HIST363: Global Perspectives on Industrialization

For this paper, we need a formulation sufficiently general that it can encompass changes in factor endowments, changes in technology, and changes in price policies. The dual approach popularized by Dixit and Norman and by Woodland provides this flexibility. The production side of the economy can be represented using a restricted profit function specifying the value of net output in the economy as a function of the domestic prices of outputs and intermediate inputs:

(1) $\pi=\pi(\mathrm{p}, v)=\max _{\mathrm{x}}\{\mathrm{p} . \mathrm{x} (\mathrm{x}, v)$ feasible $\}$ where $\pi$ is the value-added accruing to the vector of quasi-fixed factors, $v$, in the economy given the vector of domestic prices, $\mathrm{p}$, for gross outputs and intermediate inputs of the vector of produced goods, $\mathrm{x}$. The vector $v$ includes economy-wide stocks of mobile factors, any sector-specific factor inputs, and public goods such as infrastructure, that may not be readily allocable to particular sectors. The function $\pi$ is non-negative, and nondecreasing, linearly homogenous, and concave in $v$, and nondecreasing, linearly homogeneous, and concave in $p$.

As Dixit and Norman note, the specification in equation (1) represents all of the properties of the production technology. It is extremely general, being able to represent many different types of technology depending upon the particular functional form used to specify the GDP function. These specifications may include the familiar 2*2 Heckscher-Ohlin model with two factors and two outputs, and no intermediate inputs, through a range of specifications of much greater generality. It may also include specifications such as the Leamer model in which there are more goods than factors, and small, open economies move between different cones of diversification in which the set of commodities produced change. The specification is also sufficiently general to include forward and backward linkages induced by input-output linkages and transport costs. 

Over the range where the profit function is differentiable, its derivatives with respect to the prices of output yield a vector of net output supplies:

(2) $\pi_{\mathrm{p}}=\pi_{\mathrm{p}}(\mathrm{p}, v)$

Depending upon the specification of the profit function, it may be possible to identify the gross outputs of each good, and the quantities of these goods used as intermediate inputs in production. For some purposes, such as estimating the incentives created by a protection structure, it is very important to be able to identify the net outputs. 

The derivative of the profit function with respect to the factor endowments gives the vector of factor prices. 

$\pi_{k}(\mathrm{p}, v)$

One additional important expression is the matrix of Rybczynski derivatives. Differentiating the vector of price derivatives, $\pi_{p_{w}}$ by the vector of resource endowments (or, equivalently by Young's theorem, differentiating the vector of factor prices by the price vector) yields a matrix, $\pi_{p u}$, of changes in the net output vector resulting from changes in factor endowments. This matrix is clearly critical for our analysis, but its exact structure depends heavily upon the underlying production technology.

The most difficult case to analyze is the realistic situation in which there are more goods than factors. Leamer and Leamer, Maul, Rodriguez and Schott provide an extremely useful analytical framework for analyzing this problem where there are three factors and many goods. In simple cases, countries with three factors will specialize in the production of three goods. Over some range, the features of the Rybczynski theorem will hold and changes in factor endowments will result in changes in the mix of output without changes in factor prices. However, changes beyond that point will result in shifts into a new cone of diversification, with a change in the mix of output and a fall in the return to the factor whose relative supply is being augmented. As Leamer points out the location of these cones of diversification depends upon commodity prices, and hence is not merely a function of the technology. 

In the case of resource-poor economies, Leamer et al show that the adjustment path associated with accumulation of human and physical capital is likely to be relatively smooth, with increases in the supply of capital raising the demand for raw labor as the economy moves through different cones of diversification. For resource-abundant economies, however, the path may involve reductions in unskilled labor as the economy moves from, say, peasant farming to resource-based systems involving greater use of capital. This move may be associated with reductions in the returns to unskilled labor that increase income inequality. 

For some problems, such as situations where some goods are nontraded, we need to consider the consumption side of the economy as well as the production side. The consumption side of the economy can be represented similarly using an expenditure function:

(3) $\mathrm{e}(\mathrm{p}, \mathrm{u})$

where e represents the expenditure required to achieve a specified level of utility, $u$, and represents all of the economically relevant features of consumer preferences. Assuming differentiability of the expenditure function, the vector of consumer demands can be obtained as:

(4) $e_{p}(p, u)$

An important feature of real-world consumer preferences is their nonhomotheticity, with commodities like basic food having small or negative responses to income increases, while luxury goods have large positive income effects. The vector of Marshallian income effects can be derived from (4) as:

$\mathrm{c}_{\mathrm{W}}=\left(\mathrm{e}_{\mathrm{pu}} / \mathrm{e}_{\mathrm{u}}\right)$

where $\mathrm{e}_{\mathrm{u}}$ is the marginal impact of a change in utility on expenditure, and $\mathrm{e}_{\mathrm{pu}}$ is the marginal impact of a change in utility on the consumption of each good.

The vector of net imports of commodities is given by $m$, which is the difference between the vector of consumption and the vector of net outputs:

$\mathrm{m}=\mathrm{e}_{\mathrm{p}}-\mathrm{r}_{\mathrm{p}}$

World prices of traded goods are determined by the market clearing condition that the sum of the net trade vectors for all regions must equal zero. Where some goods are non-traded, the relevant sub-vector of m is exogenously equal to zero and equilibrium in the market for these goods is achieved by adjustments in the prices of these goods. Similarly, where trade in some goods is determined by binding quotas, the relevant subvector of m is set exogenously at the quota level and equilibrium is achieved by endogenous determination of these prices.

Trade policy distortions can be represented very simply as creating a difference between the vector of domestic prices, $p$, and world prices, $p_{w}$ for a small open economy. It is frequently useful to define a net expenditure function $z=(e-r)$. The derivative of this function with respect to prices, $z_{p}=\left(e_{p}-r_{p}\right)$ is also equal to the vector of net imports. This function also provides a compact way of representing the revenues accruing from trade distortions as $\mathrm{R}=\left(\mathrm{p}-\mathrm{p}_{\mathrm{w}}\right) \cdot \mathrm{z}_{\mathrm{p}}$

Finally, the welfare impacts of any exogenous shock can be represented using the balance of trade function. This function takes into account the effects of trade distortions on the cost of expenditures, the revenue to producers, and the revenues from trade distortions (or domestic taxes, which are levied on only expenditures or producer revenues). The specification of this function is based on the assumption that all revenue from trade distortions is returned to the representative consumer. If this is not the case, the function needs to be modified to take into account losses of such revenues to, for example, foreign governments or foreign traders. The balance of trade function, $B$, can be specified as:

(5) $\mathrm{B}=\mathrm{z}(\mathrm{p}, v, \mathrm{u})-\mathrm{z}_{\mathrm{p}}\left(\mathrm{p}-\mathrm{p}_{\mathrm{w}}\right)-\mathrm{f}$

where $\mathrm{f}$ is an exogenously specified financial inflow from abroad. When $\mathrm{u}$ is held constant, and changes are made in any of the exogenous variables of the system, changes in $\mathrm{B}$ show the change in the financial inflow needed to maintain the initial level of utility in the face of the changes in the exogenous variables. This change in income is a measure of the compensating variation associated with the change.

Before the system can be used to analyze the consequences of changes in productivity, we need to augment this standard system to include the impacts of technical change on producer behavior and producer profits. As noted by Martin and Alston, there is a number of ways in which this might be done, but perhaps the most appealing in terms of flexibility and consistency with economic theory is to represent technological change as resulting in a distinction between actual and effective units of an input or output. In the case of an output-augmenting technological advance, such a change might be one that increases the actual output achieved from the same bundle of inputs - such as an increase in the grain available for consumption from a given amount available for harvest in the field. In the case of an input-augmenting technological advance, the change might be one that reduces the actual quantity of the input required to achieve the same outcome - such as a reduction in the amount of labor needed to complete a task. Product quality improvements and promotion policies might create a similar augmentation of the product from the viewpoint of the user - a product augmentation, rather than a process augmentation.

Such technological changes have two important impacts on behavior and profitability. The first is the direct response of output associated with the initial level of inputs in the case of an output-augmenting technical change, or the change in required inputs to achieve a given level of output. The second impact is the induced impact resulting from changes in the effective prices of inputs. In representing such technical changes, it is necessary to take into account both the direct impacts on output/inputs, and the indirect impacts working through induced changes in the effective prices of outputs or inputs. 

In the case of output-augmenting technical change, we can define effective output i as:

(6) $\mathrm{x}_{\mathrm{i}}=\mathrm{x}_{\mathrm{i}} \cdot \tau_{\mathrm{i}}$ where $\tau$ is a technical change parameter equal to unity before the technological change.

We can define a corresponding output price as:

(7) $p_{i}=p_{i} \cdot \tau_{i}$

In the case of an output-augmenting technical change, the effect of the technological change is to increase the effective output associated with any given bundle of inputs, and to raise the effective price of output. Clearly, both of these effects operate in the same direction, tending to increase output at any given output price. The first does so by increasing the outputs obtained from any given level of inputs, and the second by drawing additional inputs into production of this good. In the case of an input-augmenting technical change, the direct effect is to reduce the inputs required to achieve a given level of output, while the indirect effect is to increase output as producers substitute the input whose effective price has fallen for other inputs. In this case, the effect on input use is ambiguous, depending upon whether the direct input-saving effect is outweighed by the substitution effect. 

Rewriting equation (2) in terms of effective prices and quantities as defined in equations (6) and (7) allows us to assess the impacts of an improvement in technology in sector $i$ on output from that sector in a small, open economy. Differentiating the supply of output in actual units with respect to $\tau$ yields:

$\frac{\partial r_{p}}{\partial \tau}=r_{p}^{*}+\tau \frac{\partial r_{p}^{*}}{\partial \tau}$

which can be rearranged to yield:

(8) $\frac{\partial r_{p}}{\partial \tau} \cdot \frac{\tau}{r_{p}}=1+\eta_{i i}$

where ηii is the own-price elasticity of supply for good i. The intuition behind equation (8) is that a technological advance proportionately increases the output generated by the resources originally committed to production of the good. In addition, it increases the effective price of the output, and hence induces an additional increase in output equal to the own-price elasticity of supply. 

Another influence on the response of output and resource use is the impact of the technological change on the actual price of output. In a small, open economy, the actual of output, and to increase demand for the through the associated reduction in the effective price of the input price is unaffected by technological changes, unless the technical change is global, when it will affect world prices. However, for a closed economy, technical changes can be expected to affect the price of output. The higher the elasticity of consumer demand in this situation, the smaller the decline in the actual price of output, and the more likely it is that input use will rise when production of a particular output benefits from a technological advance. Matsuyama distinguished between an open-economy situation in which improvements in agricultural technology increased input use in agriculture, and a closed-economy case in which improvements in agricultural technology allowed the demanded level of output to be produced with fewer inputs. When trade in a good is quantity-constrained, either for natural reasons such as transport costs or because of policy constraints such as quotas, we can readily modify the derivation of equation (8) to take the consequent changes in actual output prices into account. For a single non-traded good in an undistorted economy, the (compensated) impact on prices is given by:

(9) $\mathrm{dp} / \mathrm{p}=\left(1+\eta_{\mathrm{ii}}\right) /\left(\varepsilon_{\mathrm{ii}}-\eta_{\mathrm{ii}}\right) \mathrm{d} \tau / \tau$

where $\varepsilon_{\mathrm{ii}}$ is the compensated elasticity of demand for good $i$.

One informative limiting case is the one where the elasticity of demand is very small relative to the elasticity of supply. While this case appears very restrictive, it is probably a realistic approximation in many cases, since general-equilibrium supply elasticities for a single industry in a Heckscher-Ohlin setting are determined only through impacts of changes in its output on factor prices and are likely to be very much larger in absolute value than demand elasticities. In this case, (9) reduces to

$\mathrm{dp} / \mathrm{p}=-\left(1+1 / \eta_{\mathrm{ii}}\right) \mathrm{d} \tau / \tau$

This identifies two components of the price reduction. The unit impact is the price reduction required to exactly offset the impact of the technical change on the effective price of output, and hence on the supply of actual output. The second is the decline in the domestic price needed to offset the direct stimulus to supply (at any given level of inputs) resulting from the technical change. Given the dramatic growth rates feasible in some export-oriented sectors, this difference could result in very large differences in the welfare benefits obtainable from technical change.

We focus on compensated impacts as these are simpler, and more relevant to the calculation of compensated measures of welfare change in distorted economies.