Properties of the CU Model

Capacity Competition Externality and Inefficient Capital Accumulation

Consider a symmetric equilibrium. Let qt denotes the value of capital. A representative firm’s investment condition is given by

q_t= Et mul-tiplier of the capacity constraint. Equation (2.90) shows that the return of capital comes from two sources. First, the capacity constraint can be relaxed by capacity expansion:

µt+1. Second, profitable demand can be stolen by capacity expansion: Λut+1.

Note that firms are able to charge a positive net markup because buyers are not fully attentive to prices. Hence, if a firm expands its capacity while other firms do not, it steals profits from others by stealing demand. Because of this capacity competition externality, capital has an extra value in addition to the value of relaxing the capacity constraint.

If the capital accumulation decisions made by firms are partially efficient, the present value

of aggregate costs should be minimized: subject to the law of motion for capital (2.89), where the aggregate variable cost function C is the same as the variable cost function at the firm-level: C (y, k, l_f; wt) = wtαvy, for all y ≤ min {Ak,^lf/α_f}.

However, the optimal investment condition obtained from the minimization problem above is given by

which shows that the only value of capital in aggregate is to relax the capacity constraint.

Hence, the capital accumulation decisions made by firms are not partially efficient.

Chronic Excess Capacity and Capital Resource Slackness

If the unit cost of processing information is sufficiently large, the capacity competition externality will be strong enough to cause chronic excess capacity.

Particularly, if and only if Λ ∈ ^r+δ_A + wα_f, 1
, capacity at the firm-level will be under-utilized in steady state. Since the aggregate variable cost function C is the same as the variable cost function C at the firm-level, aggregate capacity is also underutilized. Thus, the economy exhibits chronic excess capacity.

Finally, in the steady state where capacity is in excess, a marginal decrease in capital will not cause an increase in the real MC as the real MC curve is flat: C_yk = C_yy^y_k = 0. Thus, capital resources are slack locally around the steady state.

2.5.3 Calibration

To exam the model dynamics quantitatively, the model parameters are calibrated to match the statistics observed in the U.S. data. An advantage of the calibration exercise conducted here over the Bayesian estimation conducted in section 1.5 of Chapter 1 is that the pa-rameters values are determined more transparently.

We have twelve parameters to calibrate. The depreciation rate δ is calibrated to match the average ratio of gross private domestic investment to private fixed assets from 1947 to 2016 in the NIPA published by the BEA.

The curvature of the consumption utility function γ is chosen to give an elasticity of inter-temporal substitution of 0.5. In fact, because of a locally flat real AS curve, the value of γ does not matter for the local dynamics of the CU model and consumption is directly determined by consumption demand: ˆc_t= ˆz_c,t.

The persistence of consumption demand shocks ρ_c can be calibrated to match the auto-regressive coefficient of the consumption series published by the BEA from the first quarter of 1948 to the first quarter of 2017.

The rest of the nine parameters are jointly calibrated to achieve the following targets.

The price in utils in steady state λ is normalized to 1. This target is mostly associated with the scaling parameter φ in the representative household’s utility function.

The size of output in steady state y is normalized to 1. This target is mostly associated with the dis-utility of labor ¯ω, which affects the size of the economy through the supply of labor.

The capacity utilization rate in steady state u is matched to the average of the total industry capacity utilization rate from 1967 to 2016 reported by the Federal Reserve Board (FRB). This target is mostly associated with the subjective discount factor β, which affects the opportunity cost of holding capacity.

The investment to output ratio in steady state ⁱ/y is matched to the average ratio of gross private domestic investment to GDP from 1947 to 2016 in the NIPA. This target is mostly associated with the productivity of capital A, which affects the capital to output ratio.

The labor underutilization rate in steady state is defined as one minus the ratio of the labor hours actually utilized to the total labor hours that the representative household can potentially supply: 1 − l, where the total hours that the representative household could supply is normalized to one. I choose the average of U-5 and U-6 from 1994 to 2016 published by the BLS as a target for the labor underutilization rate.This target is mostly associated with the direct labor required per unit of output αv, which affects the demand for labor.

The labor share of income in steady state ^wl/y is matched to the average labor share of

income estimated by the BLS from 1946 to 2016. This target is mostly associated with the unit cost of processing information Λ, which affects the size of the monopolistic profit and thus the labor share of income.

The cyclicality of the labor productivity generated by the model is matched to the same statistics calculated based on the output and hours data published by the BEA and the BLS from the first quarter of 1948 to the first quarter of 2017. This target is mostly associated with the indirect labor required per unit of capacity α_f. Note that locally around the steady state, the fluctuation of hours is given by

ˆl_t=

Ignoring the cyclical fluctuations of capital, we have that the cyclicality of the labor productivity is closely linked to the share of indirect labor hours in total hours worked:

β_y^LP ≡

The relative volatility of investment to consumption generated by the model is matched to the same statistics calculated based on the consumption and investment series published by the BEA from the first quarter of 1948 to the first quarter of 2017. This target is mostly associated with the curvature of the capital adjustment cost φ_k, which dampens the response of investment to consumption demand shocks.

The auto-correlation of investment generated by the model is matched to the same statis-tics calculated based on the investment series published by the BEA from the first quarter of 1948 to the first quarter of 2017. This target is mostly associated with the curvature of the investment adjustment cost φi, which helps generate a hump-shaped response of investment to consumption demand shocks.

Table 2.5 summarizes the calibrated parameter values and their mostly associated targets.