Reuben Gronau s Model of Time Allocation and Home Production

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1 Econ 301: Topics in Microeconomics Sanjaya DeSilva, Bard College, Spring 2008 Reuben Gronau s Model of Time Allocation and Home Production Gronau s model is a fairly simple extension of Becker s framework. This is a good example of how economists add to the literature by incrementally tweaking the existing models by relaxing assumptions to obtain new insights. Gronau s point is that not everything we do in the household can be considered a utility-generating activity. If you recall, Becker s formulation, the individual did two activities like playing games and eating in the household. Both these activities provide utility to this individual. Both activities needed time and goods, and the goods were purchased in the market. In order to purchase the goods, the individual gave up some time he could have used for playing and eating, and spent some time working to earn the money to buy the goods he needs for eating and playing. This market work did not bring him any utility. Gronau argues that the dichotomy between working in the market, and playing at home is unrealistic, particularly in the case of women and households in less developed economies where a large amount of time at home is spent on productive activities that do not generate direct utility. For example, we generate utility from eating but perhaps not from cooking or cleaning the pots and pans. We derive utility from watching our children grow, but not from changing diapers and cleaning after them. Many of the activities performed at home are just like what we do at work, productive activities that generate goods and services we like, but do not generate utility directly. One could think of these as chores. Going back to the example of eating, we engage in the utility generating activity of eating using cooked food and the time spent eating. The cooked food, however, can be purchased in the market using our earnings from market work, or can be produced at home using our time. We 1

2 don t necessarily generate utility directly from this act of cooking (except when some of us cook for fun!), but the home produced good is then used in a utility generating activity of eating. Gronau builds a simple model to illustrate this. Suppose we derive utility from an activity Z (e.g. eating, parenting, watching movies) that is produced with goods and time. U = U(Z(X, t Z ) (1) Here, it is assumed that U > 0, U < 0. Recall that Becker s model had two activities in the utility function. We can easily include another activity here, but don t need to do so to tell that story that Gronau wants to tell. In building model, we need to be parsimonious, i.e. use the simplest possible model to clearly illustrate an argument. The good X can be purchased in the market or produced at home. X = X M + X H (2) Here, by writing a linear function, Gronau assumes that the market produced good and the home produced good are perfect substitutes. Moreover, he assumes that one unit of the market produced good generates the same amount of the activity as one unit of the home produced good. For example, for activity of eating, we could bring take-out food or use home cooked food that are considered exactly equivalent perfect substitutes. For the activity of child rearing, we could use a home made toy or buy a plastic toy from the market. Again, the two goods are considered perfect substitutes. The production function for the home produced good is X H = f(t H ) (3) where f > 0, f < 0 ensures concavity (i.e. diminishing marginal product). Notice also that Gronau include only time in the home production of the good. In reality, we probably need some other inputs as well. It this is food, we can think of land as a 2

3 fixed input that is an endowment of the household. Because there is a fixed amount of land, there is diminishing returns to the time spent growing food at home. The budget and time constraints are P X M = wl + V (4) L + t H + t Z = T (5) Combining these two constraints as usual, we get the full income constraint, P X M + w(t H + t Z ) = wt + V (6) Going back to the utility function, note that U is monotonically increasing in Z, maximizing U is the same as maximizing Z. Note that the additive nature of the X function leads to the possibility of corner solutions, X M = 0 or X H = 0. With the reasonable assumption that f(0) = 0, the second constraint is the same as t H = 0. We need to add the nonnegativity constraints, and consider the three possibilities separately. Therefore constrained maximization problem can be expressed as the following Lagrangian, max Z(X M + f(t H ), t z ) + λ 1 (wt + V P X M w(t H + t Z )) + λ 2 X M + λ 3 X H (7) t H,t Z,X M If X M > 0 and X H, t H > 0, we have the full interior solution with the following FOC s. X = λ 1P X f (t H ) = λ 1 w = λ 1 w P X M + w(t H + t Z ) = wt + V We can manipulate the FOC s to get some interesting insights. Dividing the first FOC by the second, we get f (t H ) = w P 3 (8)

4 The household will allocate time to home production until the marginal product of the home produced good is equal to the real wage (or the marginal cost of the market produced good). Graphically, the optimum amount of home production is where the home production function is tangent to a straight line with the slope equal to the real wage. Dividing the first FOC by the third, we get X = X Z = w P This condition tells us that the optimal amount of goods and time consumed by the household for the activity Z is found at the point where the TRS in the activity production function is equal to real wage. Just like in Becker s generalized model, we allocate time and goods to the activity based on the technical trade-off (TRS) and the market trade-off (relative price). These two results together demonstrates one of most important assertions in neoclassical theory that a household s consumption and production decisions are separable when labor markets operate. Here, the household first decides how much time and goods it should allocate to its consumption activity, and then separately decides how much of this good should be produced at home vis-a-vis purchased in the market. The first decision depends on the Z(.) function but not the f(.) function, and the second decision depends on the f(.) function but not the Z(.) function. Both decisions are determined, however, by the relative factor prices. We will see shortly that the separation property does not hold when the household does not supply any labor to the labor market. If X M = 0, the FOC s are λ 2 = λ 1 P X f (t H ) = λ 1 w = λ 1 w X M = 0 w(t H + t Z ) = wt + V 4

5 Here, the first FOC is useless. When we divide the second FOC by the third, we get a somewhat different conclusion on how time is allocated by a household that does not participate in the labor market. X = X Z = f (t H ) This result is completely different from the interior solution. Here, we see that the time allocation between home production and consumption is found at the point where the TRS of the consumption activity is equal to the marginal product of home production. Now, any change in the home production function will influence the household s consumption choices, and any change in the consumption technology will influence the home production choices. For example, if the leisure activities become less time intensive, home production time may increase. Similarly, if the home production technology of improves, more time will be freed up and allocated to leisure activities. This stands in contrast to the previous example with households that supply labor to the market, all the time that is freed up at home will be allocated to the market rather than to the other home-based activity. If X H, t H = 0, the FOC s are X = λ 1P λ 3 = λ 1 w = λ 1 w t H = X H = 0 P X M + wt Z = wt + V Here, the second FOC is useless. From the first and the third FOCs, we get an analogous condition to what we saw in the first case where the allocation of market purchased goods and time to the activity Z is determined simply by equating the slope of the budget constraint, i.e. the real wage, to the TRS of the consumption activity. Because there is no home production here, this result is the same as what we found in Becker s time allocation model. 5

6 By assuming reasonable functional forms for Z and X H, we can obtain explicit closed-form solutions to this problem. Let s assume that Z = X α t β Z X H = ln(t H ) Question: Using these functional forms, find closed form solutions for time allocation to market work, home production and consumption activity. Do comparative static exercises for each case. Question: Examine the conditions under which the two corner solutions and the interior solution hold. Demonstrate that for X H = 0, the real wage must be higher than some number, and for X M = 0, the real wage must be lower than some number. Both these numbers should be expressed in terms of the parameters of the model. 6