} increases from *0.25* to *1*. Because the *MRS* increases rather than diminishes, Mariah’s indifference curves must be concave rather than convex. For further review see section “Indifference Curves”.