Reference no: EM13334344
1. Write down a simple labour supply model for an individual. Assume initially that the person works a positive number of hours.
a) Specify the utility function and the budget constraint.
b) Write down the Lagrange Function and solve for the first order conditions.
c) Using the first order conditions, solve for the marginal rate of substitution condition between consumption and leisure.
d) Assuming that the utility function is of the following form: U=βCaL(1-a). Use this functional form to solve for an exact expression for both the first order conditions and the marginal rate of substitution between consumption and leisure.
e) Solve for the labour supply function for the individual.
f) Derive a mathematical expression for the impact on the individual's hours of work of an increase in unearned income.
g) Now, relax the assumption that the person must have positive hours of work. Write out the new problem and use the Kuhn -Tucker method to solve for the necessary conditions.
2. Assume that:
1) the person lives for two periods and works for the wage w1 in period 1 and w2 in period 2;
2) unearned income is the same in each period;
3) the person works a positive number of hours in each period;
4) lifetime utility is the sum of utility from each time period (and that utility in each time period has the same functional form as in part d) of question 1);
5) the interest rate is zero; and
6) the price of consumption is 1 in each period.
a) Write out the person's two-period utility maximization problem.
b) Solve for the first order conditions for the problem in part a).
c) Using the first order conditions from b), solve for a marginal rate of substitution condition between consumption in period 1 and consumption in period 2.
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