We are going to imagine that a worker has the preferences U(c,l) =a(c)+'v[l) (9.1.1) where c stands for consumption and 5 stands for leisure. The function u. (c) describes how much the worker enjoys consumption and the function 1: {4!} describes how much the worker enjoys ded- icating time to non-market activities [we call these \"leisure\" but they could include non-market production such as doing laundry}. We are going to imagine that both 1:: (c) and \"U (l) are concave functions. This means that the household experiences diminishing marginal utility of both leisure and consumption. The worker has a total of one unit of time, so the amount of time he spends working is given by L = 1 l Youill sometimes see preferences over consumption and leisure expressed in terms of disutility from working rather utility from leisure? with a function of the form: UieL) = MC) - Z{L) Setting 2 (L) = o (1 L) makes the two formulations exactly equivalent. We'll stick to expression {9.1.1}. The worker has to decide how much of his time to dedicate to market work and how much to dedicate to leisure. One way to interpret this decision is literally: imagine that the worker has a job that allows him to choose how manj,r hours to work (for instance, the worker is an Uber driver} and think about how the worker makes this choice. More broadly, there are many decisions that involve trading of f higher income against less leisure: choosing between a full-time job and a part-time job; choosing between a high-stress, highly paid job and a lower-paid, more relaxed job; choosing at what age to retire; choosing how many members of a many-person household will be working in the market sector, etc. We can think about the choice of \"leisure" as sumnlarizing all of these decisions. The worker gets paid a wage in per unit of time, so the total amount he can spend on con- sumption goods is given by the budget: c :1 w (1 i :1 The worker solves the following problem: Ingaxu (c) + a (l) ' (9.1.2) at. cgwi) This is a two-good consumption problem like the ones you know from nlicroeconomics. The two goods here are time and consumption. The only thing to keep in mind is that the household is initially endowed with one unit of time, and it has to choose how much of it to sell in order to buy consumption. Figure [9.1.1:] shows the solution to problem (9.1.2). The worker will choose the highest indif- ference curve he can afford, which implies that it will pick a point where the indifference curve is tangent to the budget constraint. For any wage, the budget constraint always goes through the point [1, U): the worker can always choose to enjoy its entire endowment of time in the form of leisure and consume zero. The slope of the budget constraint is 'w: w is the relative price of time in terms of consumption goods. When to is high, time is expensive relative to goods, so the budget constraint becomes steeper. We can also find the solution to problem {9.1.2} through its rst order conditions. The La- grangian is:l L(c,l,)\\)=u{c)+u{l)A[cm(ll)] 1This problem is sufficiently simple that we don't need to use a Lagrangian to solve it. We could just as easin replace c: = w {1 E} into the objective function and solve which gives us the same solution. 9.1. STATIC MODEL The first order conditions are: maxu(w(1l))+u(l) 153 Updated osmuame CHAPTER 9. CONSUMPTION AND LEISURE \"b" (i) = to (9.1.3) Equation [9.1.3] describes how the worker trades of f dedicating time to market work or to leisure. If the worker allocates a unit of time to leisure, he simply enjoys the marginal utility of leisure o\" (l). [f instead the worker spends that time at work, he earns 10, so he is able to increase his consumption by 21:; this gives him to times the marginal utility of consumption u' (c). At the margin, the worker must be indifferent between allocating the last (innitesimal) unit of time between these two alternatives, so {9.1.3} must hold. (9.1.3) is also an algebraic representation of the tangency condition shown in Figure (9.1.1). The slope of the indifference curve is given by the marginal rate of substitution between leisure and consumption: 3%. The slope of the budget constraint is to, so (9.1.3) says that the two are equated. The effect of wage changes Letis imagine the wage to changes. How does the worker change his choice of leisure and consump- tion? The answer to this question is going to play an important role in some of the models of the entire economy that we'll analyze later. For now, we are going to study the question in isolation, just looking at the response of an individual worker to an exogenous change in the wage. For concreteness, letis imagine that the wage rises. Let's take a first look at this question graphically. A change in wages can be represented by a change in the budget constraint, as in Figure 9.1.2. The new budget constraint still crosses the point (1, 0), but the slope of the budget constraint is steeper. As with any change in prices, this can have both income and substitution effects. The substitution effect is straightforward: a higher wage means that time is more expensive. Other things being equal, this would make the worker substitute away from leisure (which has become relatively expensive) towards consumption [which has become relatively cheap). This makes the worker work more. In addition, the higher wage unambiguoust helps the worker: the worker is selling his time so a higher price is good for him. In other words, there is a positive income effect. For the consumption choice, the income effect reinforces the substitution effect since both push the worker to consume more. For the leisure choice, this goes in the opposite direction as the substitution effect, as the worker becomes richer, he wants more of everything, including leisure. In the example depicted in Figure 9.1.2, the substitution effect dominates. The worker ends up at a point to the left of where he started, showing he has decided to work more (and get less leisure) when wages rise, but this could easin go the other way. Figure 9.1.3 shows an example where the income effect dominates so the worker decides to work less {enjoy more leisure) when the wage rises. An explicit example Suppose that the utility function takes the following form: 510" I'L[:c}=1cr SE 1_+e 3(a)=1+(1s) where I9 and E are parameters. For this case, we can get an explicit formula for how much con- sumption and leisure the worker will choose. Marginal utlility of consumption and leisure are, 156 Updated osmuams 9.1. STATIC MODEL CHAPTER 9. CONSUMPTION AND LEISURE respectively: 1;" (c) = c\"r so replacing in equation (9.1.3) we get Using the budget constraint to replace c we get: a [1 r)? _ w (10(1 - UT" 1 .E = 49me (9.1.4) Equation (9.1.4) gives us an explicit formula for how much the worker will choose to work depending on the wage and the parameters in the utility function. Will this worker work more or less when the wage is higher? Mathematically, this depends on whether the exponent on 1U is positive or negative. If J 1. Solve for the hours worked and consumption under this new assumption. Compare your answer with the previous part (where (,0 = 1). Does the consumer wants to work more or less? 3. (5 points)[modemte] The government fears that a virus will spread in the population, so it decides to limit the number of hours the consumer can work. We model this by assuming that the maximum hours worked can be C, with 0