Question

1. Imagine that there is a technological advancement which increases the productivity of capital. Which variable in the model should we increase? Show the effect of this on steady-state capital and output on the usual graph.

2. It turns out that the effect of a change in the savings rate on the steady-state consumption is ambiguous. Let's try to show this with the power of algebra! Let A=2, L=100, and d=1/4 but do not pick a value for s. Solve for the steady-state values of K, Y and C by just leaving an "s" in there. This means that you will get functions of s as your answer! You should be able to look at K*(s) and Y*(s) and see easily that they will be increasing in s. C*(s), should be a parabola. Economists call the savings rate which maximizes this function the "golden rule" savings rate. You will learn how to find it explicitly in intermediate macro (you have to use calculus). Notice what the value of C* is if s=0 and what it is if s=1.

Answer 1

Q 1) If there is a technological advancement resulting in increase of productivity of capital, Firms usually add capital to the point where the value of marginal product is equal to the rental rate of capital.

A) Capital is a factor of production, along with labor and land. It consists of the infrastructure and equipment used to produce goods and services. Capital can include factory buildings, vehicles, plant machinery, and tools used in the production process. Firms may buy, rent, or lease infrastructure and tools in the capital market, but even if the firm owns these factors of production, the opportunity cost of using this capital is the foregone rent that the firm could receive if it rented the capital to somebody else rather than using it for production. Because of this, we say that the price of capital is the rental rate.

B) A firm decides how much of each factor input to use and how much output to produce based on the market prices for outputs and inputs, as well as exogenous technological determinants represented by the production function. The production function describes the relationship between the quantity of inputs used in production and the quantity of output. It can be used to derive the marginal product for capital, which is the increase in the amount of output from an additional unit of capital.

C) The value of marginal product (VMP) of capital is the marginal product of capital multiplied by price. The downward-sloping demand curve for capital, which is equal to the VMP of capital, reflects the fact that the production process exhibits diminishing marginal product. A firm will continue to add capital up to the point where the rental rate is equal to the value of marginal product of capital, which is the point of equilibrium.

D) Total factor productivity measures the residual growth in total output of a firm, industry, or national economy that cannot be explained by the accumulation of traditional inputs such as labor and capital. Increases in total factor productivity reflect a more efficient use of inputs, and total factor productivity is often taken as a measure of long-term technological change or dynamism brought about by such factors as technical innovation.

E) Total factor productivity cannot be measured directly. Instead, it is a residual which accounts for effects on total output not caused by inputs. In the Cobb-Douglas production function, total factor productivity is captured by the variable A:

Y=AK^α L^β

F) In the equation above, Y represents total output, K represents capital input, L represents labor input, and alpha and beta are the two inputs’ respective shares of output. An increase in K or L will lead to an increase in output. However, due to to the law of diminishing returns, the increased use of inputs will fail to yield increased output in the long run.

G) The quantity of inputs used thus does not completely determine the amount of output produced. How effectively the factors of production are used is also important. Total factor productivity is less tangible than capital and labor inputs, and it can account for a range of factors, from technology, to human capital, to organizational innovation.

H) When a country is able to increase its total factor productivity, it can yield higher output with the same resources, and therefore drive economic growth.

Q 2) This question comes under Statistics.