Let p be the home price of manufactures in terms of food. Let p_w be the exogenous world price of manufactures in terms of food. Then,

p = [1 + τ] p_w

τ = rate of tariff

Let food output, F, be a function of the amount of labour in the food sector, L_F.

F = F(L_F )

Let manufactures output, M, be a function of the amount of labour in the food sector, L_M.

M = M(L_M )

Labour in food and manufactures sum to the exogenously given supply of labour, L.

L_F + L_M = L

Labour is mobile between the two sectors. Therefore, the marginal productivity of labour in food must equal the marginal product of labour in manufactures (evaluated in terms of food), as long as both food and manufactures are produced.

F '(L_F) = M '(L_M)p

These five equalities in five unknowns imply the impact of a tariff on living standards. Defining f as the wage in terms of food, F’(L_F), and m as the wage in terms of manufactures, M’(L_M), then,

Thus the impact of protection will be to increase the wage in terms of food, and diminish the wage in terms of manufactures. Thus the impact of the tariff of the welfare of the worker is ambiguous.

There are, however, two conditions under which the welfare of the worker has unambiguously increased. The first we call ‘The Brigden Condition’: constant marginal productivity of labour in manufactures.

M " = 0

This is conveyed diagramatically in Figure 4.2.

The Brigden Condition implies,

These equalities imply that an increase in the price of manufactures will increase the food wage and not reduce the manufactures wage. The marginal product of labour in food is increased and the marginal product in manufactures is the same. In other words, a tariff has necessarily increased the utility of any worker who consumes both food and manufactures.

The second condition for a tariff to increase the welfare of a worker we will call ‘The Sidgwick Condition’: a discontinuous fall in marginal productivity of labour in Food. This is conveyed diagrammatically in Figure 4.4, and mathematically as,

F" = –∞

The Sidgwick Condition, like the Brigden Condition, implies,

These equalities state that an increase in the price of manufactures will increase the food wage and not reduce the manufactures wage. The logic is that an increase in p increases the food value of the marginal product in manufactures, but labour will not reallocate from food towards manufactures, owing to the discontinuity. Consequently, the marginal product of labour in manufactures will be the same. But the food wage (= the wage measured in food) must increase because the food value of the marginal product in manufactures has increased.

The upshot is that the manufactures wage is unchanged and the food wage is increased. In other words, a tariff has necessarily increased the living standard of the worker.

Under these two conditions, therefore, any increase in a tariff increases worker living standards

The contention that the population must be ‘appropriately large’ if protection is to increase real wages can be easily shown if we make the Brigden Assumption. Figure 4.6 plots L on the horizontal axis and the marginal product of labour in manufactures evaluated in food (using free trade prices), M’p_w, on the vertical axis. The plotting of M’p_w is horizontal, since M’ and p_w are parametrical to L. The marginal product of labour in food if all labour is applied to food, F’(L), is also plotted in Figure 4.6. At L = L*, F’(L) = M’p_w, and the marginal product of labour in Food (assuming all labour is applied to food) equals the marginal product of labour in manufactures, evaluated in terms of food at free trade prices.

If L > L*, a tariff will increase real wages, by the argument previously rehearsed.

But if L < L* a tariff may not increase real wages. If L< L* then there would be no manufacturing sector under free trade, as all labour will be allocated to food.^[26] In these circumstances any merely ‘small’ tariff (that is, one that is insufficient to create a manufacturing industry) would merely raise the price of manufactures, without making local production of manufactures viable. Consequently, the wage in food is unchanged and the tariff has succeeded only in reducing the manufactures wage (that is, the amount of manufactures that can be purchased by the marginal product of labour in food, = F’(L)/ p_w[1+t]).