In the previous section, the calibration shows that there is no indeterminacy in my model under traditional parametrization. However, according to previous studies, in-determinacy should be more likely to be generated in a two-sector model than in a one-sector model. The reason is that a two-sector model allows the agents in the econ-omy to much more freely choose the allocation of their resources; hence, the impact from self-fulfilment expectation would be enhanced in a two-sector model.
For instance, if the agents believe that it would be higher return in the first sector, then the agents would increase the capital and labor inputs in the first sector. As for labor allocation, since the aggregate labor supply of the agents remains constant, increasing the labor supply in one sector must crowd out the labor supply in the other sector.
This crowding-out phenomenon results in the decline of the marginal product of capital in the second sector; thus, the agents now would put more capital into the higher-return sector. This belief would lead to a new adjustment of the optimal allo-cation of capital across the two sectors. However, since the steady state is determined by the fundamental economy variables, the agents would finally realize that such an increase is only temporary and expect a depreciation in the “higher-return” sector. By then the agents start to re-allocate their resources to the original steady state level.
The whole process would create a new convergence path toward the equilibrium, thus indeterminacy appears.
In my model, I transform the traditional leisure term into a second sector produc-tion. This modification should have made our model much prone to obtaining an inde-terminate equilibrium; nonetheless, the calibration results indicate that the dynamic characteristics of the equilibrium would be a saddle under the traditional combination of parameters. Consequently, in section 3.4, Figure 5 illustrates the numerical range of χ and θ for indeterminacy. However, the numerical range for indeterminacy is unob-servable in reality and section 3.3 also shows the unrobustness of Benhabib and Farmer
(1994)’s model. Apparently, once I increase the value of β by a very small amount, the original conclusion of Benhabib and Farmer (1994) can not continue to hold. A possible reason why the result of Benhabib and Farmer (1994) was not robust might be due to the adjustment mechanism of labor supply. It is because the agents are allowed to reallocate the quantity of labor supply in each sector that an endogenous shock, say a belief, could lead to a indeterminate equilibrium in a two-sector model.
Compared with one-sector models, leisure can not directly increase the agents’
utility in my model since there is no “pure” leisure term in the utility function; in other words, an increase in GDP resulting from increasing returns would also lead to an increase in the two production goods (y1, y2). Thus, no stabilizer such as leisure in one-sector model drives the agents to forming the reverse belief forcing the economy to return to the steady state level determined by fundamental factors. On the other hand, in comparison with two-sector models such as Benhabib and Farmer (1996) or Harrison (2001), the second sector production, in fact, simultaneously has the role of consumption and investment. Once some expect that there is higher return in one sector, this will drive the labor supply out from the lower-return sector and marginal production of capital in that sector would decrease, too. This crowding out effect may induce a large fall in this sector production; thus, the welfare gain from a higher-return sector may not be able to compensate for the welfare loss caused by the decrease in a lower-return sector. For this reason, an optimistic belief might not have equally strong impact like in other two-sector models.
5 Summary and Conclusion
Most previous literature claims that a two-sector model would generate indeterminacy much more easily than a one-sector model. However, this study has examined a model with different production functions in each sector and it turns out that indeterminacy is absent under a reasonable parametrization. Furthermore, I found that the result of Benhabib and Farmer (1994) was not robust as long as the second sector was introduced having a role as a home production sector. These results may suggest that the plausible minimum level obtained from previous studies do not hold in general cases.
Nonetheless, due to lack of a suitable stabilizer and enough utility compensation in our model structure, a self-fulfilment effect might not be able to drive the business cycles, thus producing indeterminacy.
Appendices
A Maximizing Problem
Since this paper follows Benhabib and Farmer (1994)’s model structure, here I use Hamiltonian Equation to solve the maximizing problem as follows:
H= e−ρt
In this model the endogenous variables are {c1, n, s, k}, thus four first order condi-tions are
Note that equation (A.4) is exactly the same as equation (2.3); similarly, equation (A.3) is the same as equation (2.4). Besides, by rearranging the equations above one can derive the relative price (P ) of y1 and y2, here take equation (A.4) as example :
Now that the first order conditions were obtained, one can decide the steady state conditions. Since all of the dynamics equations should equal zero under steady state;
hence, the first steady state condition is the law of motion
c∗1+ δk∗ = [A(s∗k∗)α(1+θ1)(n∗)(1−α)(1+θ2)]
c∗1+ δk∗ = y1∗. (B.1)
Next, because equation (A.2) implies c˙c1
I combine equation (B.1) and (B.2) to obtain the ratio of consumption of the capital-intensive sector over the aggregate capital input under steady state that
c∗1
For analyzing the dynamic properties of this model, it needs to compute the elements of the Jacobian Matrix. First, I combine equation (A.2) and (A.3) to derive the partial derivative of labor with respect to consumption of the capital-intensive sector and the aggregate capital input, the answers are as follows:
∂n
Note that Ω is, in fact, identical to the necessary condition for indeterminacy in Benhabib and Farmer (1994); that is, Ω represents the difference between the slope of labor demand curve and the slope of labor supply curve, the details will be introduced in the following section.
Secondly, I differentiate equation (2.6) and (2.8), thus the elements of the Jacobian
Finally, I use these elements to calculate the trace and determinant of the Jacobian Matrix. The results are the following:
T R = −δ + s[y1
In the section 3, these two equation are what we use to compute the calibration results.
D Duplication of Benhabib and Farmer (1994)
Since this model follows the basic settings of Benhabib and Farmer (1994), we can simply duplicate their model by proper setting of the parameters. Keeping in mind that there is only one production sector in Benhabib and Farmer (1994), this implies that the representative agent has no need to decide how to allocate the capital allocation.
That is, s = 1. In addition, the production function of y2 also abides by the Cobb-Douglas setting. As long as the capital share equals to zero, y2 can be also considered as the term of leisure in Benhabib and Farmer (1994). That is, I let β be equal to zero.
The representative agent maximizing problem becomes
max
And the one-sector production function would be as follows
y = Akαn1−α(k)αθ1(n)(1−α)θ2. (D.2)
In addition, choose variables in this model are now consumption (c), labor (n), capital (k) and the shadow price. From the first order conditions for labor market, one could derive that the slope of the aggregate labor demand and supply curve are
∂log w
Next, for the dynamic analysis I directly plug β = 0 and s = 1 into Equation (C.7) and (C.8), thus obtaining the duplicated trace and determinant as follows
T RD = −δ + α(y
Note that φD1 and φD2 are just coincident with the slope of labor demand and supply curve severally. Hence, in this section I will present how to get the identical condition for indeterminacy as Benhabib and Farmer (1994). Since indeterminacy would be
generated only if T rD < 0 and DetD > 0 are simultaneously satisfied; thus, this requirement could be described as the following inequality system:
−1 − φD2 φD1 − φD2
<0
[α(1 + θ1) − 1] −1 − φD2 φD1 − φD2
>0.
At first, I could simplify these two inequalities as one single condition; that is,
1 + φD2
φD1 − φD2
>0. (D.7)
To decide how this inequality would be satisfactory, there are two cases that needs to be discussed: (1) 1 + φD2 <0 and φD1 − φD2 <0 or (2) 1 + φD2 >0 and φD1 − φD2 >0.
As mentioned before, φD2 is also the slope of the labor supply curve. Besides, following the model setting, χ > 0 and n ∈ (0, 1) need to be satisfied, too. This indicates that 1 + φD2 should always be greater than zero; hence, the former case just contradicts and the latter case would lead to indeterminacy. That is, φD1 − φD2 > 0 would be the necessary and sufficient condition for indeterminacy. One can notice that this term is just consistent with the condition which Benhabib and Farmer (1994) proposed.
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