4.7 Implications

Following Campbell, assume that each period corresponds to a quarter, and take for baseline parameter values a = , g = 0.5%, n = 0.25%, S = 2.5%, PA = 0.95, PC = 0.95, and G, p, and b such that (G/Y)* = 0.2,-r* = 1.5%, and =

priate in the log-linearized version of (4.23) is not E,[Ztti], but ln£fle+l- Campbell (1994) addresses this difficulty by assuming that Z is normally distributed with constant variance; that is, has a lognormal distribution. Standard results about this distribution then imply that InEtle*] equals E,IZ,, i] plus a constant (see, for example, Mood, Graybill, and Boes [1974] or any other statistics textbook). Thus we can express the log of the right-hand side of (4.23) in terms of E, [Z,+i and constants. Finally, Campbell notes that given the log-linear structure of the model, if the underlying shocks-the Eas and fos m (4.9) and (4.11)-are normally distributed with constant variance, his assumption about the distribution of Z,+, is correct.

See Problem 4.10 for the implications of these parameter values for the balanced growth path.

the as; this is enough to determine the as in terms of the underlying parameters.

Unfortunately, the model is sufficiently complicated that solving for the as is tedious, and the resulting expressions for the as in terms of the underlying parameters of the model are complicated. Even if we wrote down those expressions, the effects of the parameters of the model on the as, and hence on the economys response to shocks, would not be transparent.

Thus, despite the comparative simplicity of the model and our use of approximations, we must still resort to numerical methods to describe the models properties. What we will do is choose a set of baseUne parameter values and discuss their implications for the as in (4.43)-(4.44) and the bs in (4.52). Once we have determined the values of the as and bs, equations (4.43), (4.44), and (4.52) specify (approximately) how consumption, employment, and capital respond to shocks to technology and government purchases. The remaining equations of the model can then be used to describe the responses of the models other variables-output, investment, the wage, and the interest rate. For example, we can substitute equation (4.44) for L into the log-Unearized version of the production function to find the models implications for output:

Yt = akt + a - am + At)

= akt +{1- a){aLKkt + + QLcGt + At) (4.53)

= [a + (l- a)aLK]kt + (1 - a)(l + aLA)Af + aLcGf

l.Or

2 4 6 8

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Quarters

A-K-L-

FICURE 4.2 The effects of a 1% technology shock on the paths of technology, capital, and labor

The Effects of Technology Shocks

One can show that these parameter values imply Ala ~ 0.35, aix = -0.31, ac\ - 0.38, - 0.59, - 0.08, and - 0.95. These values can be used to trace out the effects of a change in technology. Consider, for example, a positive 1% technology shock. In the period of the shock, capital which is inherited from the previous period) is unchanged, labor supply rises by 0.35%, and consumption rises by 0.38%. Since the production function is KiAL), output increases by 0.90%. In the next period, technology is 0.95% above normal (since pa = 0.95), capital is higher by 0.08% (since . - 0.08), labor supply is higher by 0.31% (0.35 times 0.95, minus 0.31 times 0.08), and consumption is higher by 0.41% (0.38 times 0.95, plus 0.59 times 0.08); the effects on A, K, and L imply that output is 0.86% above normal. And so on.

Figures 4.2 and 4.3 show the shocks effects on the major quantity variables of the model. By assumption, the effects on the level of technology die away slowly. Capital accumulates gradually and then slowly returns to normal; the peak effect is an increase of 0.60% after 20 quarters. Labor supply jumps by 0.35% in the period of the shock and then declines relatively rapidly, falling below normal after 15 quarters. It reaches a low of -0.09% after 33 quarters, and then slowly comes back to normal. The net result of

l.Or

0.4

-0.2

I i I

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Quarters

FIGURE 4.3 The effects of a 15 consumption

> technology shock on the paths of output and

the movements in A, A, and I is that output increases in the period of the shock and then gradually returns to normal. Consumption responds less, and more slowly, than output; thus investment is more volatile than consumption.

Figure 4.4 shows the percentage movement in the wage and the change in percentage points in the interest rate at an annual rate. The wage rises and then retums very slowly to normal. Because the changes in the wage (after the unexpected jump at the time of the shock) are small, wage movements contribute little to the variations in labor supply. The annual interest rate increases by about one-seventh of a percentage point in the period of the shock and then returns to normal fairly quickly. Because the capital stock moves more slowly than labor supply, the interest rate dips below normal after 14 quarters. These movements in the interest rate are the main source of the movements in labor supply.

To understand the movements in the interest rate and consumption, consider for simplicity the case where labor supply is inelastic, and recall that r = a{ALJK)° - S. The immediate effect of the increase in A is to raise r. Since the increase in A dies out only slowly, r must remain high unless increases rapidly. But since depreciation is low, a rapid rise in would require a large increase in the fraction of output that is invested. But