1 The consumption-leisure framework Suppose that the representative consumer has the following utility function over consumption (c) and labour (n): u(c, l) = ln c − A 1 + � n 1+� (1) where, as usual, c denotes consumption and n denotes the number of hours of labour the consumer chooses to work, The constants A and � are outside the control of the individual, but each is strictly positive. Suppose the budget constraint (in real terms) faced by the individual is given by: c = (1 − t) · w · n (2) where t is the labour tax rate, w is the real hourly wage rate, and n is the number of hours the individual works. Remember, as seen in class, n + l = 1 is always true. Using the static consumption-leisure framework, answer the questions that follow. 1. State the utility maximization problem, and state carefully the choice variables in the problem. [2] 2. Write the Lagrangian function for this problem. [2] 3. Using the Lagrangian function from above, derive the first order condition with respect to c. [2] 4. Using the Lagrangian function from above, derive the first order condition with respect to n. [4] 5. Compute the first order condition with respect to the Lagrange multiplier [2] 6. From the first order conditions obtained above, use the appropriate conditions to derive the consumption-leisure optimality condition for this problem. (Be sure to eliminate the multiplier from your final answer, and simplify as much as possible)
1 The consumption-leisure framework Suppose that the representative consumer has the following utility function over consumption (c) and labour (n): u(c, l) = ln c − A 1 + � n 1+� (1) where, as usual, c denotes consumption and n denotes the number of hours of labour the consumer chooses to work, The constants A and � are outside the control of the individual, but each is strictly positive. Suppose the budget constraint (in real terms) faced by the individual is given by: c = (1 − t) · w · n (2) where t is the labour tax rate, w is the real hourly wage rate, and n is the number of hours the individual works. Remember, as seen in class, n + l = 1 is always true. Using the static consumption-leisure framework, answer the questions that follow. 1. State the utility maximization problem, and state carefully the choice variables in the problem. [2] 2. Write the Lagrangian function for this problem. [2] 3. Using the Lagrangian function from above, derive the first order condition with respect to c. [2] 4. Using the Lagrangian function from above, derive the first order condition with respect to n. [4] 5. Compute the first order condition with respect to the Lagrange multiplier [2] 6. From the first order conditions obtained above, use the appropriate conditions to derive the consumption-leisure optimality condition for this problem. (Be sure to eliminate the multiplier from your final answer, and simplify as much as possible)
Chapter16: Labor Markets
Section: Chapter Questions
Problem 16.10P
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1 The consumption-leisure framework Suppose that the representative consumer has the following utility function over consumption (c) and labour (n): u(c, l) = ln c − A 1 + � n 1+� (1) where, as usual, c denotes consumption and n denotes the number of hours of labour the consumer chooses to work, The constants A and � are outside the control of the individual, but each is strictly positive. Suppose the budget constraint (in real terms) faced by the individual is given by: c = (1 − t) · w · n (2) where t is the labour tax rate, w is the real hourly wage rate, and n is the number of hours the individual works. Remember, as seen in class, n + l = 1 is always true. Using the static consumption-leisure framework, answer the questions that follow. 1. State the utility maximization problem, and state carefully the choice variables in the problem. [2] 2. Write the Lagrangian function for this problem. [2] 3. Using the Lagrangian function from above, derive the first order condition with respect to c. [2] 4. Using the Lagrangian function from above, derive the first order condition with respect to n. [4] 5. Compute the first order condition with respect to the Lagrange multiplier [2] 6. From the first order conditions obtained above, use the appropriate conditions to derive the consumption-leisure optimality condition for this problem. (Be sure to eliminate the multiplier from your final answer, and simplify as much as possible)
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