Macroeconomic Theory Optional Assignment-

On the Volatility of Investment-

Consider an economy that lasts for infinite periods. In every period, there is a consumer that lives for one period. Each consumer is the daughter of the previous consumer that inhabited this economy, of whom she inherits the stock of capital. Consumers care not only about their own consumption, but also about the stock of capital that they leave to the consumer that will live in the next period. The consumer that lives in period t has preferences represented by:

u (c_t, k_t+1) = ln (c_t) + γ ln (k_t+1)

where γ > 0 is a parameter, ct is consumption and k_t+1 the stock of capital left as bequest. To consume, the agent uses the capital inherited from the previous agent, but can also use this capital to invest in new capital goods, and therefore generate new capital for the next period. Then, capital follows the law of motion:

k_t+1 = i_t + (1 - δ) k_t

with δ ∈ [0, 1]. The consumer rents the capital at a rental rate r_t.

Each period there is a firm that hires capital to produce according to the technology y_t = A_t√(k_t), with A_t > 0. The firm pays r_t for the use of a unit of capital, and distributes any profits made to the consumer.

a) Solve for the optimal decisions of how much to consume and how much capital to leave to the next agent as a function of prices.

b) Solve for the optimal decisions of how much capital to hire and how many profits to make as a function of prices.

c) For every period t define a competitive equilibrium and solve for the equilibrium quantities and prices.

d) Assume that business cycles are driven by At. In equilibrium, are consumption and investment procyclical, acyclical or countercyclical? Which one is more responsive to the business cycle?

e) Choose a country other than the US. Download data for consumption, investment and GDP, measured in real and annual terms. A good source of data is http://databank.worldbank.org/. Compute the cyclical components of these three variables. To do this for variable X, follow this procedure:

1. Take logarithms of the variable X.

2. Compute the slope (α) and intercept (β) of a linear regression of ln (X) against time. In Excel and in Google Sheets, this is done with the commands Slope and Intercept.

3. Construct the trend component as X^Trend_t = α + βt, where t is the time variable.

4. Construct the cyclical component as X_­^Cycle_t = ln (X_t) - X^Trend_t.

f) Compute the correlation between consumption and GDP, and the correlation between investment and GDP, using the cyclical components. Which one is more correlated with GDP? Also graph the three cyclical components against time and analyze.

g) In terms of the model, what restriction does your answer to the previous point impose on the parameters?