Lecture 17: The Investment Schedule

Firms

• Firms borrow in period 1 to invest in capital goods, which become productive in period 2.
• $I_1$ denotes investment in period 1.
• In period 2, firms use the capital to produce final goods using the technology: Output in period 2 = $f(I_1)$.
  • The production function $f(·)$ is increasing and concave.
• In period 2 firms repay their loans in the amount $(1 + r<em>1)I</em>1$.
• In period 1, firms choose the level of investment, $I_1$, to maximize profits.

The Production Function

• The production function indicates that output in period 2 is an increasing and concave function of investment, $I_1$.

The Firm’s Profit Maximization Problem

• $\Pi_2$ denotes the profits of the firm in period 2.
• Profit is the difference between output, $f(I<em>1)$, and the cost of investment, including interest, $(1 + r</em>1)I_1$.
• $\Pi<em>2 = f(I</em>1) − (1 + r<em>1)I</em>1$
• Firms choose investment to maximize profits.
• The optimization problem of the firm is $\max {I<em>1} [f(I</em>1) − (1 + r<em>1)I</em>1]$, taking $r_1$ as given.

First-Order Condition

• The first-order condition associated with the firm’s profit maximization problem is obtained by taking the derivative of profits with respect to $I<em>1$ and equating the result to zero: $\frac{\partial \Pi</em>2}{\partial I_1} = 0$
• This yields $f'(I<em>1) − (1 + r</em>1) = 0$, or $f'(I<em>1) = 1 + r</em>1$
• $f'(I<em>1)$ is the derivative of $f(I</em>1)$ with respect to $I_1$. It represents the marginal product of capital, measuring the increase in output due to a unit increase in capital.
• The right-hand side of the firm’s optimality condition, $1 + r_1$, is the marginal cost of capital. It measures the increase in the firm’s cost resulting from a unit increase in capital.
  • To increase its capital by one unit in period 1, the firm must borrow 1 in period 1. In period 2, it must pay back the amount borrowed, 1 unit, plus interest, $r<em>1$. Thus, the marginal cost of capital is $1 + r</em>1$.

Effect on Investment of an Increase in the Interest Rate

• An increase in the interest rate from $r<em>1$ to r1' > r1 causes a fall in investment from $I</em>1$ to $I<em>1' < I</em>1$.
• The increase in the interest rate makes some investment projects that were profitable before the interest rate hike to become unprofitable. As a result, the firm abandons those projects.

The Investment Schedule

• There exists a negative relationship between $r<em>1$ and $I</em>1$, referred to as the investment schedule:
• $I<em>1 = I(r</em>1)$, with $I'(r<em>1) < 0$, where $I'(r</em>1)$ denotes the derivative of $I(r<em>1)$ with respect to $r</em>1$.

Effect on Profits of Changes in the Interest Rate

• At the optimum level of investment, profits are given by the function: $\Pi(r<em>1) ≡ f(I(r</em>1)) − (1 + r<em>1)I(r</em>1)$
• To see how $\Pi(r<em>1)$ changes in response to a change in $r</em>1$, take the derivative of $\Pi(r<em>1)$ with respect to $r</em>1$ to obtain:
• $\Pi'(r<em>1) = f'(I(r</em>1))I'(r<em>1) − I(r</em>1) − (1 + r<em>1)I'(r</em>1)$
• By the first-order condition of the profit maximization problem, $f'(I<em>1) = 1 + r</em>1$. This means that the first and last terms on the right hand side of the above expression cancel each other.
• \Pi'(r1) = −I(r1) < 0
• This expression indicates that when the interest rate rises, profits go down.
• An increase in $r_1$ raises the financial cost of investment, thereby reducing the profitability of the firm.
• Firms are owned by households. Thus profits will be part of the period-2 income of households. So when the interest rate increases, households’ income falls.

Summary of Firm Behavior

• Investment Demand Schedule: $I<em>1 = I(r</em>1)$ with I'(r_1) < 0
• Profits: $\Pi<em>2 = \Pi(r</em>1)$ with \Pi'(r_1) < 0

An Example

• Suppose that the production function is of the form $f(I<em>1) = 2\sqrt{I</em>1}$.
• This production function is positive, strictly increasing, and concave.
• The marginal product of capital is given by $f'(I<em>1) = \frac{1}{\sqrt{I</em>1}}$.
• The marginal product of capital is decreasing in $I<em>1$. When the marginal product of capital is decreasing in $I</em>1$, we say that the production technology displays diminishing marginal product of capital.
• Equating the marginal product of capital to the marginal cost of capital, we obtain $\frac{1}{\sqrt{I<em>1}} = 1 + r</em>1$.
• Solving this optimality condition for $I_1$ yields the investment schedule:
• $I(r<em>1) = \frac{1}{(1 + r</em>1)^2}$
• The optimal level of investment is a strictly decreasing function of the real interest rate.
• To obtain the optimal level of profits as a function of the real interest rate, start with the definition of profits and then replace investment for its optimal value:
• $\Pi(r<em>1) ≡ f(I(r</em>1)) − (1 + r<em>1)I(r</em>1) = \frac{2}{\sqrt{(1 + r<em>1)^2}} − (1 + r</em>1) \frac{1}{(1 + r<em>1)^2} = \frac{1}{1 + r</em>1}$
• The optimal level of profits is a decreasing function of the real interest rate, which is in line with our previous result.

Summing Up

• Firms borrow in period 1 at the interest rate $r<em>1$ to invest in physical capital, $I</em>1$.
• The optimal level of investment occurs when the marginal product of capital, $f'(I<em>1)$, equals the marginal cost of capital, $1 + r</em>1$.
• The optimal level of investment is a decreasing function of the real interest rate $r<em>1$, called the investment schedule and denoted $I(r</em>1)$.
• In period 2, firms use the capital built in period 1 to produce goods using the production function $f(I<em>1)$, which is increasing and concave in $I</em>1$.
• In period 2, firms make profits, denoted $\Pi(r_1)$. Profits are a decreasing function of the real interest rate.
• In period 2, firms distribute profits to households, who own the firms. We will focus on households in the next lecture.