In the US, households with income before taxes between $40,000 and $49,999 spend about 12.5% of their net income on food. Two thirds of this expenditure is spent on food consumed at home while one third goes towards food consumed away from home. We are interested in nding a utility function that is consistent with this behavior. (U S Bureau of Labor Statistics. Consumer Expenditure Survey (CEX), Table 1203). Let's assume that Such a household has preferences for food eaten at home(x) and food consumed away from home (y) summarized by the utility function U(x,y) = x2/3y1/3. 1. Write the expression of the consumer's marginal utility from food consumed at home (MUx) and from food consumed away from home (MU y). Is MUx decreasing in x? Is MUy decreasing in y. What is the economic interpretation of these mathematical results? 2, Write the expression of the consumer's Marginal Rate of Substitution (MRS). What is the economic interpretation of this rate? Now, assume that the consumer's yearly food budget is M = $6,000 and that the average cost of a meal consumed at home is Px = $5 while the average price of a meal consumed away from home is $10. 3. Write the expression of the consumer's budget constraint. What is the opportunity cost of a meal in terms of meals consumed out of the house? 4. In an X/Y diagram draw the consumer's budget constraint. 5. Using the Lagrangian method, nd the consumer's optimal bundle of meals consumed at home and meals consumed away from home. 5.1 Write the consumer's Utility Maximization Problem (UMP). 5.2 Write the expression of the consumer's Lagrangian function. 5.3 Write the rst order conditions for the Lagrangian (rst derivative test. . .don't forget to also take the rst derivative with respect to the Lagrangian multiplier). 5.4 Solve the rst order conditions for x*, y*, and N". 6. To your diagram, add an indifference curve to illustrate the consumer's optimal bundle. 7. How much money does the household spend on food consumed at home? How much money does the household spend on food consumer away from home? 8. Does the suggested utility function capture the observed fact that this type of family spends 2/3 of its food budget on food consumed at home and 1/ 3 on food consumed away from home? What relationship do you notice between these expenditure shares and the coefcients of x and y in the utility function? Households with income before taxes between $100,000 and $149,999 spend 10% of their net income on food. Of this total expenditure 60% is spent on food consumed at home while 40% is spent on food consumed away from home. 9. Given the previous results, what utility function could you use to represent their preferences for food consumed at home and food consumed away from home