Optimal Consumption Bundle Analysis in Economics

Surname 1 Name Instructor Institution Course Date Accounting Essay a) The equation of Sarah's budget constraint can be written as follows: 4B + 5C = 100 where B is the quantity of bread and C is the quantity of chicken. The intercept values of the budget constraint are (25,0) and (0,20), which represent the maximum quantity of bread and chicken Sarah can purchase with her $100, respectively. The graph of the budget constraint is as follows: (b) Since Sarah is a utility maximizer, her optimal bundle will lie on the highest indifference curve that is still affordable given her budget constraint. Suppose her utility function is U(B,C) = B^(1/2) * C^(1/2).

Surname 2 At her current consumption of 10 loaves of bread and 12 pounds of chicken, her utility is U(10,12) = (10)^(1/2) * (12)^(1/2) = 34.64. We can find her optimal bundle by finding the combination of bread and chicken that maximizes her utility, subject to the budget constraint. At the optimal bundle, the slope of the indifference curve (MRS) should be equal to the slope of the budget constraint (Pc/Pb). The MRS can be found using the marginal utilities: MRS = MU(B)/MU(C) = (1/2) * (C/B)^(1/2) At the optimal bundle, this should be equal to Pc/Pb, which is 5/4. So we have: (1/2) * (C/B)^(1/2) = 5/4 Solving for C/B, we get: C/B = (5/4)^2 = 1.5625 This means that for every unit of bread she gives up, she should consume 1.5625 units of chicken in order to maximize her utility. We can use the budget constraint to find the affordable combinations of bread and chicken. 4B + 5C = 100 Solving for C, we get: C = (100 - 4B)/5 Substituting the expression for C/B into this equation, we get: (100 - 4B)/5B = 1.5625 Solving for B, we get: B = 16

Surname 3 Substituting this into the budget constraint, we get: 4(16) + 5C = 100 C = 8 So Sarah's optimal bundle is (16,8), which we label "E" on the graph: Her budget is not exhausted because she spends $80 on bread and chicken, which is less than her budget of $100. (c) Sarah's new budget constraint can be written as: 4B + 5C = 80 The intercept values of the new budget constraint are (20,0) and (0,16), which represent the maximum quantity of bread and chicken Sarah can purchase with her new budget of $80, respectively. The graph of the new budget constraint is as follows: (d) Following the change in income, Sarah cannot consume the same bundle E because it is no longer affordable given her new budget constraint. We can find her new optimal bundle by following the same procedure as before, but using the new budget constraint:

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Surname 4 4B + 5C = 80 (1/2) * (C/B)^(1/2) = 5/4 Solving for C/B, we get: C/B = (5/4)^2 = 1.5625 Substituting this into the budget constraint, we get: 4B + 5(1.5625)B = 80 B = 12 Substituting this into the budget constraint, we get: 4(12) + 5C = 80 C = 8 So Sarah's new optimal bundle is (12,8), which we label "E1" on the graph. (e) Following the decrease in her income, Sarah's total utility will likely decrease because she can no longer afford to purchase the same bundle that she previously did. However, it is possible that her total utility may increase if she is able to find a new optimal bundle that gives her higher utility per dollar spent. (f) Quantity MU(B) MU(C) 1 0.50 1.00 2 0.47 0.95 3 0.45 0.90 4 0.44 0.87 5 0.43 0.84 6 0.42 0.82 7 0.42 0.80 8 0.41 0.79 MU(B) represents the marginal utility of pineapple and MU(C) represents the marginal utility of green tea.

Surname 5 (g) To determine whether Archie is consuming the optimal bundle, we need to compare the marginal utilities of the two goods with their respective prices. The price of pineapple is $5 and the price of green tea is $2.50. At the current consumption bundle of 4 pineapples and 2 bottles of green tea, the marginal utility per dollar spent on pineapple is 0.44/5 = 0.088, and the marginal utility per dollar spent on green tea is 0.82/2.5 = 0.328. Since the marginal utility per dollar spent on green tea is higher than the marginal utility per dollar spent on pineapple, Archie should consume more green tea and less pineapple to maximize his utility. One possible optimal bundle would be to buy 2 pineapples and 4 bottles of green tea, where the marginal utility per dollar spent on both goods is equal: MU(B)/Pb = MU(C)/Pc 0.47/5 = 0.82/2.5 This bundle would give Archie a total utility of U(2,4) = (2^(1/3)) * (4^(1/3)) = 3.17.

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