1. Introduction of another instrument of savings, namely, capital

Transcription

1 Chapter 7 Capital Main Aims: 1. Introduction of another instrument of savings, namely, capital 2. Study conditions for the co-existence of money and capital as instruments of savings 3. Studies the effects of inflation on capital accumulation 4. Introduction to credit

2 The Production Economy So far we have studied an endowment economy. Now we move to the production economy. The availability of goods (supply) in the economy is no longer fixed, but depends on the decisions of individuals. Suppose now that each individual possesses a production technology, which converts k t amount of goods used as capital input in time t to f (k t ) amount of goods at time t + 1. The production function is strictly increasing and concave in capital k t, i.e., f 1 (k t ) > 0 and f 11 (k t ) < 0. Rest of the environment remains the same as before. Now an individual can finance his/her consumption in old period through capital accumulation (or by savings in terms of capital in his/her first period).

3 Golden Rule Allocation First we analyze the golden rule allocation in such an economy. The golden rule allocation consists of c 1,t, c 2,t, k t which maximize the utility of future generations. Suppose each young saves k t 1 amount of goods in period t 1. Then the total amount of goods available (supply) at time t is The total demand for goods is The feasibility set is given by yn t + f (k t 1 )N t 1. (6.1) c 1,t N t + k t N t + N t 1 c 2,t. (6.2) c 1,t N t + k t N t + c 2,t N t 1 yn t + f (k t 1 )N t 1. (6.3)

4 Golden Rule Allocation In the stationary environment where c 1,t = c 1, c 2,t = c 2, k t = k, t, (6.3) can be written as [ ] [ 1 1 c 1 + k + c 2 y + n n The central/social planner problem is to ] f (k). (6.4) max U(c 1, c 2 ) (6.5) c 1,c 2,k subject to the feasibility constraint (6.4). The first order conditions are and f 1 (k) = n (6.6) U 1 (c 1, c 2 ) = n. (6.7) U 2 (c 1, c 2 )

5 Golden Rule Allocation (6.6) equates the marginal product of capital to the growth rate of population. (6.7) equates the marginal rate of substitution between the current and the future consumption to the growth rate of population. Notice that by combining (6.6) and (6.7), we get f 1 (k) = U 1(c 1, c 2 ) = n. (6.8) U 2 (c 1, c 2 ) (6.8) shows that the central/social planner allocates goods in such a way that the marginal rate of substitution between the current and the future consumption equals the marginal rate of transformation. (6.4), (6.6), and (6.7) characterize the golden rule allocations c 1, c 2, k.

6 The Competitive Equilibrium Without Money Now consider the competitive equilibrium without money. Later we will consider the competitive monetary equilibrium. The first-period budget constraint of an individual is The second-period budget constraint is c 1,t + k t y. (6.9) c 2,t+1 f (k t ). (6.10) The life-time budget constraint can be written as c 2,t+1 f (y c 1,t ). (6.11) In the case of linear technology f (k t ) = xk t, (6.11) simplifies to c 1,t + c 2,t+1 x y. (6.12)

7 The Competitive Equilibrium An individual s problem is to choose c 1,t, c 2,t+1, k t to maximize his/her utility. max U(c 1,t, c 2,t+1 ) (6.13) c 1,t,c 2,t+1,k t subject to (6.11). By putting (6.9) and (6.10) in (6.13), the problem can be recast as follows: The first order condition is max k t U(y k t, f (k t )). (6.14) f 1 (k t ) = U 1(c 1,t, c 2,t+1 ) U 2 (c 1,t, c 2,t+1 ). (6.15)

8 The Competitive Equilibrium In the stationary environment (6.15) reduces to f 1 (k) = U 1(c 1, c 2 ) U 2 (c 1, c 2 ). (6.16) (6.16) equates the marginal product of capital (or the marginal rate of transformation in this case) to the marginal rate of substitution between the current and the future consumptions. (6.16) together with (6.9) and (6.10) characterize the competitive equilibrium, which constitutes of c 1, c 2, k. Now we compare the golden rule allocation with the allocations of the competitive equilibrium.

9 Over and Under Capital Accumulation (6.8) shows that the golden rule allocation satisfies two conditions. First, the marginal rate of transformation is equal to the marginal rate of substitution, f 1 (k) = U 1(c 1,c 2 ) U 2 (c 1,c 2 ). Second, both the marginal rate of transformation and the marginal rate of substitution are equal to the population growth rate n. (6.16) shows that in the competitive equilibrium, the first condition is satisfied. However, there is nothing in the model that ensures that the second condition is satisfied. Thus, in general the competitive equilibrium is inefficient. The capital accumulation in the competitive equilibrium can be higher (over-accumulation) or lower (under-accumulation) than the golden rule level.

10 Over and Under Capital Accumulation One can find out whether an economy has over or under accumulation of capital by comparing the marginal product of capital, f 1 (k) to the rate of population growth, n. However, the marginal product of capital is not observed. But it can be shown that in a competitive equilibrium the marginal product of capital equals the real rate of interest, r. Thus by comparing the real rate of interest, r, with the population growth, n, one can tell whether there is over or under accumulation of capital. 1. If r f 1 (k) > n then there is under-accumulation. 2. If r f 1 (k) < n then there is over-accumulation.

11 The Competitive Monetary Equilibrium Now we introduce fiat money in the economy. With the introduction of fiat money, the economy has two assets: money and capital. An individual can save both in terms of money and capital. Given these two alternatives, the question is whether an individual would choose to save only in terms of money or capital or both. In order to answer this question, one has to compare the rate of return from these two assets. We already know that the rate of return on capital is its marginal product or the real rate of interest, f 1 (k) r. The rate of return on money is v t+1 v t = n z.

12 The Competitive Monetary Equilibrium Three cases can emerge: 1. Case I: n z > f 1(k) r, then an individual will save only in terms of money. 2. Case II: n z < f 1(k) r, then an individual will save only in terms of capital. 3. Case III: n z = f 1(k) r, then an individual will save both in terms of money and capital. We will only consider case III where both money and capital are used for saving.

13 The Tobin Effect Using the equilibrium condition that the rate of return on money and capital are equal, n z = f 1(k) r, one can derive the effects of changes in the growth rate of money supply z on capital accumulation, k. Note that the inflation rate, p t+1 p t = z n. Thus, for a given population growth rate, n, any change in the growth rate of money leads to proportionate changes in the inflation rate. Note that the rate of return on money n z is decreasing in the growth rate of money supply z. If the growth rate of money supply z rises, the rate of return on money, n z, falls. Since in equilibrium the rate of return on money is equal to the rate of return on capital, the rate of return on capital, f 1 (k), must fall. Given the concavity of the production function, f 11 (k) < 0, this implies that the capital stock rises. Thus, a higher growth rate of money supply/inflation rate leads to higher capital accumulation. This positive association between the growth rate of money supply/inflation rate and the capital accumulation is known as the Tobin effect.

14 Nominal and Real Interest Rates The nominal interest rate refers to the interest rate in terms of dollars or units of money. The real interest rate refers to the interest rate in terms of purchasing power or in terms of goods. Suppose that an individual lends out one dollar at time t and receives R t dollars at time t + 1, then R t is the gross nominal interest rate. In general, the rate of return on any asset is given by the ratio of its pay-off to its price. The nominal rate of interest is simply the rate of return on one period loan in nominal terms. The pay-off for lending one unit of dollar is R t next period. Thus the nominal rate of interest is R t = R t 1. (6.17)

15 Nominal and Real Interest Rates The nominal rate of interest does not tell us about the rate of return in terms of goods or real terms. In order to derive the rate of return in real terms, we have to convert the price of an asset as well as its pay-off in real terms by using the price level, p t. The real rate of return is simply the ratio of pay-off in real terms to the price of the asset in real terms. Thus we can derive the real rate of interest, r t as follows: r t = R t p t+1 / 1 p t. (6.18) The numerator of (6.18) is the pay-off in real terms. Notice that we have divided the nominal pay-off, R t, by the price level of the next period, p t+1, since the lender receives the pay-off next period. The denominator is the amount lent in real terms.

16 Nominal and Real Interest Rates (6.18) can be rewritten as ( ) ( ) pt+1 pt+1 R t 1 = (r t 1) (r t 1) 1. (6.19) p t p t (6.19) states that the net nominal interest rate R t 1 equals the net real interest rate, r t 1, plus the net inflation rate and the product of the two. The relationship between the net nominal rate of interest and the net real interest rate and the net inflation rate is known as the Fisher s relationship. In general, the last term is ignored and the net nominal interest rate is approximately given by ( ) pt+1 R t 1 (r t 1) + 1. (6.20) p t

17 Inflation and Nominal and Real Interest Rates: Fisher Effect We can use (6.18) to analyze the effects of inflation or changes in the growth rate of money supply on the nominal and real interest rates. Since the real rate of interest r = f 1 (k) and the inflation rate p t+1 p t = z n, (6.18) can be written as R = f 1 (k) ( z n ). (6.21) (6.21) shows that for a given real rate of interest or the marginal product of capital, a higher inflation rate leads to a proportionate increase in the nominal interest rate. This is called the Fisher effect. However, if the Tobin effect is present, then a higher inflation rate does not lead to a proportionate increase in the nominal interest rate, since the marginal product of capital, f 1 (k) and thus the real rate of interest falls.

18 Risk So far we have assumed that assets (money or capital) yield known pay-offs next period. But this need not be the case. For instance, a borrower may default on loan next period. Similarly, there may be some uncertainty about the production next period (think of hurricane Katrina). In such cases, in order to find return on an asset, we need to find the expected pay-off, E(Y ). Suppose that an asset yields Y 1 with probability π 1 and Y 2 with probability π 2 next period, where π 1 + π 2 = 1. Then the expected pay-off of the asset is E(Y ) = π 1 Y 1 + π 2 Y 2. (6.22) Suppose that the price paid for the asset is P in the current period, then the expected return on the asset, E(y), is given by E(y) = E(Y ) P. (6.23)

19 Savings with Risky Assets Suppose now that an individual can save in terms of a safe asset (known pay-off) and a risky asset (random pay-off). Which asset he/she would choose. The answer depends on the rate of return on both the assets and the attitude of the individual towards risk. In general, we assume that individuals are risk-averse (strictly concave utility function ensures that). In other words, an individual strictly prefers safe asset, if the rate return on a safe asset, y safe, equals the expected rate of return on the risky asset, E(y). In order to induce such an individual to hold the risky asset it must be the case that the rate of return on the risky asset is strictly higher than the rate of return on the safe asset, E(y) > y safe. (6.24)

20 Risk Premium The excess of rate of return that is required in order to induce a risk-averse individual to hold the risky asset is known as the risk-premium; risk premium = E(y) y safe. (6.25)

21 A Model of Private Debt/Credit Now we modify the environment of our model in order to allow buying and selling of goods on credit. In the previous models considered, buying and selling on credit is not possible due to the endowment and age structure. The young can sell their goods to the old on credit in the current period, but the old would not be around next period to repay their loans. In order to allow for the circulation of credit, we need to introduce heterogeneity in terms of endowment. Suppose that there are two types of individuals: borrowers, with no endowment when young and endowment y when old, and lenders with endowment y when young and no endowment when old. With this structure of endowment, borrowers would like to borrow when young while lenders would like to lend in order to finance their consumption when old.

22 Preferences and Constraints of Lenders Let c 1,l, c 2,l, and l denote the first-period consumption, second-period consumption, and the amount of lending of a lender respectively. The first-period budget constraint for a lender is The second-period budget constraint is c 1,l + l = y. (6.26) c 2,l = rl (6.27) where r is the real rate of interest. The life-time budget constraint is given by c 1,l + c 2,l = y. (6.28) r The lender chooses c 1,l, c 2,l, l in order to maximize subject to (6.28). U(c 1,l, c 2,l ) (6.29)

23 Optimal Choices of Lender Putting (6.26) and (6.27) in (6.29), we have The first order condition is max U(y l, rl). (6.30) l U 1 (c 1,l, c 2,l ) = r. (6.31) U 2 (c 1,l, c 2,l ) (6.31) equates the marginal rate of substitution between current and future consumption to the rate of interest. Using this equation, we can derive the amount lent, l, as a function of interest rate r, l(r). Normally we assume that utility function is such that lending, l, is an increasing function of the real interest rate, r, i.e., l 1 (r) > 0. Using (6.26), (6.27), and (6.31), we can derive c 1,l, c 2,l, l as a function of interest rate r.

24 Preferences and Constraints of Borrowers Let c 1,b, c 2,b, and b denote the first-period consumption, second-period consumption, and the amount of borrowing of a borrower. The first-period budget constraint for a borrower is The second-period budget constraint is c 1,b = b. (6.32) The life-time budget constraint is given by c 2,b = y rb. (6.33) c 1,b + c 2,b = y r r. (6.34) The lender chooses c 1,b, c 2,b, b in order to maximize subject to (6.34). U(c 1,b, c 2,b ) (6.35)

25 Optimal Choices of Borrower Putting (6.32) and (6.33) in (6.35), we have The first order condition is max U(b, y rb). (6.36) l U 1 (c 1,b, c 2,b ) = r. (6.37) U 2 (c 1,b, c 2,b ) (6.37) equates the marginal rate of substitution between the current and the future consumption to the real rate of interest. Using this equation, we can derive the amount borrowed, b, as a function of the real interest rate r, b(r). Normally we assume that utility function is such that the borrowing, b, is a decreasing function of the interest rate, r, i.e., b 1 (r) < 0. Using (6.32), (6.33), and (6.37), we can derive c 1,b, c 2,b, b as a function of interest rate r.

26 Determination of the Interest Rate (6.31) and (6.37) give the amount of lending l and borrowing b as a function of the interest rate r. In order to find out actual amount of lending and borrowing, we need to determine the real rate of interest, r. The real rate of interest is given by the condition that the amount borrowed should be equal to the amount lent: l(r) = b(r). (6.38) Once we have determined the equilibrium real rate of interest, we can derive the allocations c 1,l, c 2,l, l, c 1,b, c 2,b, b.