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Understanding the budget constraint is crucial for making informed decisions about how to allocate limited resources. The budget constraint shows the combinations of goods that someone can purchase given their budget and the prices of these goods. In Chrissy's case, she spends her income on fishing lures and guitar picks.

The budget constraint can be expressed as the equation \( 2L + G = 30 \), where \( L \) and \( G \) are the quantities of lures and guitar picks, respectively. This equation ensures that Chrissy's spending does not exceed her budget of $30.

The coefficients before \( L \) and \( G \) represent the prices of lures and guitar picks. By understanding and setting up this constraint, Chrissy can determine how many items she can afford to buy without overspending.

To maximize utility, Chrissy needs to consider the Marginal Rate of Substitution (MRS). MRS indicates how many guitar picks she is willing to give up for one additional lure while keeping her utility level constant. This is especially useful when making decisions between two goods.

Mathematically, MRS is calculated using the ratio of the marginal utilities of the two goods: \[ MRS = \frac{MU_L}{MU_G} = \frac{0.5L^{-0.5}G^{0.5}}{0.5L^{0.5}G^{-0.5}} = \frac{G}{L}. \] This tells us that the MRS is equal to the current amount of guitar picks over lures. If the MRS differs from the actual price ratio of the goods, Chrissy should alter her consumption to increase utility.

The price ratio is a key factor in utility maximization, as it helps balance the trade-off between different goods. In Chrissy's scenario, the price ratio is \( \frac{Price\, of\, G}{Price\, of\, L} = \frac{1}{2} \).

For utility maximization, Chrissy needs the MRS to equal this price ratio, implying that she values the goods in proportion to their prices. At the optimal point, we have MRS \( = \frac{G}{L} = \frac{2}{1} \), meaning Chrissy should consume twice as many guitar picks as lures to maintain maximum utility with her current prices.

Understanding this balance is crucial as it indicates whether she should buy more of one good and less of another if their prices change.

When prices change, maintaining the same level of utility often requires an income adjustment. After the price of guitar picks doubles to \(2, Chrissy's budget constraint shifts, now expressed as \( 2L + 2G = I \). To maintain her utility level of 3.87, Chrissy should solve this new equation.

By using the condition \( G = 2L \) (from the MRS and price ratio matching), the constraint becomes \( 2L + 4L = I \), simplifying to \( 6L = I \). With \( L = 7.5 \), Chrissy finds that she needs an income of \)45.

This adjustment ensures that Chrissy can continue buying the same quantities of goods and enjoy the same satisfaction level, despite the price hike.