Q1: Tobin's *q* and the Cost of Capital

Q2: Why is the Investment Function Negatively Sloped?

**
The cost of capital falls and Tobin's q increases with a fall in the
nominal interest rate. True or False?
Explain your answer.**

False, because investment depends upon the real interest rate and not the nominal interest rate.

Suppose however, to make things interesting, we ask whether the cost of
capital falls with a fall in the real interest rate. The cost of capital,
usually called the *user cost*, is the opportunity cost of the funds
that must be used to buy the capital plus the amount that must be set aside
to cover the depreciation of the capital. This would be an amount

(1) r Ck + D/K

where D/K is the rate of depreciation as a proportion of the stock of capital. Obviously, a fall in r reduces the cost of capital to the firm. Investment will expand to where the marginal product of capital, denoted by dY/dK, with Y being output, K the capital stock, and dY/dK representing the small change in Y that will result from a small change in K, is equal to the cost of producing new capital goods plus the adjustment costs of harmonizing them with labour and already-employed capital. Letting z(I) be these adjustment costs, per unit of capital stock, as a function of the level of investment I and assuming no capital gains (which could be treated as either an increase in the return to capital or a reduction in its cost) the equilibrium condition will be

(2) dY/dK = r Ck + D/K + z(I) r Ck

Letting the cost of producing a unit of capital goods and putting it in place as investment equal

(3) Pk = Ck + z(I)Ck = Ck (1 + z(I))

we can write the equation (2) as

(4) dY/dK = r Pk + D/K

By assuming that capital goods can be produced at constant cost in terms of consumer goods and choosing the units of capital so that this relative cost Ck equals unity, we can manipulate the above equation to yield

(5) r = (dY/dK - D/K)/(1 + z(I))

From this we can see that if adjustment costs increase with increasing
investment (i.e., dz(I)/dI > 0$), the interest rate will decline as
the level of investment increases, giving the investment function its
negative slope. Tobin's *q* is the ratio of Pk to Ck. In equilibrium,
Pk is the present value of capital, which must equal the cost of producing
a unit of capital goods, Ck, plus the adjustment costs of putting it in place.
Tobin's *q* thus equals

(6) *q* = Pk/Ck = (dY/dk - D/K)/Ck = (1 + z(I))Ck/Ck = 1 + z(I).

As the real interest rate falls, the present value of capital (and its total cost) increases relative to the cost of producing new capital goods because investment expands and increases the cost of adapting the new capital to the previous capital and labour inputs. These adjustment costs thus drive a wedge between the present value of capital and the cost of producing capital goods. Equation (4) can be manipulated to yield

(7) (dY/dK - D/K)/ r = Pk.

It can be easily seen from this equation that a fall in the real interest rate
will increase the present value of capital and thereby raise Tobin's *q*.

**
The investment function slopes downward to the right because the marginal
product of capital falls as the stock of capital expands. It shifts
to the right with a shift in income because the marginal productivity of
capital is positive. True or False? Explain your answer.**

The second sentence is true but the first is not true. The demand curve for capital stock with respect to the price of capital stock slopes downward because an increase in the capital stock reduces the marginal product of capital. But an increase in investment, assuming it is small in relation to the existing capital stock, will have a trivial effect on the stock of capital and, hence, a trivial effect on the marginal product of capital. The investment function slopes downward because of the relation between the interest rate and the level of investment given by equation (5) in the answer to the previous question. The relevant equation is

r = (dY/dK - D/K)/(1 + z(I))

The interest rate falls when investment increases because of the effect of investment on adjustment costs, given by z(I), and not because of any effect it might have on dY/dK.

An increase in income accompanied by an increase in the employment of labour causes the marginal product of capital, dY/dK, to increase. As can be seen in the above equation, this will increase the level of r associated with any given level of I or increase the level of I associated with any given level of r. The investment function thus shifts to the right.

The relationship between investment and the stock of capital can be seen
in **FIGURE 1** below.

The market for the stock of capital is given in the left panel and the market for the flow of additions to the stock of captital in the right panel. The vertical axes in the two panels are the same and measure the market price of capital goods measured in units of consumer goods. The scales on the horizontal axes are different, with investment measured in thousands of units and capital stock in millions of units. A fall in the rate of interest or an increase in output and employment and, hence, in the marginal product of capital shifts the demand for capital upward. This increases the present value of the initial capital stock K0. The desired stock of capital at the old market price of capital becomes K1. At the higher present value and market price of capital, investment expands from Io to Ia along the supply curve of new investment. That supply curve slopes upward because, even when capital goods can be produced at constant cost in terms of consumer goods, the costs of harmonizing new capital with the original labour and capital inputs increases as the flow of additions to the capital stock gets larger.

As time passes the increased level of investment will cause the capital stock to grow faster relative to the demand for capital. The vertical supply curve of capital stock will thus shift to the right through time faster than the demand curve for capital stock and the present value and market price of capital will fall back towards its initial level. As this happens, the level of investment will also gradually decline.

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