1. [35 marks] An individual derives utility from consumption spending C and leisurel according to the following utility function: U(C,1)=C"1¹-a where 0>x>1. Leisure time in hours is given by: 1=T-H where T is hours of total time available and H is hours of work. The consumer's real income is given by: C=w (T-1)+N where w is real wage and N is real non-labour income. a) Derive the optimal values of C, I and the Lagrangian multiplier if the individual wishes maximise utility subject to the to constraint: C=w (T-1)+N. [15 marks]
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- 2. Utility maximization - Trading off labor and leisure The indifference curves on the following graph show Amy's preferences for leisure and consumption for increasing levels of utility, such that her utility increases in both consumption and leisure. Assuming that Amy spends 68 hours each week sleeping, she has a maximum of 100 hours available to her for leisure if she does not work at all. Initially, she works 45 hours (and thus has 55 hours of leisure) and earns $250 per week. Use the grey point (star symbol) to indicate Amy's initial leisure/consumption bundle. Dashed drop lines will automatically extend to both axes. ? CONSUMPTION (Dollars per week) 1000 900 800 700 600 500 400 300 200 100 0 0 10 20 30 40 50 60 70 80 LEISURE (Hours per week) 90 100 110 Initial bundle $200 Unemployment Benefits Equivalent Bundle, $100 Unemployment Benefits Equivalent Bundle Now, suppose that Amy is laid off, but she is eligible to collect unemployment benefits of $200 per week. Use the black point…The nearby graph shows Claudia's budget constraint for purchases of shoes and purses. Claudia's monthly budget for shoes and purses is $100. Claudia's parents have told her that she has too many shoes and is not allowed to buy more than 1 pair a month. If she buys 1 pair of shoes and 1 purse, she will have $_________ left each month. Number of 4.5 purses 4.0 3.5 3.0 2.5 O $50 O $10 O $75 O $25 2. A 1.5 1.0 0.5 0 0.2 0.4 0.6 0.8 1.0 1.2 Number of pairs of shoesQuestion 35 Suppose real estate analysts expect that 100,000 homes will be needed in a particular community by 2014. If the current number of homes in the community is only 50,000, we can expect to see a significant increase in the demand for investment. True O False
- 8. (Continuing with the situation described in the preceding question.) Consider the economic model of an individual's labor-leisure choice with the following components: good w represents the hourly wage rate Y represents nonlabor income T represents total time available U(C, L) represents the individual's utility function MUC denotes the marginal utility of consumption MUL denotes the marginal utility of leisure Unless otherwise instructed, assume that consumption and leisure are normal goods. Whenever graphing the model, put C on the vertical axis and L on the horizontal axis. 4 C represents units of the consumption good L represents hours of leisure The individual maximizes his utility by choosing the optimal consumption bundle C*=90, L*=15, and H*=9. At this consumption bundle, MUC=4 and MUL=12. If p=2, what is w? 8 H represents hours of paid work p represents the unit price of the consumption O 10 O There is not enough information to determine w.25 Columns 1 through 4 in the following table show the marginal utility, measured in utils, that Ricardo would get by purchasing various amounts of products A, B, C, and D. Column 5 shows the marginal utility Ricardo gets from saving. Assume that the prices of A, B, C, and D are, respectively, $9, $3, $2, and $6 and that Ricardo has an income of $71. Column 1 Column 2 Column 3 Column 4 Column 5 Units of Good A Units of Units of Units of MU MU MU MU Good B Good C Good D Number of Dollars Saved MU 1 72 1 24 1 15 1 36 1 5 2 54 2 15 2 12 2 30 2 4 3 45 3 12 3 8 3 24 3 3 4 36 4 9 4 7 4 18 4 2 5 27 5 7 5 5 5 13 5 1 6 18 6 5 6 4 6 7 6 0.50 7 15 7 2 7 3.5 7 4 7 0.25 8 12 8 1 8 3 8 2 8 0.13 Instructions: Enter your answers as a whole number. a. What quantities of A, B, C, and D will Ricardo purchase in maximizing his utility? 4 units of A 3 units of B units of C units of D b. How many dollars will Ricardo choose to save?3. [30+10] A consumer who consumes two goods and has the following utility function: U(x₁, x₂) = log(x₁ + 1) + log(x2 + 1), where ₁ and ₂ denote the monthly consumption of two goods. The consumer maximizes her utility subject to the following constraints: P₁x1 + P₂x2 ≤m, x₁ ≥ 0, X₂ ≥ 0, where p₁ > 0, p2 > 0 and m > 0. [Hint: Note that log f(x) = f(df (2).] 1_df f(x) dx (a) Write down the Kuhn-Tucker conditions for a point (x₁, x2) to solve the problem, and find all points that satisfy the conditions. Do any of those points solve the problem? (b) Show the above results by drawing indifference curves and budget set on the x₁x2 plane.
- (7 points) Suppose Backbeat BBQ, Dave Grohl's one-man catering company, offers two types of membership plans. Plan A has an entry fee of $20 with a price of $2 per unit. Plan B has an entry fee of $40 with a price of $1 per unit for using the service. Let x be expenditure on all other goods and y be the units of consumption of Backbeat BBQ's services, and suppose you have an income of $100. A. Write down the budget equation that you would have after you paid the entry fee for each of the two plans. B. If your utility function is xy, how much y would you choose in each case? C. Which plan would you prefer? Explain.2. Utility maximization - Trading off labor and leisure The indifference curves on the following graph show Amy's preferences for leisure and consumption for increasing levels of utility, such that her utility increases in both consumption and leisure. Assuming that Amy spends 68 hours each week sleeping, she has a maximum of 100 hours available to her for leisure if she does not work at all. Initially, she works 45 hours (and thus has 55 hours of leisure) and earns $250 per week. Use the grey point (star symbol) to indicate Amy's initial leisure/consumption bundle. Dashed drop lines will automatically extend to both axes. CONSUMPTION (Dollars per week) 1000 900 800 700 600 500 400 300 200 100 0 0 + 10 20 30 40 50 60 70 80 LEISURE (Hours per week) 90 U3 100 110 Initial bundle + $200 Unemployment Benefits Equivalent Bundle, $100 Unemployment Benefits Equivalent Bundle ?Economics Suppose that in the year 2015, Oceanaire, Inc. planned to produce 525,000 units of its lightweight scuba tanks. Of the 525,000 it planned to produce, a total of 35,000 units would be added to the inventory at its new plant in Arizona. Also assume that these units have been selling at a price of $250 each and that the price has been constant over time. Suppose further that this year the firm built a new plant for $4 million and acquired $3.5 million worth of equipment. It had no other investment projects, and to avoid complications, assume no depreciation. Now suppose that at the end of the year, Oceanaire had produced 525,000 units but had only sold 485,000 units and that inventories now contained 40,000 units more than they had at the beginning of the year. At $250 each, that means that the firm added $10,000,000 in new inventory. This year Oceanaire actually invested $. Oceanaire planned to invested $_? ?
- 11. (Originally #7 on Homework #3) Consider a consumer with the following information: At the optimal consumption bundle this consumer consumes 3 boxes of apples and 3 boxes of oranges. If the Marginal Rate of Substitution of Apples for Oranges is 6 at the optimal consumption bundle (MRSA0= 6), and the price of a box of Oranges is $1.80, what is the price of a box of Apples? Show how you found your answer.Principles of Economics 1 | S1 21/22 Time left 0:5) When the price of one good increases and the price of the other good and income are held constant, the budget line Select one: shifts parallel to the original budget line so that the new budget line is farther from the origin O b. shifts parallel to the original budget line so that the new budget line is closer to the origin rotates so that the intercept is farther from the origin on the axis representing the good that has experienced an increase in price O a. O c. O d. rotates so that the intercept is closer to the origin on the axis representing the good that has experienced an increase in price Next page5. (12) Max is stocking up on ice cream and cookies; ice cream costs $2 and cookies cost 50 cents ($.50). The marginal utility gained from each treat is given in the table below. If Max is a rational utility maximizer and has $8 to spend, how many of each treat will they purchase, and in what order? Show your work. Qice cream MU of ice Qcookies MU of cream cookies 1234567 40 1 9 32 2 9 26 3 7 18 4 5 8 5 3 6 6 3 4 7 1