hw1_fig - h t to be equal to zero On the other hand as a...

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ECON 831 Homework #1 due in class Sept. 24th Provide detailed calculations and justi cations to get full credit. Partial credit may be given. Exercise 1: (Human capital accumulation) Consider a worker who enters the labor market at time t = 1 with human capital stock k 1 and retires at t = T . Her earnings depend on the number of hours worked, h t and the wage rate, w t , which is proportional to the worker's human capital k t . That is: w t = αk t . For simplicity, we assume α = 1 . Given the discount factor β = 1 / (1 + r ) where r is the interest rate, the worker's objective is to maximize the present value of her lifetime earnings. The worker can use all the time at her disposal to work and earn income in which case her human capital diminishes or can devote part or all of her time to enhancing the capital stock which would diminish earnings. In general, we can write: h t = φ ± k t +1 k t with φ 0 < 0 Moreover the worker's human capital can grow at most at the rate λ , for which she has to devote all her time to education causing
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Unformatted text preview: h t to be equal to zero. On the other hand, as a result of working full time, h t = 1 , the worker's human capital would diminish at most at the rate δ . i) What is the present value of her lifetime earnings? ii) What are the state and control variables? iii) What is the equation of motion? iv) Write the dynamic optimization problem. 1 f ( X 1 ,X 2 ) f ( X 1 ,X 2 ) + v ( X 2 ) min[ f + v ] D E DF EF B 8 9 13 16 13 C 11 11 16 18 16 From the vantage point of stage 1, therefore, optimum paths and costs are: Path Cost BDF 13 CDF 16 We need only concern ourselves with the 2 paths listed in the table above. Thus we eliminate from consideration all the other paths. Stepping back to stage 0, we repeat the exercise. f ( X ,X 1 ) f ( X ,X 1 ) + v ( X 1 ) min[ f + v ] B C BDF CDF A 7 6 20 22 20 Thus the optimal path obtained by the recursive method is ABDF and the cost is 20. 4...
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