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The Budget Constraint Here we explore the combinations of consumption and leisure the individual can obtain

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Budget constraint The budget constraint for an individual shows combinations of consumption and leisure that can be attained. The constraint will be a line in a diagram where we put the hours of leisure on the horizontal axis and the dollar amount of consumption on the vertical axis. When an individual takes more leisure here we are assuming that less time is spent as labor.

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**initial assumptions Initially let’s assume**

1) the individual gets the same wage for every hour of work, 2) we are looking at a 24 hour day and the day is spent either working of enjoying leisure (where leisure is all those things we do when not working), and 3) the dollar amount of consumption we can have only comes about by earning income first in the labor market and so C = wage times hours worked. Since total hours = 24 = hours worked + hours in leisure (L), then hours worked = 24 – L and C = w(24 – L) = 24w – wL.

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**This budget line consumption is drawn here assuming**

Budget constraint This budget line is drawn here assuming the wage rate is $1. In one day, one could have 24 hours of leisure and no consumption, or no leisure and $24 consumption, or any other combo on the line. consumption (0, 24) leisure (24, 0) Here C = 24 – L.

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**the negative of the wage rate. For every unit of leisure **

Budget constraint consumption Note how the slope of the budget line is the negative of the wage rate. For every unit of leisure given up, $1 would be earned and used as consumption. slope=(change in con) (change in leis.) =1/-1= -1 = -wage. (0, 24) leisure (24, 0)

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**axis now has to indicate $48 in consumption if one worked all day. **

(0, 48) Budget constraint consumption If the wage should rise, to say $2 per hour, the budget constraint rotates clockwise. The point on the vertical axis now has to indicate $48 in consumption if one worked all day. The slope of the new budget is - $2. (0, 24) leisure (24, 0)

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**consumption (0, 24) nonlabor income leisure (24, 0)**

Budget constraint consumption (0, 24) nonlabor income leisure (24, 0) If the individual has nonlabor income called V, the budget line shifts up by the amount of nonlabor income. The slope remains as was - remember the slope is the negative of the wage rate.

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**Let’s come up with the equation for the budget constraint.**

The dollar volume of consumption – C - we can undertake is made up of our income from work and any nonlabor income we have. If the wage is w, h is the number of hours we work, and the nonlabor income is V, then C = wh + V.

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**C = wh + V and with T = h + L, or h = T – L,**

C = w(T – L) + V = wT – wL + V = wT + V - wL Say T = 24 hour day, then the amount spent in work is 24 minus the time spent in leisure (L). So, C = w(24 - L) + V = 24w + V – wL. Note: 1) If L=0, meaning all time is spent working, C = 24w + V. 2) If L = 24, meaning all time spent in leisure, C = V. 3) If V = 0, meaning no nonlabor income, C = w(24-L) = 24w –wL.

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**Note here that when the nonlabor income changed the budget shifted in a parallel fashion.**

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The budget line represents the most of consumption and leisure the individual can get. The individual can be inside the budget, but not outside it (meaning farther out in the northeast direction from the budget.) The opportunity set of a worker is the set of all leisure/consumptions “baskets” a worker can afford to buy. We see the opportunity set is from the budget line back to the origin in the southwest direction. Definition: marginal wage rate – this is the wage rate received for the last hour of work. We have assumed the worker receives the same rate for all hours of work. In this case the marginal wage rate is the existing wage. But, later we can see other cases. As an example, you know about time and a half after 40 hours of work. This will change the budget constraint.

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**definition and example**

Endowment point – this is the point on the budget line where the person is spending all their time on leisure and can only consume the amount of nonlabor income V. It is the point farthest to the bottom right on any budget constraint. Example: T = 24, w = 7.25 per hour of work, V = 25. The budget line is C = wT + V – wL = 7.25(24) + 25 – 7.25L. We can look at many points on the line by doing the following: If L = 24, C = 25. This is the endowment point. If L = 23, C = If L = 22, C = 39.50, and .... If L = 0, C = This is the vertical intercept.

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**example in graph C (0, 199) The endowment point (24, 25) L**

The vertical intercept L

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Slope The slope of a line is rise over run, right? On the previous slide this would be change in C divided by change in L between any two points. We could use the endowment and intercept to get Slope = change in C/change in L = (25 – 199)/(24 – 0) = -7.25 So, again we see the slope of the budget line is the negative of the wage rate. This means if an hour of leisure is given up the extra hour of work will mean consumption can go up by 7.25.

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**Something to think about**

In econ we like to think about where people will end up on their budget. I have a more complete story later, but want to anticipate some of that story now. Say many people have the budget I just had in the example. If I did a survey of these people do you think they would all end up in the same spot on the budget? I dare say not all would end up at the same place on the budget. Some would stay at the endowment point. Some would move up the budget a little and some would move up the budget a lot.

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**Something to think about**

Let’s think about people who stay at the endowment point. They could give up an hour of leisure and get back some dollar amount of consumption equal to the wage. Moving from the endowment means 2 things: Give up leisure, Get back consumption. We assume people get happiness (what I will call utility later) from both leisure and consumption. Thus, movement from the endowment means “trading” things that are both liked. The person who stays at the endowment must like the leisure more than the consumption that could be acquired at the going wage rate.

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**Something to think about**

Those that leave the endowment point must get more happiness from the consumption than the leisure they give up. People move out from the endowment until the additional consumption is not liked as much as the leisure given up. Different people stop at different points.

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