Week 3: Consumer and Firm Behaviour: The Work-Leisure Decision and Profit Maximization

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1 AROEOOIS 2006 Week 3: onsumer and Firm Behaviour: The Work-Leisure Decision and Profit aximization Questions for Review 1. How are a consumer s preferences over goods represented? By utiity functions: U(,L), h. 4, p What three properties do the preferences of the representative consumer have? Expain the importance of each. ore is preferred to ess (non-satiation assumption) onsumer ikes diversity in her own consumption unde onsumption and eisure are norma goods h. 4, p What two properties do indifference curves have? How are these properties associated with the properties of the consumer s preferences? Downward soping (due to the non-satiation assumption) onvexity (due to the diversity assumption) h. 4, p Why is the margina product of aour diminishing? Short-run: fixed input (capita) As more and more amounts of aour is introduced, these new aour has to share the existing fixed capita (Law of diminishing margina product) h. 4, p Proems 1. Last week, we showed an exampe where the consumer has preferences for consumption with the perfect compements property. Suppose, aternativey, that eisure and consumption goods are perfect sustitutes. In this case, an indifference curve is descried y the euation u = a + where a and are positive constants, and u is the eve of utiity. That is, a given indifference curve has a particuar vaue for u, with higher indifference curves having higher vaues for u. a. Show what the consumer s indifference curves ook ike when consumption and eisure are perfect sustitutes, and determine graphicay and ageraicay what consumption unde the consumer wi choose. Show that the consumption unde the consumer chooses depends on the reationship etween a and w, and expain why. 1

2 u = a + : Indifference urve: = u a sope of the indifference curve = Budget onstraint: = w( h ) + π T sope of the udget ine = w a orner soution Use reativey cheaper good (since they are perfect sustitutes) ompare w (reative price of eisure to consumption good) with RS L,. o w>rs sope of B > sope of I choose o w<rs sope of I > sope of B choose o w=rs sope og B = sope of I choose any point on the indifference curve that touches the udget ine.. Do you think it ikey that any consumer woud treat consumption goods and eisure as perfect sustitutes? o! We assumed that the consumer has a preference for diversity in her own consumption unde. c. Given perfect sustitutes, is more preferred to ess? Do preferences satisfy the diminishing margina rate of sustitution property? ore is sti preferred to ess We sti choose the highest possie indifference curve that touches the udget ine. Preferences do not satisfy diminishing RS RS is constant aong the indifference curve 2

3 2. Suppose that a consumer can earn a higher wage rate for working overtime. That is, for the first hours the consumer works, he or she receives a rea wage rate of w 1, and for hours worked more than he or she receives w 2, where w 2 > w 1. Suppose that the consumer pays no taxes and receives no non-wage income, and he or she is free to choose hours of work. a. Draw the consumer s udget constraint, and show his or her optima choice of consumption and eisure. Budget constraint: Since π T = 0 = w( h ) if h-< then = w1 ( h ) with sope= -w1 if h-> then = w ( h ) with sope =-w 2 2 This consumer works overtime whie this one does not h- h-. Show that the consumer woud never work hours, or anything very cose to hours. Expain the intuition ehind this. K For the consumer to chose point, we need RS, to e eua to the wage rate (either w 1 or w 2 ). But, if the consumer has convex indifference curves (a preference for diversity), then she aways has a higher attainae indifference curve ecause of the inwards kink. K 3

4 The kinked udget ine is composed of two ines K and, which have different sopes. Whether the consumer has ue of red indifference curves aove (with different RS at point ) point is aways a corner soution which the consumer wi not prefer. The ony case there the kink point coud e a soution is if, were perfect compements! c. Determine what happens if the overtime wage rate w 2 increases. Expain your resuts in terms of income and sustitution effects. You wi need to consider the case of a worker who initiay works overtime, and a worker who initiay does not work overtime. A worker who initiay works overtime wi not switch to not overtime portion ecause that woud not e consistent his previous choice of choosing overtime at previous ower overtime wage rate. She wi increase her consumption for sure (consumption is a norma good). The change in eisure depends on the income and sustitution effect. A worker who initiay does not work overtime may or may not shift overtime working depending on the shape of its RS. Red one does not shift to overtime region (nothing changes) Green one shifts to overtime region eisure decrease (he sustituted for more consumption good) consumption increase (works more, has higher income, consumption is a norma good) 4

5 3. Suppose a firm has a production function given y Y = zk 0. a) If z=1 and K=1, graph the production function. Is the margina product of aour positive and diminishing? With z=1 and K=1, the production function ecomes: Y= 0.7. Y P P = = 0.7 > 0 and 1.3 = 0.21 < 0 So, the P is positive and diminishing Y Y=2 0.7 Y= 0.7 ) ow graph the production function when z=2 and K=1. Expain how the production function changed from part a). For z=2 and K=1, the production function ecomes: Y=2 0.7 and is graphed in the aove diagram. ompared to part a) the production function shifts upwards and the sope increases as we. c) Given this production function graph the margina product of aour for (z, K)=(1,1), (2,1), (1,2) and expain what you get. i. (z,k)=(1,1): Y= 0.7 and ii. (z,k)=(2,1): Y=2 0.7 and P iii. (z,k)=(1,2): Y= and P = P = 0.7 = 1.4 P

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