How Effective is Lithium Recycling as a Remedy for Resource Scarcity?

Abstract

We investigate to what extent recycling can remedy resource scarcity, and whether market intervention is desired. For doing so, we develop a dynamic model of the global lithium market. An efficient market for resource waste allows consumers to internalize the waste value when they buy the resource. Without a market for lithium waste, we show that the efficient outcome can alternatively be realized through a proper set of subsidies to either buyers or sellers of both virgin and recycled lithium. We find that optimal subsidies may become quite substantial in the second half of this century. The size of these subsidies depends, however, on several uncertain assumptions such as technological progress in recycling, quality-grade of recovered lithium, and demand elasticity.

Appendices

Appendix 1

1.1 Cost Function

To make the model fit better with the current situation in the lithium market, we add two cost elements for lithium extractors. In the numerical simulations, the cost function looks like:

$$c^{E} \left( {A_{jt} } \right) = c_{j0} e^{{\eta_{j} A_{jt} - \tau_{E} }} x_{jt} - c_{j}^{T} x_{jt} - c_{jt}^{add} \left( {x_{jt} ,x_{j0} } \right)x_{jt}$$

(27)

So total cost depends on three parts. The first part, $(c_{j0} e^{{\eta_{j} A_{jt} - \tau_{E} t}} x_{jt} )$ adopted from e.g. Grimsrud et al. (2016), assumes unit extraction costs (starting at $c_{j0}$) increase with accumulated supply $(A_{jt}$) and decrease with (exogenous) technological progress $\left( {\tau_{E} = 0.05} \right)$. The parameter $\eta_{j}$ represents the rising costs rate as accumulated production increases and is calibrated to the initial stock levels of lithium resources for each producer. The second part, ($c_{j}^{T} x_{jt}$), is simply linear transportation costs to the world market. The third part, $c_{jt}^{add} \left( {x_{jt} ,x_{j0} } \right),$ is a quadratic term to consider that it is costly to ramp up production substantially in the short to medium term, and also that sunk costs make sudden output reductions less profitable. One particular example is Bolivia, which has enormous and profitable lithium resources, but where production is close to zero due to institutional barriers such as constraints on property rights and on foreign investments. The term $c_{jt}^{add} \left( {x_{jt} ,x_{j0} } \right)$ equals zero if production equals the base year output, is quadratic in deviation from the base year output, and reduces gradually over time.

The first order condition for producer j, subject to a positive outcome $(x_{jt} > 0)$ is:

$$\frac{{\delta c^{E} \left( A \right)}}{\delta x} = c_{j0} e^{{\eta_{j} A_{jt} - \tau_{E} }} - c_{j}^{T} - c_{jt}^{add} \left( {x_{jt} ,x_{j0} } \right)$$

(28)

Table 5 displays the value of a range of parameters applied in the numerical model.

Table 5 Parameters applied in the model

Appendix 2

2.1 Utility and Demand Functions

We assume the following standard utility function for use of lithium $y_{it}$ (in all consuming sectors i):

$$U_{it} \left( {y_{it} } \right) = \varPhi_{i} + \frac{{\alpha_{i} }}{{1 + \alpha_{i} }}y_{i0} \sigma_{it} \left( {\frac{{y_{it} }}{{y_{i0} \sigma_{it} }}} \right)^{{\frac{{1 + \alpha_{i} }}{{\alpha_{i} }}}}$$

(29)

where $\varPhi_{i}$ is a constant, $\alpha_{i}$ represents the (long-term) price elasticity of demand, and $y_{i0}$ denotes the initial demand level. The factor $\sigma_{it}$ is an exogenous growth function reflecting the underlying growth in demand. The elasticity $\alpha_{i}$ is − 0.5 in the benchmark scenarios.¹³ This gives the following marginal utility function:

$$U_{it}^{ '} \left( {y_{it} } \right) = \left( {\frac{{ y_{it} }}{{y_{i0} \sigma_{it} }}} \right)^{{\frac{1}{{\alpha_{i} }}}}$$

(30)

And thereby the derived demand function that we use in the model numerical simulations:

$$y_{it} - y_{i0} \sigma_{it} \left( {\frac{{p_{t} }}{{p_{0} }}} \right)^{{\alpha_{i} }} \ge 0 \bot y_{it} \ge 0$$

(31)

The elasticity $\alpha_{i}$ is − 0.5 in the benchmark scenarios.¹⁴ The growth function $\sigma_{it}$ is calibrated based on projections from the IEA (2018a, b) for the medium term to 2030, and Kushnir and Sanden (2012) for the longer term, using a logistic functional form with several parameters. Obviously, the long-run growth in demand is highly uncertain.

The demand growth factor $\sigma_{it}$ is calibrated based on the growth rates shown in Table 6, and the following functional form:

$$\sigma_{it} = \frac{{\sigma_{i1} }}{{\sigma_{i2} + \sigma_{i3} e^{{ - \sigma_{i4} t}} + \sigma_{i5} e^{{ - \sigma_{i6} t^{2} }} }}$$

(32)

Table 6 Annual Growth rate in lithium demand in sector i (given price in 2015)

The calibrated parameters are displayed in Table 7.

Table 7 Parameters in the demand growth function

Appendix 3

3.1 Recycling Cost Function

From the recycling costs function (18) we derive the following first order conditions:

$$\frac{\delta CR}{\delta w} = e^{{ - \kappa_{t} }} \left[ {cr_{0} - \ln \left( {1 - \left( {\frac{w}{l}} \right)^{\rho } } \right)} \right] + \frac{{\rho e^{{ - \kappa_{t} }} \left( {\frac{w}{l}} \right)^{\rho } }}{{1 - \left( {\frac{w}{l}} \right)^{\rho } }} = P^{L}$$

(33)

$$\frac{\delta CR}{\delta l} = e^{{ - \kappa_{t} }} \left[ { - \rho \frac{{\left( {\frac{w}{l}} \right)^{\rho + 1} }}{{1 - \left( {\frac{w}{p}} \right)^{\rho } }}} \right]$$

(34)

Or equivalent to

$$\frac{\delta CR}{\delta l} = e^{{ - \kappa_ t}} \left[ {\rho w\frac{{\left( {\frac{w}{l}} \right)^{\rho } }}{{\left( {\left( {\frac{w}{p}} \right)^{\rho } - 1} \right)l}}} \right]$$

Table 8 displays the parameters value applied in the numerical calculation of equations (18), (33) and (34).

Table 8 Parameters in the recycling cost function

Appendix 4

4.1 Resources and Subsidies Analysis

Figure 8 shows accumulated extraction for individual countries in the benchmark scenario. It also shows, for each country, when accumulated extraction surpasses the currently identified resources. We see that Australia and Chile, the two biggest producers today, will run out of identified resources around 2065. Chile has large resources and continues as one of the largest producers throughout the century, whereas Australia has rather limited resources compared to the others. In the first half of our time horizon, Argentina and China also have large identified resources and are important suppliers. In the second half of this century, USA and the rest of the world become important suppliers and Bolivia is the biggest producer of lithium at the end of the century. Up to date, these estimates are susceptible to change when more resources are identified, and producers tend to accelerate their extraction rates.

Fig. 8

Accumulated production and identified resources of lithium

Figure 9 shows the optimal subsidy rates in the transport sector across scenarios. We see that the subsidy rates in 2100 vary between 14 and 30 USD per kg of LCE. As a comparison, the initial price of lithium in 2016 is 8.7 USD per kg of LCE. However, until 2050 all subsidies remain below 7 USD per kg of LCE.

Fig. 9

Optimal subsidies (shadow price of lithium in use) in the Transport sector across scenarios

Figure 10 shows how net present value (NPV) of welfare gains (relative to unregulated market) and subsidy payments are affected if the optimal subsidies end prematurely in different years—in percent of the corresponding numbers in the efficient solution. For instance, whereas the welfare gains in the efficient solution (vis-à-vis market solution) is 4.5 billion USD, the welfare gain drops by almost 90 percent, to 0.5 billion USD, if the subsidies end in 2060. Total subsidy payments drop from 100 to around 25 billion USD. Hence, ending the subsidies prematurely implies that most of the welfare gains are lost. This shows the importance of establishing a market for recycled lithium, so that subsidies are no longer needed. The figure also shows the share of lithium being recycled in different years in the efficient solution. We see e.g. that in 2060, this share is 90 percent

Fig. 10

NPV of welfare gains and subsidy payments if subsidies end prematurely in the given year. Percent of efficient solution (ES). Share of lithium being recycled in the efficient solution in the different years