Introductory Macroeconomics, Hilary Term 2024

Week 3 Problem Set

  1. Suppose that output in an economy depends on labour (L), natural resources (R), and technology (A):
    Y = ARαL1-α. L grows at constant rate n, while R grows at rate 0, i.e. is constant.

    1. At what rate does the ratio of natural resources per worker, R/L, grow?
    2. At what rate does productivity, Y/L, grow? Can growth be sustained in the long run?
    3. Does the model have a steady state in the familiar output per effective worker, Y/AL?

  2. Consider a two period version of the Romer model. Technology at the beginning of period 1 is at a given exogenous level A1. In period 1, agents spend proportion (1 - l1) of their time in productive activities, with production function:

    Y1 = A1(1 - l1) L,

    where L ≥1 is the total labour supply in the economy. Agents spend the remaining share of their time l1 in period 1 producing ideas, according to the production function:

    A2 - A1 = zA1 l1L,

    with z ≥ 1 a parameter measuring the productivity of workers in producing ideas. In the final period 2, workers spend proportion (1 - l2) of their time in productive activity, producing:

    Y2 = A2(1 - l2) L

    1. Assuming that agents only value output produced in periods 1 and 2, what would be the best proportion of time to spend producing ideas in period 2?
    2. Make a plot of the combinations of Y1 and Y2 that are possible when agents choose different proportions of their time to spend on production in period 1. To start you off, what happens to Y1 and Y2 if l1 = 0, so that all time in period 1 is spent producing ideas? What about intermediate cases where l1 is between 0 and 1, for example 1/2?
    3. Agents ideally want to consume production goods in both periods 1 and 2. One way to model this is through a multiplicative utility function:

      U = Y1 Y2

      Use the three production functions given in the introduction to this question to solve out for output Y1 and Y2 as functions of l1, A1, z, and L alone. What amount of time should be spent producing ideas in period 1 to maximise utility? Add indifference curves to your plot in part (b) to illustrate your answer.
    4. How does the optimal allocation of time in period 1 change as the productivity of hours spent producing ideas, z, increases?

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