33. Non-Convex Adjustment Cost

In this section, we will be considering a model for each individual firm that faces a wedge between the buy price \(p_b\) and the sell price \(p_s\) of capital where \(p_b\ge p_s\). And the adjustment cost comes from the capital price wedge in the sense that firms cannot freely convert back and forth between capital and goods (e.g. to sell and then repurchase one unit of capital, the cost is \(p_b-p_s\)). But such cost does not involve a convex function \(C\!\left(\frac{k_{t+1}(s^t)}{k_t(s^{t-1})}\right)\) as in section 32.

33.1 Set-up

• Each firm faces a buy price \(p_b\) and sell price \(p_s\) of capital where \(p_b\ge p_s\).
• Firms are hit by productivity (or demand) shock \(z_t(s^t)\) in every period.
• The firm's production function (revenue function) is \(k_t(s^t)^{\alpha}z_t(s^t)^{1-\alpha}\).
  • Only capital \(k_t(s^t)\) is involved in production in every period (so there is no labor in the model); the firm's production (revenue) has diminishing returns to scale since \(0<\alpha<1\), but is h.o.d. < 1 in \((k_t,z_t)\).
• The cost of purchasing capital in period \(t\) is given by \(p_b\times\max\{k_t(s^t)-k_{t-1}(s^{t-1}),0\}\).
• The payoff from selling capital in period \(t\) is given by \(p_s\times\max\{k_{t-1}(s^{t-1})-k_t(s^t),0\}\).
• The firm will discount profits at rate \(\beta<1\).
• We also assume that the productivity (demand) shock ratios \(\gamma_{t+1}(s^{t+1})=\dfrac{z_{t+1}(s^{t+1})}{z_t(s^t)}\) are i.i.d., and assume \(\beta\mathbb{E}[\gamma]<1\).
• Here the history \(s^t\) is included simply to indicate the measurability property, i.e. each variable is uniquely pinned down given history \(s^t\).

33.2 The value function and recursive representation

33.2.1 The sequence problem

Firm's period profit is the production plus revenue from capital sale minus cost from capital purchase, i.e.

$$ V(k_0,z_0)=\max_{\{k_t\}_{t=1}^{\infty}}\sum_{t=1}^{\infty}\beta^t\big((k_t)^{\alpha}z_t^{1-\alpha}+p_s\max\{k_t-k_{t+1},0\}-p_b\max\{k_{t+1}-k_t,0\}\big) \tag{33.1} $$

By exactly the same argument as in the proof for proposition 32.1, \(V(k,z)\) is h.o.d. 1 in \((k,z)\) (i.e. \(V(\alpha k_0,\alpha z_0)=\alpha V(k_0,z_0)\); see (33.2), (33.3)).

33.2.2 Recursive formulation

The original Bellman equation for the firm's value function \(V(k,z)\):

$$ V(k,z)=\max_{k'}\big\{(k')^{\alpha}z^{1-\alpha}-p_b\max\{k'-k,0\}+p_s\max\{k-k',0\}+\beta\mathbb{E}[V(k',z')]\big\} \tag{33.4} $$

where \(z\) is the current period shock, \(k\) is the begin-of-current-period capital, \(k'\) is the current-period-production (begin-of-next-period) capital, and \(z'\) is the next period shock. Now define \(x=\dfrac{k}{z}\), \(x'=\dfrac{k'}{z}\), \(z'=\gamma z\), and \(v(x)\equiv v(\frac{k}{z})\equiv V(\frac{k}{z},1)=\frac{V(k,z)}{z}\). Divide \(z\) through (33.4) to obtain a simplified version of Bellman equation:

$$ v(x)=\max_{x'}\big\{(x')^{\alpha}-p_b\max\{x'-x,0\}+p_s\max\{x-x',0\}+\beta\mathbb{E}\big[\gamma v(x'/\gamma)\big]\big\} \tag{33.5} $$

with single state variable \(x\) and single choice variable \(x'\).

33.3 Model with no adjustment costs (\(p_b=p_s=p\))

$$ v(x)=\max_{x'}\big\{(x')^{\alpha}-p(x'-x)+\beta\mathbb{E}[\gamma v(x'/\gamma)]\big\} $$

The envelop condition is \(v'(x)=p\) (33.6). The first-order condition for \(x'\) is \(\alpha(x')^{\alpha-1}-p+\beta\mathbb{E}[v'(x'/\gamma)]=0\), and using the envelop condition \(v'=p\),

$$ \alpha(x')^{\alpha-1}=p(1-\beta) $$

So, with no adjustment cost, \(x'=x\) always (firm adjusts to the optimal every period).

33.4 Model with adjustment costs (\(p_b>p_s\))

33.4.1 Conjecture of the solution

Assume that \(\gamma\in\{e^{-\Delta},1,e^{\Delta}\}\) i.i.d. with probabilities \(\pi,1-2\pi,\pi\) respectively. This implies \(\mathbb{E}[\ln\gamma]=-\Delta\pi+0+\Delta\pi=0\) and \(\text{Var}(\ln\gamma)=2\Delta^2\pi\). The assumption \(\beta\mathbb{E}[\gamma]<1\) here implies that \(1>\beta(\pi e^{-\Delta}+(1-2\pi)+\pi e^{\Delta})\). To solve for an equilibrium, we will conjecture that the solution \(x'\) satisfies

$$ x'=\begin{cases}\underline x & \text{if }x<\underline x\\ x & \text{if }\underline x\le x\le\bar x\\ \bar x & \text{if }x>\bar x\end{cases} \tag{33.7} $$

which is an (S,s) policy.

33.4.2 Use the conjecture to solve the value function and verify the conjecture

For \(x\in[\underline x,\bar x]\), \(x'=x\), and the Bellman equation (33.5) implies

$$ v(x)=x^{\alpha}+\beta\big[\pi e^{-\Delta}v(xe^{\Delta})+(1-2\pi)v(x)+\pi e^{\Delta}v(xe^{-\Delta})\big] \tag{33.8} $$

Smooth pasting conditions: consider \(x<\underline x\): at \(x=\underline x\) if the firm experiences a positive shock \(e^{\Delta}\) to \(z\) (i.e. a negative shock \(e^{-\Delta}\) to \(x\) making it become \(\underline xe^{-\Delta}\)), the firm will buy capital until \(x=\underline x\), i.e. \(v(\underline xe^{-\Delta})=-p_b(\underline x-\underline xe^{-\Delta})+v(\underline x)\); the first-order Taylor expansion around \(\Delta=0\) gives \(v'(\underline x)=p_b\). Similarly, consider \(x>\bar x\): \(v(\bar xe^{\Delta})=p_s(\bar xe^{\Delta}-\bar x)+v(\bar x)\) gives \(v'(\bar x)=p_s\). So

$$ v'(\underline x)=p_b,\qquad v'(\bar x)=p_s $$

are called smooth pasting conditions (this says the value of capital below the threshold \(\underline x\) is what the firm doesn't have to buy when adjusting up to \(\underline x\), i.e. the cost of purchasing capital \(p_b\); the value of capital above the threshold \(\bar x\) is what the firm can earn from the market buy price when adjusting down (selling) to \(\bar x\), i.e. the revenue of selling capital \(p_s\)).

Figure 20 (Marginal Value of Capital, paraphrased): \(v'(x)\) versus \(x\): for \(x<\underline x\), \(v'(x)=p_b\) (flat); for \(x\in[\underline x,\bar x]\), \(v'(x)\) decreases from \(p_b\) to \(p_s\); for \(x>\bar x\), \(v'(x)=p_s\) (flat). Smooth pasting + super contact make the curve smoothly tangent at the thresholds.

33.4.3 The stationary distribution of \(x'\): discrete uniform distribution

For simplicity we impose the assumption that there are integer number of steps for shocks to get the state variable from \(\underline x\) to \(\bar x\), i.e. \(\dfrac{\bar x}{\underline x}=e^{n\Delta}\) for \(n\in\mathbb{Z}_+\). Define \(\phi_t(x')\) as the probability of a firm having \(\frac{k_t}{z_t}=x'\) (prime indicates after-adjustment) in period \(t\). Then we obtain the discrete analog of Kolmogorov forward equation:

$$ \text{for }x'\in[\underline xe^{\Delta},\bar xe^{-\Delta}]:\quad \phi_t(x')=\pi\phi_{t-1}(x'e^{-\Delta})+(1-2\pi)\phi_{t-1}(x')+\pi\phi_{t-1}(x'e^{\Delta}) \tag{33.16} $$

$$ \text{for }x'=\bar x:\quad \phi_t(\bar x)=\pi\phi_{t-1}(\bar xe^{-\Delta})+(1-\pi)\phi_{t-1}(\bar x) \tag{33.17} $$

$$ \text{for }x'=\underline x:\quad \phi_t(\underline x)=(1-\pi)\phi_{t-1}(\underline x)+\pi\phi_{t-1}(\underline xe^{\Delta}) \tag{33.18} $$

There is positive probability to reach any point in \(\mathcal{X}=\{\underline x,\underline xe^{\Delta},\ldots,\bar xe^{-\Delta},\bar x\}\) within finite steps, so there is a unique ergodic set \(\mathcal{X}\), and thus a unique stationary distribution. In steady state (positive and negative shocks are symmetric in size and probability), rewrite (33.16), (33.17):

$$ \phi(x')=\frac{\phi(x'e^{-\Delta})+\phi(x'e^{\Delta})}{2}\ \text{(33.19)},\qquad \phi(\bar x)=\phi(\bar xe^{-\Delta})\ \text{(33.20)} $$

Use (33.20) in (33.19) repeatedly to obtain \(\phi(\bar x)=\phi(\bar xe^{-\Delta})=\cdots=\phi(\underline x)\), then by normalization \(\phi(\bar x)+\cdots+\phi(\underline x)=1\), the discrete uniform distribution

$$ \phi(\bar x)=\cdots=\phi(\underline x)=\frac{1}{n+1} $$

33.4.4 Discussion on the discrete uniform distribution

• On the firm level: suppose a firm invests in period 0 (start from \(\underline x\)), asymptotically (very long time) the firm's \(x'\) follows the discrete uniform (the unique stationary distribution). This is only true when the firm thinks about a very far future. Once that future comes, the right next few periods are not discrete uniform. Around period 0, the firm is still near \(\underline x\), so its probability of investing again is still higher, and it reduces gradually, which displays a pattern of serially correlated investment (on individual firm level): when the firm think about the very far future, the probability of investing in any far future period is \(\frac{1}{n+1}\pi\); but when the firm invests in this period, its probability of investing again in right next period is much higher, which is \(\pi\).
• On the aggregate level:
  • Individual (idiosyncratic) shock: suppose there are many firms each facing its own idiosyncratic shocks, and on the aggregate level it is a distribution across firms (on individual firm level the distribution is across time). Since each grid is equally likely to be hit, in the steady state, firms are uniformly distributed at each of the \(n+1\) grids. Even though on individual firm level the distribution across time is not uniform in short run, on aggregate level, equally likely positive and negative shocks cancel with each other on average, and the distribution over firms is always uniform given that the number of firms is very large.
  • Aggregate shock: suppose that all firms face the same aggregate shocks. Again, since each grid is equally likely to be hit, in the steady state, firms are uniformly distributed at each of the \(n+1\) grids. However, once the shock takes place, each firm is moved by one grid in the same direction, so the distribution is not uniform anymore. The distribution of firms over the grids will become uniform only when looking into extremely far in the future, which then becomes uniform for the same reason as the distribution over time on the firm level.
  • Figure 21 (Before the MIT Shock: Steady State Initial Distribution, paraphrased): \(\phi(x')\) uniform at \(\frac{1}{n+1}\) over the grids \(\underline x\) to \(\bar x\).
  • Then a MIT shock (one-time unexpected shock) happens: suppose the shock is a positive shock to \(z\), which moves all firms one grid down (in \(x\)); the distribution is not uniform anymore. Only in infinitely far future without any further shocks (or with equally like up and down shocks every period) can the system restore the discrete uniform distribution. Figure 22 (After the Positive MIT Aggregate Shock: One Period Lag, paraphrased): one grid bunched at \(\frac{2}{n+1}\) with the rest at \(\frac{1}{n+1}\).
  • Suppose the MIT shock happens to \(\pi\) (the probability of up and down shocks): then the variance of all firms increases (less probability \(1-2\pi\) of staying put), and the only possible way to increase variance for a uniform distribution is to increase the range, so \(\underline x\) moves left and \(\bar x\) moves right, and the system of many firms will gradually expand to the newly created region. In the short run, the uniform distribution is broken, but after a long time, the firms will gradually spread over the new whole space, and the uniform distribution is thus restored.