**Let us make an in-depth study of the relationship between the interest rate, desired capital stock and investment. **

Using the neoclassical theory of investment, assume that the desired capital stock is a function of output and the interest rate.

By assuming output constant, the relationship between the interest rate and the desired capital stock may be plotted.

If the interest rate is plotted on the vertical axis and the desired capital stock on the horizontal, the relationship is shown in the left panel of Figure 8.10.

Thus, if the interest rate is z_{0}, the desired capital stock is K_{0}*. Should the interest rate decrease to i_{1}, the desired capital stock increases to because, with a lower interest rate, it is profitable for firms to employ more capital.

Suppose the interest rate is i_{0} and the desired capital stock is K_{0}*. If the actual capital stock, also measured on the horizontal axis, is K_{1}, there is no discrepancy between the desired and actual capital stocks. Hence, net investment is zero.

Gross investment is, however, positive, since firms must invest to replace plant and equipment that has worn out or been destroyed. Consequently, at interest rate I_{0}, investment is I_{0}, which according to equation (iii), equals δ K_{0}. This combination is shown in the right panel of Figure 8.10.

If the interest rate decreases to i_{1} the desired capital stock increases to K_{1}*. Since the actual capital stock is A_{0}, the desired capital stock exceeds the actual capital stock, and net investment is positive as firms add to their productive capacity.

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But firms do not attempt to eliminate the gap between the desired and actual capital stocks in a single period; they do so over a number of periods. For example, suppose the desired capital stock exceeds the actual capital stock by $20 billion.

If λ in equation (iii) equals one-half, net investment equals $10 billion, obtained by multiplying λ by the difference between die desired and actual capital stocks. Consequently, at interest rate i_{1}, gross investment equals I_{1}, which consists of net investment, λ (K_{1}* – K_{0}), plus replacement investment, 6 A_{0}. This combination is depicted in the right panel of Figure 8.10. Since I_{1} exceeds I_{0}, an inverse relationship exists between the market rate of interest, i, and investment, I; as the interest rate decreases, investment increases.

Over time, the discrepancy between the desired and actual capital stocks will be eliminated. For example, suppose the interest rate remains at i_{1}. So long as net investment remains positive, the capital stock increases, thereby reducing the discrepancy between the desired and actual capital stocks.

Since net investment equals λ, a constant multiplied by the discrepancy between the desired and actual capital stocks, investment will be less in succeeding periods, and the investment function of Figure 8.10 will shift to the left.

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Eventually, the actual capital stock will equal the desired capital stock, and net investment will be zero. Gross investment, however, will be greater than the original equilibrium level, 7_{0}, since the new equilibrium capital stock K_{n} (equal to K_{1}*) is greater than the original level K_{0} (equal to K_{0}*), and more investment is required for replacement purposes.

Thus, we find that investment, I, is a function of the interest rate, i, and the capital stock, K. It is also a function of output, Y, which has been assumed constant so far. If output increases, the relationship between the interest rate and the desired capital stock is altered. For example, suppose the interest rate is i_{0} and output increases from Y_{0} to Y_{1}. Based on the neoclassical theory of investment, the desired capital stock increases to K_{2}*. This implies that the interest rate-desired capital stock relationship of Figure 8.10 shifts to the right.

Since the new desired capital stock, K_{2}*, exceeds the actual capital stock, K_{0}, net investment is positive, this implies that the investment function of Figure 8.10 shifts to the right. Since only part of the discrepancy between the desired and actual capital stocks is eliminated within a single period, the investment function will shift by a smaller amount than the interest rate- desired capital stock relationship.

This analysis suggests that investment is a function of output, the interest rate and the difference between the actual and the desired stock of capital.