Julio receives utility from consuming food ( \(F\) ) and clothing \((C)\) as given by the utility function \(U(F, C)=F C\) In addition, the price of food is \(\$ 2\) per unit, the price of clothing is \(\$ 10\) per unit, and Julio's weekly income is \$50. a. What is Julio's marginal rate of substitution of food for clothing when utility is maximized? Explain. b. Suppose instead that Julio is consuming a bundle with more food and less clothing than his utility maximizing bundle. Would his marginal rate of substitution of food for clothing be greater than or less than your answer in part a? Explain.

Utility maximization is a fundamental concept in microeconomics, referring to the behavior of consumers aiming to get the highest level of satisfaction or 'utility' from their consumption choices within the limits of their budget. When consumers make decisions about how to spend their money, they typically try to choose the combination of goods and services that will maximize their happiness or satisfaction.

To quantify this intuitive idea, economists use mathematical functions called 'utility functions' which represent preferences for combinations of different goods. Consumers reach utility maximization when they cannot increase their utility by changing the consumption bundle, given their budget. This state is found where the consumer's budget constraint is tangent to an indifference curve, representing the trade-offs a consumer is willing to make between two goods. In other words, the point where the consumer has balanced the marginal utility per dollar spent across all goods.

The budget constraint in economics outlines the combinations of goods and services a consumer can purchase with a given income at specific prices. It serves as a boundary that limits a consumer's consumption choices. Represented graphically, the budget line is downward sloping, indicating that if a consumer spends more on one good, they must spend less on another to stay within their budget.

Formally, if a consumer has an income 'I,' and the prices of two goods are 'P1' and 'P2,' the budget constraint is given by the equation 'P1X1 + P2X2 = I', where 'X1' and 'X2' represent quantities of the two goods. The slope of the budget line reflects the trade-offs and is determined by the price ratio of the two goods. The consumer’s goal is to find the highest utility they can achieve without crossing this budget line.

Marginal utility is the additional satisfaction or utility that a consumer receives from consuming one more unit of a good or service. It can be thought of as the 'extra value' a consumer gets from an additional unit. In economic theory, as one consumes more units of a good, the marginal utility of additional units typically decreases—a phenomenon known as the law of diminishing marginal utility.

Mathematically, marginal utility is the derivative of the utility function with respect to the quantity of the good consumed. In the context of the problem, where the utility function is given by 'U(F, C) = F * C', the marginal utility of food, 'MU_F', is the derivative of the utility function with respect to 'F', which is 'C'. Similarly, the marginal utility of clothing, 'MU_C', is the derivative with respect to 'C', which is 'F'. Consumers maximize utility by equating the ratio of marginal utilities of two goods to the ratio of their prices.

The Cobb-Douglas utility function is a specific type of utility function widely used in economics to represent consumer preferences. It takes the form 'U(X, Y) = X^a * Y^b', where 'X' and 'Y' are the quantities of two different goods consumed, and 'a' and 'b' are parameters that reflect the relative importance of each good for the consumer.

The function implies that the goods are perfect substitutes to some degree and that the marginal rate of substitution is diminishing. This property means that as a consumer has more of 'X' and less of 'Y', they are willing to give up less of 'X' for an additional unit of 'Y'. It also results in indifference curves that are convex to the origin, which shows the trade-off between 'X' and 'Y' at different levels of utility. The Cobb-Douglas form allows for constant returns to scale; if the quantities of all goods are multiplied by a common factor, total utility is multiplied by the same factor.