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The real price of base metals exhibits a decreasing trend over time. We model base metals prices as the equilibrium of aggregate supply and demand. This allows us to study the effect of determinants of base metals prices. The trend in the price of base metals depends on technological progress, resource scarcity, natural resource taxes, and the interest rate. Under certain parameter restrictions we can explain the decreasing trend in prices over time. This phenomenon is mostly explained by the substitution effect and technological progress. We derive policy implications related to natural resource taxation.

Base metals are industrial non-ferrous metals: aluminum, copper, lead, nickel, tin, and zinc. They are used for building homes, automobiles, plants, equipment, pipes, wires, and so on. Such extensive use of base metals in industry inevitably links base metals markets to economic conditions. An article in Investopedia by Mark Riddix states that “Investors who want to know where global economies are headed should keep an eye on base metals”.

In this paper we study the determinants of trends in base metals prices. For the supply side we consider the regulated industry with a Cobb-Douglas production function for base metals. By solving the Hotelling-style problem of the regulator we find that the trend in the supply of base metals depends on natural resource tax, interest rate, technology progress and degradation of ore in nature. Under a constant return to scale (CRS) production function the supply of base metals is perfectly inelastic. Importantly, the supply is decreasing over time for realistic values of parameters. We then study the demand for base metals from the manufacturing sector. The resulting demand is also decreasing. Thus we can justify the possibility of a downward sloping price trend.

The model incorporates the following features of base metals: 1) Base metals are recyclable, and the portion of recycle materials is constant over years; 2) Producers’ incomes vary with the content of base metals distributed in the Earth’s crust; 3) Deposits are common in nature and inexpensive to access; 4) Prices show positive cross-elasticities of demand; 5) Deposits are homogeneous products, durable for storage. Our results are driven by the use of the Cobb-Douglas function for base metals production^{1}. The use of this production function reflects substitutability between inputs, particularly between capital and mineral deposits, which are degrading over time―thus making the use of capital more efficient. [

Despite the important role of base metals in the world economy there is little research on pricing. Our paper fills the gap in this literature by offering a theoretical treatment of pricing which includes the production functions for metals, manufacturers, and the set of technological and policy-relevant parameters which affect the price trend.

Theoretical Hotelling-style models predict an increasing real price for non-renewable natural resource commodities. However, empirical observations establish falling prices for these commodities ( [

The regulated industry provides the supply of base metals. The regulator maximizes the discounted difference between revenue

where ^{2}. We denote by r the cost of capital and by

Demand for base metals comes from the manufacturing sector. The manufacturing sector has a Cobb-Douglas production function

where

The industry’s cost minimization function

Using standard technique we obtain the following:

Lemma 1―The cost function of the mining sector is

where

Proof: see Appendix 5.1.

After extracting ore from the deposit, the minerals are separated from ore by the benefaction process. Smelters produce base metals from refined minerals. Production output

The regulator maximizes industry profit by choosing the extraction path. By (4) it is equivalent to choosing the rate of ore degradation

Solving the maximization problem yields the following (we denote by

Proposition 1

1) The (inverse) supply function for base metals from the mining sector is

2) If a + b = 1 then

Proof: see Appendix 5.2.

The following Corollary follows from Proposition 1:

Corollary 1

1) The price elasticity of supply is

2) If a + b = 1 then the supply of base metals is perfectly inelastic.

Proof: see Appendix 5.3.

In (9),

Evaluation of the returns to scale in metals production is a difficult task. We follow a somewhat implicit method to justify the constant returns to scale hypothesis. Empirically, [

1) High total factor productivity and good quality of ore in nature keeps the supply of base metals relatively low;

2) The growth rate of technology

3) The degradation progress of ore in nature ^{3}.

To simulate the time path of supply consider the values ^{4}. Substituting these values into (8) we obtain decreasing supply over time.

Now considering the parameters of the economy

Proposition 2

Proof: Immediate from (7) and (8).

There are several policy implications from Proposition 2: 1) If the natural resource tax

These results depend on the assumption of a Cobb-Douglas production function. According to the Inter- national Council on Mining and Metals [^{5}.

The demand for base metals comes from the manufacturing sector. Rewriting the production function (2) in per- capita form we have

Lemma 3

1) The (inverse) demand function for base metals from the manufacturing sector is

2) The price elasticity of demand is

3) The rate of change of

Proof: see Appendix 5.4.

Equation (11) implies that ceteris paribus: 1) Technological progress reduces the demand for base metals; 2) Economic growth and interest rate increase the demand.

The expression (12) decomposes

Equation (13) implies that technological progress has a positive impact on the rate of change of

In the long run the capital growth rate equals zero in the steady state:^{,}

To see the trend of equilibrium price we present supply and demand in the three-dimensional coordinate system

The

^{9}. These trends illustrate our findings.

Notice that the prices of Aluminum and Copper spike during World War I (1914-1918). The price of base metals in periods of economic crises during the 1930s and in 1996 was below trend. During the financial crisis of 2007-2009 the prices of almost all base metals were highly volatile which was depicted clearly and explained in [

The theoretical literature has no comprehensive consideration of all relevant factors that impact the price of base metal. Normally the price is defined only by the mining industry [

This paper answers the question what drives the trends of base metals prices. By modeling the industry regulator’s problem for extraction of base metal minerals in combination with the demand from the economy, we show that in the long-run the price of base metals is a function of total factor productivity of the mining industry, the availability of metals in nature, natural resource tax, interest rate and demand from economy. We decompose the price elasticity of supply and demand into responsiveness of price to changes of price determinants. Assuming constant returns to scale, the price elasticity of the supply of base metals is relatively small (

The authors thank Zhiqi Chen, Jean-Francois Tremblay, Gamal Atallah for careful reading of the manuscript and many helpful comments. Our sincere thanks go to Margaret Slade, Nguyen Van Quyen for useful suggestions and David Stambrook for extensive discussions. We also thank participants of CEA Annual Conference at Ryerson 2015 for comments and suggestions. The paper is a part of a bigger project: “Trends and Fluctuations in Base Metals Prices”, which is awarded the prize for the best paper at Vietnam Economist Annual Meeting 2015 in Thai Nguyen City, Vietnam. The authors thank SSHRC for financial support.

Nguyen BaoAnh,AggeySemenov, (2015) Trends of Base Metals Prices. Theoretical Economics Letters,05,531-540. doi: 10.4236/tel.2015.54062

The cost minimization problem is

Ignoring subscript t and solving (15) for D we have

Substituting (16) into (14) yields

The first order condition is

Thus we obtain the conditional demand function for K

The conditional demand function for D is

Substituting (18) and (19) into (14) yields the cost function of the industry

Finally denote

The Hamiltonian is

where

The derivative of revenue equals to the price of base metals

The price change over time is the derivative of p with respect to time (we ignore superscript s)

The derivative of

Substituting (22) into (21) and rearranging the RHS yields

Considering the cost function in Equation (3), in case of non-CRS, or

Assuming a constant returns to scale production function

We obtain:

Substituting (27) and (28) into (23) yields

Substituting (26) into (20) yields

Taking the logarithm of both sides of (24) yields

Differentiating both sides of (29) leads to

Dividing both sides of (30) by

Note that

We have

The inverse demand function is

Taking the derivative of p with respect to time yields

Dividing both sides of (35) by (34) we obtain (13). Finally (12) can be obtained as in the proof of Corollary 1 above.