Question

Let T = 24 be total available hours(note that T=24, not T=12),L be hours of leisure, and n be hours of work. Let w >0 be the hourly wage and p >0 be the price of the consumption good. Finally, let C be consumption. Treat each part of this question as a separate and independent scenario. Assume the utility function is U(L, C) = LC.Suppose the agent receives a lump-sum amount of B dollars if he works 8 hours or more (ie. n > 8). Find (L*,C*) and illustrate graphically your solution

Answer 1

To find (L*, C*), we need to maximize the utility function U(L, C) = LC subject to the constraint on hours of work.

First, let's set up the problem mathematically.
Maximize U(L, C) = LC
Subject to the constraint n > 8

Since T = 24 and n represents hours of work, we have L + n = T
Therefore, L + n = 24

We can rewrite the constraint as n > 8 as n - 8 > 0

Now, let's solve the problem graphically.

Step 1: Drawing the budget line:
Given that the agent receives a lump-sum amount of B dollars if he works 8 hours or more, we can set up the budget line.
The agent's income will be given by the hourly wage w multiplied by hours of work n.
Income = w * n

If n > 8, the agent will receive an additional B dollars.
Therefore, the budget line equation is:
Income + B = w * n

Step 2: Finding the indifference curve:
The utility function U(L, C) = LC represents the agent's preferences. Since we are maximizing utility, we can draw indifference curves to find the highest level of satisfaction.

Step 3: Maximizing utility:
To find the optimal point (L*, C*), we need to find the point where the indifference curve is tangent to the budget line.

Graphically, the optimal point (L*, C*) will be where the indifference curve has the same slope as the budget line.

The slope of the indifference curve is given by the marginal rate of substitution (MRS), which is the ratio of the marginal utilities of leisure (MU_L) to consumption (MU_C). In this case, MRS = MU_L / MU_C = 1.

The slope of the budget line is the negative ratio of the wage w to the price of the consumption good p. In this case, the slope of the budget line is -w / p.

Therefore, at the optimal point (L*, C*), we have:
- w / p = 1

Solving for C*, we get:
C* = -p / w

Since L + n = 24, we can substitute n = 24 - L into the budget line equation to solve for L*.

Income + B = w * (24 - L)

Substituting Income = C* * p, we have:
C* * p + B = w * (24 - L)

Solving for L*, we get:
L* = 24 - [(C* * p + B) / w]

With L* and C*, we have found the optimal values of leisure and consumption.

Please note that the graph would depend on specific values for B, w, and p, and the exact shape and position of the indifference curves will vary based on individual preferences.