Output and Real Interest Rates
The Production Function Again
Y = A F(K, N) (1)
In the short run K is approximately constant:
Y = A F(K, N) (2)
Features of (2):
1. Output (Y) is increasing in employment (N). (Figure 1)
2. Diminishing returns to labor: (Figure 2)
MPN = dY/dN = A dF(K, N)/dN
MPN = Marginal Product of Labor
Shifts of the production function:
1. Increases in A (due to technological development) shift the production function upward.
2. Decreases in A (due to negative shocks such as oil price shocks) shift the production function downward.
3. Increases in the capital stock (K) shift the production function upward.
Shifts of the curve versus shifts along the curve.
The Labor Market
Labor Demand
An extra unit of labor produces MPN extra units of output
The value of this additional output is P x MPN
The cost of an additional unit of labor is W (the wage rate)
A profit maximizing firm will employ labor N up to the point where:
P x MPN = W or MPN = W/P (W/P = real wage)
P x MPN = Value of the marginal product of labor
Example:
| N | MPN | P | PxMPN | W |
| 1 | 11 | 2 | 22 | 14 |
| 2 | 10 | 2 | 20 | 14 |
| 3 | 9 | 2 | 18 | 14 |
| 4 | 8 | 2 | 16 | 14 |
| 5 | 7 | 2 | 14 | 14 |
| 6 | 6 | 2 | 12 | 14 |
| 7 | 5 | 2 | 10 | 14 |
So the otpimal amount of employment is 5 workers because:
2 x 7 = P x MPN = W = 14
Note that the MPN curve is the (inverse) demand for labor.
Labor demand depends also on:
1. Productivity A. If A increases, MPN increases at every level of N. So, labor demand increases.
2. The stock of capital (K). Higher K increases MPN at every level of N. So, labor demand increases.
3. An oil shock reduces A and reduces the demand for labor.
4. Taxes and fringe benefits. With a payroll tax (at the rate f), the cost to a firm of an additional unit of labor is W (1+f). So:
P MPN = W (1+f) or MPN/(1+f) = W/P
So a payroll tax increases real labor costs and reduces labor demand.
Labor Supply
Labor supply increases when the real wage (W/P) goes up:
NS = F(W/P)
Caveat on substitution and income effects.
Other variables affecting the labor supply:
1. Demographic factors (immigration, population growth, labor force participation rates).
2. Income taxes NS = F ( (1-t)W/P )
t = income tax rate
Equilibrium in labor market:
ND = NS (labor demand = labor supply) (Figure 3)
The equilibrium is affected by:
1. Income tax rates (t). Higher t leads to lower N, higher W/P and lower after tax real wages (1-t)W/P.
2. Payroll tax rates (f). Lower f leads to higher N, higher W/P and lower after tax cost of labor for firms (1+f)W/P.
3. Changes in the stock of capital (K). Higher K leads to higher employment N and higher real wages (W/P).
4. Changes in the level of productivity A. Lower A leads to lower N, lower W/P and lower output.
Output, factor payments and income distribution
From the national income accounts:
Output = Value Added = Income (Payments to the factors of production)
Nominal value of output = P x Y = W N + R K = Wages + Total Profits
or: Y = (W/P) N + (R/P) K = w N+ r K
where:
w = real wage (W/P)
r = real rental (return) to capital (R/P)
Dividing by Y, we get:
1 = (WN)/(PY) + (RK)/(PY) = sl+ sk
where:
sl= (WN)/(PY) = Share of wages in total income
sk= (RK)/(PY) = Share of profits in total income
sl= (WN)/(PY) = (W/P)/(Y/N) = (real wage) / (average labor productivity)
If the share of labor in income (sl) remains constant over time, it must be the case that the growth rate of the real wage (W/P) is equal to the growth rate of labor productivity (Y/N) or:
d(W/P)/(W/P) = d(Y/L)/(Y/L)
If real wages grow faster than productivity [(d(W/P)/(W/P) > d(Y/L)/(Y/L)], the share of wages in income goes up and the share of profits in income falls.
If real wages grow slower than productivity [d(W/P)/(W/P) < d(Y/L)/(Y/L)], the share of wages in income goes down and the share of profits in income goes up.
Note: the increase in the last decade in the share of profits in total output is consistent with the view that real wages have increased slower than their productivity. So many of the productivity gains have gone to benefits corporate profits.
However, over long periods of time (two decades) the share of wages in income appears to be constant suggesting that, over the long-run, real wages have increased at a rate equal to labor productivity growth.
Note also that, if each factor of production is paid according to its marginal product:
dY/dN = MPN = W/P = w
dY/dK = MPK = R/P = r
where MPK = dY/dK = Marginal Product of Capital
we get that:
Y = (W/P) N+ (R/P) K = MPN N + MPK K
Application: The 1974-75 Oil Price increase
1. Real energy prices went up by 70%.
2. Productivity A fell by 5.7%
3. Real wages fell by 9%
4. Profits, employment and output fell
The oil shock is equivalent to a negative productivity shock (A) that shifts downward both the production function and the demand for labor.
Savings, Investment and the real interest rate
Production (output) is determined on the supply side.
Consider the determination of the "demand" for output.
National income accounts imply that:
GDP = C + I + G + NX
Y = GNP = C + I + G + CA
Sp = I + DEF + CA
S = Sp +Sg = I +CA
where:
Y = GNP = GDP + ixNFA
Sp = Y - T - C
DEF = G - T = - Sg
NX = X - M
CA = NX + ixNFA
Assume for simplicity:
1. Closed economy (no trade: X = M = 0, NX=0, CA=0)
2. No government policies ( G - T = DEF = 0)
Determinants of (Private) Savings:
1. Current (net of tax) income (Y-T)
2. Future income
3. Taxes and transfers, present and future
4. Wealth
5. The real interest rate. If r is higher you will save more since the return to your savings is higher. Note: r = i - p
6. Taxes on interest
Savings function relates savings (positively) to the level of the real interest rate (see figure 4).
Shifts along the curve occur because of changes in r.
Shifts of the curve occur because of changes in factors 1,2,3,4 and 6.
Determinants of Investment (in plant and equipment):
1. Productivity of future capital. If the productivity of capital is high, the return from investment is higher and I will be higher.
2. The real interest rate. At lower rates of interest, firms will invest more since borrowing costs will be lower.
3. Corporate taxes lower the after-tax return from investment and reduce the amount of investment.
Investment function relates investment (negatively) to the level of the real interest rate (see figure 4).
Shifts along the curve occur because of changes in r.
Shifts of the curve occur because of changes in factors 1 and 3.
Short-Run Equilibrium and Long-Run Dynamics
Assume first that CA=0 and G=T so that DEF=0.
In the short-run there is only one real interest rate (r) at which national savings are equal to national investment (see Figure 4).
Now add government:
1. Effects of an increase in government spending (G). Suppose that starting from a balanced budget (G=T), there is an increase in the government spending G that leads to a budget deficit (DEF = G-T >0). Since a budget deficit is equivalent to negative public savings, the budget deficit lead to a fall in total national savings (S=Sp +Sg) and shifts the S curve upward to the left (see Figure 5): total national savings are reduced at every level of r. In the new short-run equilibrium, the real interest rate is higher, investment is lower, national savings are lower, public savings are lower, private savings are higher (since r has increased) and consumption is lower (since C = Y-T -Sp).
2. Effects of an increase in taxes (T). Public savings increase as the increase in T reduces the budget deficit. As a higher T implies a lower disposable income (Y-T), private consumption fall. Therefore, the fall in private savings is smaller than the increase in taxes. So total national savings increase (the S curve shifts downward to the left) as public savings increase and private savings decrease by less than the change in public savings.
Long-Run Effects of an increase in G (see Figure 6). Lower short-run investment means that, inthe long-run, the capital stock will be lower. Such reduction in K leads to a shift downward in the production function and a shift downward of the demand for labor (as a lower K means that each worker is less productive). The result in labor markets is lower wages and slightly less employment. That means output is lower for two reasons: there are fewer people working and (most important) each person is less productive.
Caveat: Effects of productive G (government investments in infrastructure, R&D, education) which may raise output by increasing A.
Application: Are Low Real Interest Rates Good for the Economy ?
A misleading view of real interest rates is that high real interest rates are bad because they choke off investment while low real interest rate are good as they stimulate investment. This view is imprecise since real interest rates will be high in boom times while they will be low during recessions.
The economy is a good place to invest during booms since these are periods where the profitability of capital is high and firms want to invest a lot. So, booms are associated with high demand for funds by firms represented by a rightward shift of the I schedule. Booms are also associated with an increase in private savings (as income is higher in a boom both savings and consumption increase) represented by a rightward shift of the S curve. Since investment demand is more cyclical than income and savings (it increases more than income and savings in booms), a boom period will be characterized by a rightward shift of the I curve that is larger than the rightward shift of the S curve. (see Figure 7). So, we get higher real interest rates, higher saving, higher investment in booms.
The reverse happens during recessions (see Figure 8): in a recession we observe lower real interest rates, lower savings and lower investment.
For a more detailed discussion of the points above read the article in The Economist "How Low Low Can They Go?" (included in the Reading package).
For a detailed analysis of supply-side economics, see the home page on the controversy on Supply Side economics.
The current account CA measures, as well as trade in products and services, changes in the net asset position of the US vis a vis the rest of the world. If CA is negative, the US is borrowing abroad. We can write the connection between trade in goods and assets as CA + KA = 0, where KA is net new foreign borrowing (net means net of foreign lending) or the net capital inflows from abroad. Thus, if we are running a trade deficit we can think of the rest of world as an additional source of funds for the government and firms to borrow from.
CA = S - I
How, is the current account determined ? Suppose that the country we are considering is small and open to international capital markets. This means that the country can borrow or lend in international capital markets at the exogenously given world rate of interest (say r).
Then, suppose that at the given world interest rate, domestic savings are below domestic investment (this is represented in Figure 9). Unlike the case of a closed economy where S= I, in an open economy where the country can borrow and/or lend, S can differ from I since S = I + CA. So, if domestic savings are below domestic investment, the country will run a current account deficit equal to S-I.
Countries tend to run trade (and current account) deficits during booms, surpluses during recessions. Why is that? The economy is a good place to invest during a boom, by foreigners as well as domestic residents. Booms are associated with high demand for funds by firms represented by a rightward shift of the I schedule and an increase in private savings (as income is higher in a boom savings increase) represented by a rightward shift of the S curve. Since investment demand increases more than income in booms, we get a rightward shift of the I curve that is larger than the rightward shift of the S curve (see Figure 10) That leads to higher saving, higher investment and a current acount deficit (higher foreign borrowing) in booms.
The converse occurs in a recession: a recession period will be characterized by a leftward shift of the I curve that is larger than the leftward shift of the S curve (see Figure 11). Therefore, in a recession we will observe lower savings, lower investment and a current acount surplus, all of which we see in the data for a typical recession.
In short, trade deficits typically indicate that the country is doing well and is a good place to invest. In that sense a deficit is not a bad thing.
For a more detailed discussion of the determinants of the current account read the articles in The Economist "In Defence of Deficits" and "Global Capital Flows: Too Little, Not Too Much" (included in the Reading package).
There's a popular suggestion that one of the causes of the massive trade deficit in the US in the 1980s was the fiscal deficit. Even more suggestive, they're connected by the identity: Sp= I + Def + CA. So can we say that government spending was the source of the trade deficit?
Table 1 tells us that this explanation is partially correct: in the 1980s we had large budget deficits and large current account deficits even if the correlation between budget deficits and current account deficits is not perfect. Moreover, in the 1990s the improvement in the U.S. budget deficit has not been associated with an improvement of the current account of the same magnitude.
Decade Averages for US Expenditure Shares
Entries are percentages, computed from ratios of nominal variables. Saving is S=Y-C-G (our comprehensive measure).
| Variable | 1950s | 1960s | 1970s | 1980s |
| Net Exports | 0.8 | 1.1 | 0.8 | -0.8 |
| Saving | 16.4 | 16.7 | 17.2 | 14.9 |
| Investment | 15.1 | 14.5 | 15.6 | 15.3 |
| Consumption | 63.6 | 62.7 | 62.7 | 65.2 |
| Government | 20.0 | 20.7 | 20.1 | 19.9 |
Figure 1. The Production Function
Figure 2. The Marginal Product of Labor
Figure 3. Labor Market Equilibrium
Figure 4. Savings and Investment Schedules
Figure 5. Short-Run Equilibrium (Effect of An Increase in G)
Figure 7. High Real Intrest Rates During an Economic Boom
Figure 8. Low Real Interest Rates during of a Recession
Figure 9. Determination of the Current Account
Figure 10. The Current Account Deficit in a Boom
Figure 11. The Current Account Surplus in a Recession
