3. Labour Demand

KAT.TAL.322 Advanced Course in Labour Economics

Nurfatima Jandarova

September 1, 2025

Static model

Single factor input

Production function where and

FOC:

Downward-sloping labour demand

We begin from the static theory of labour demand.

A firm produces good using as input and technology . The production function is increasing in , but every additional unit of labour adds less and less to the production. Think of a bakery that initially had only one baker. Let’s say she could make 100 buns a day. If another person is hired, she could maybe help prepare twice as much dough. However, the bakery still has one oven. Therefore, maybe the total production of buns rises by only 50 more buns a day. And so on.

Note that in this version of the model there is no capital. This is equivalent to introducing capital in the model that is fixed and cannot be adjusted. In the example, the bakery needs to have physical facilities such as building and oven. Typically, physical capital is not easy to adjust on a day-to-day basis. For this reason, you can also use the model without capital to describe short-run behaviour of firms. On the next slide we will see the model where capital can be adjusted together with labour input.

The firm then sells its product at price and pays wages to its personnel. As usual, we assume that firms’ objective is profit maximisation. We also assume here that firms are price-takers with respect to both and . We can easily right down firm’s maximisation problem and first-order condition.

The FOC indirectly characterizes the labour demand function (if you assume a specific functional form of you can derive the labour demand function explicitly). In particular, it also helps us see that it is decreasing marginal product of labour in the production function that is key for negative slope of the labour demand. Let . Then, using implicit function differentiation

Static model

Two factor inputs: conditional factor demand

Production function where

Cost minimization problem: s.t.

Conditional demand: and

Now, let’s consider how does firm behaviour changes with more than one input. Firm produces output using production function . Again, is increasing in its inputs, meaning that higher quantities of either or will increase total output. We also assume that production function is concave in its inputs, meaning that marginal product of and is decreasing. But we will not put any restrictions on cross-productivities. In mathematical notation, it means that we do not restrict the sign of and second-order derivatives.

We can solve firm’s optimisation problem in two ways:

we can write down expression for profit and maximise it with respect to and
we can write down cost function and minimise it holding production at certain level.

As in the labour supply lecture, you will see that both of these solutions will be useful in disentangling substitution and scale effects.

If we follow the second approach, we will get so-called conditional factor demands, i.e., conditional on level of production. After writing down the Lagrangean and taking first-order derivatives, we obtain optimal solution conditions. It can again be interpreted as the tangency point between an isoquant line (all combinations of and that produce same output level ) and cost function. The slope of the isoquant line is also called technical rate of substitution between capital and labour.

As in the labour supply lecture, the tangency point is exactly the point where there is no room for further improvement. If you increase labour beyond this point, you can still produce the required level of output , but at an unnecessarily higher cost. If you decrease labour below the tangency point, then you will not be able to produce the required output .

Static model

Two factor inputs: conditional demand elasticities

Own-price elasticities: and
Cross-price elasticities: and
Elasticity of substitution

It is also possible to show that

where is labour share in total cost

Once we have conditional factor demand functions, we can study how they change with respect to factor prices. Notice that there are four different price elasticities: own-price elasticities and cross-price elasticities. In Chapter 4 in Cahuc (2004) (or Chapter 2 in Cahuc, Carcillo, and Zylberberg (2014)) you can find derivations that prove the properties of the conditional demand functions. However, we can borrow the intuition from the last lecture. Since conditional demand functions fix the production scale, their price elasticities capture substitution effects and have unambiguous signs.

Own-price elasticities always negative again due to concavity of the production function.

If wages go up, each unit of labour becomes more expensive and firms want to substitute away from labour towards capital.
Cross-price elasticities are positive

This is simply saying that substitution effect is symmetric. We already said that if , then firms want to decrease and substitute away from towards . It means also that when , then demand for rises. Symmetrically, if , firms want to substitute away from towards .

Even though these two elasticities tell us what happens to optimal and when one or both of the factor prices change, it can still be difficult to understand what happens to the relative demand of production inputs. Imagine that increased by 5% and decreased by . We will know that should decrease by %, and also by %. Similarly, should increase by % and by %. You can agree, that it is still difficult to understand what exactly are the values of new optimal pair.

For this reason, we often use elasticity of substitution. It describes how the relative conditional factor demands change with respect to relative factor prices. In the previous example, the relative factor prices increases by 17%. Then, the relative factor demand increases by 17%. Note that inverse relationship between factor demands and their prices is preserved here.

Naturally, the elasticity of substitution has to be consistent with own-price and cross-price elasticities. You can show that own- and cross-price elasticities, in fact, can be written as functions of and factor shares. These expressions can be useful empirically and also offer more convenient reasoning. If and are extremely substitutable (high ), then any variation in factor prices will result in large adjustments of input demand. Conversely, for a given , if labour share in total cost is already super high, any change in factor prices is unlikely to generate large changes in labour demand.

Static model

Two factor inputs: unconditional factor demand

Second step:

Solution:

Total elasticities decomposed into substitution and scale effects:

Once we have expressions for the conditional factor demands, we can write down the conditional cost function

Then, the second-step of optimisation is to find output level that would maximise firm’s profit. The optimality condition is given by equality of marginal revenue and marginal costs. Each unit of output can be sold at price . It is easy to show that the cost is increasing in . Thus, if we produce where , we could clearly increase profit by producing more. If we had production where , we could also clearly increase profit by producing less and cutting cost.

Once, you know the optimal , plug it back into conditional factor demands to get the unconditional demand functions. Now, we can study the properties of the unconditional factor demands, i.e., the quantities we will observe in the data.

Even though you could get unconditional factor demand by solving profit maximisation from the start, the two-step procedure makes it easier to separate between substitution and scale effects. See Chapter 4 in Cahuc (2004) (or Chapter 2 in Cahuc, Carcillo, and Zylberberg (2014)) for the derivations of the relationship between total and conditional demand elasticities.

Consider the own-price elasticities of labour demand. We’ve seen on previous slide that : firms want to substitute away from labour if wages increase. What happens to output (scale)? Higher means that producing any output level is now costlier (you can show that ). Therefore, marginal cost curve shifts upwards and optimal production level . Since, now firm is producing less, it requires less labour. Therefore, total wage elasticity of labour demand is negative, and even more negative than conditional elasticity - .

In case of cross-price elasticity, the sign is not know apriori. We have already established previously that substitution effect is always positive, . However, the scale effect can be of any sign. As in the own-price elasticity, the scale effect depends on whether marginal cost of output is increasing or decreasing in input prices. However, unlike in own-price elasticity, the scale effect of cross-price elasticity depends on marginal cost with respect to both input prices. And we cannot assert sign of without further restrictions on the production function.

If total cross-price elasticity is positive (), we call and gross substitutes. Can you think of examples where labour and capital are substitutes?

If scale effect is sufficiently negative, then higher decreases production so much that total labour demand also drops. In this case, we call and gross complements. Can you think of examples where labour and capital are complementary?

Estimations of static model

Main issues

Endogeneity
General equilibrium
Definitions of variables

Again, the main issue with estimates from observational data is endogeneity.

First of all, the observed wages and employment levels at firms are always equilibrium quantities. The wages and employment in different firms that operate in different product and labour markets may be influenced by the market-specific conditions. For example, technological advancement, globalisation and many other factors contributed to shrinking manufacturing sectors and expanding service sectors. So, you may imagine that mining firms pay lower wages and have lower labour demand, while service sector firms - now higher more and pay higher wages. This relationship arises due to external shifts in the markets and is not informative about the shape of labour demand.

Second, there may be unobserved characteristics of the firm that influence both their labour demand and wages. For example, Google may pay much higher wages and higher more people than any other firm in the market. This again arises because of other firm-specific factors and comparison of Google to other firms will give us biased estimates of labour demand elasticities.
In the model, we just refer to as labour input. But what exactly is that is most relevant for the production process? Can it be captured adequately by number of employees? Or even by person-hours (sum hours of work done by all employees)? Do all workers enter production function in the same manner? Many studies attempt to differentiate between skilled and unskilled labour. However, the estimates can vary a lot depending on the exact way the skilled and unskilled workers are classified.
Also definition of wages becomes less obvious when looking from firm side. For example, the cost function of firms depends not only on the monetary wages paid out to workers, but also on all non-monetary benefits the firm provides to its workers (actually, same can be said for the workers: they also receive part of their consumption via non-monetary firm benefits). Therefore, if , a firm may not respond so much by reducing labour demand, but by shrinking all the non-monetary benefits.

Dynamic model

Adjustment costs

Quadratic cost:
Asymmetric convex costs:
Linear cost:
Fixed cost

The simplest way to account for these non-wage costs is to specify a separate labour adjustment cost function. There are many ways you could think about these adjustment costs. We will review some of these costs in today’s lecture.

Many papers have chosen quadratic costs, which essentially means that small changes in are not as costly as very large changes in .
However, the quadratic function assumes that cost of scaling up employment are identical to costs of shrinking employment, which may not be true in real life. Therefore, papers have started using asymmetric convex costs.
The functional form of asymmetric convex costs can make interpretation of model solutions complex. Therefore, some papers opted for simpler linear cost function, where slope is different for positive and negative adjustments.
These adjustment costs assume that if total employment doesn’t change , then firm does not need to pay any extra costs. But it can also be that firm hired and fired the same amount of workers. In this case, the total employment does not change, but firm still had to pay hiring costs and firing costs. To get around this issue, some papers advocate for inclusion of fixed cost of labour.

Nature of adjustment costs

Note that these adjustment costs are “weird”! You may get this impression when thinking longer about the rationale of fixed costs in adjustment costs. In these models, firms pay adjustment costs mostly when total employment changes. However, when we think about the nature of these costs, they arise at the point of hiring or firing workers.

The basic labour demand model introduced here is not suitable for studying hiring/firing decisions. Therefore, we cannot incorporate these costs separately and have to resort to more coarse proxies. In the next lecture, we will study job search models, which can be extended to include firm-side behaviour of looking for candidates. In these kinds of models, specification of hiring and firing costs is more organic.

Estimations of dynamic model

Some key results

Adjustments happen fast (1-2 quarters) (Hamermesh 1996, chap. 7)
Dynamic substitutes: utilization of capital increases with
Hours of work are adjusted faster than number of workers

Figure 1 from Houseman and Abraham (1993) (adjustment to demand shocks)

Papers that study adjustment of labour and try to estimate the parameters of adjustment cost conclude that it cannot be represented by quadratic function. They suggest that most likely the shape of adjustment cost curve is piecewise linear and asymmetric. This corresponds to the case when we considered linear adjustment cost function with different slopes for employment expansion and contraction. They also suggest that the cost function should include fixed non-wage cost of labour to be consistent with the patterns observed in the data.

When they estimate speed of adjustment, the conclusion is that most of it happens really fast. Most of the adjustment takes place within 1 year and median time to adjustment is about 6 months. This is another reason why adjustment costs are unlikely to be quadratic.

Interestingly, some papers in the US find that and are dynamic substitutes. This means that when deviation of labour demand from stationary level is very large, firms adjust faster. This makes sense. When the deviation of labour demand from optimal is high, the cost of adjusting the labour is at its highest. We’ve seen in the model with linear adjustment costs that in this case, firms do adjust, but only up until or , which still can be very far away from optimal . In these cases, firms have to exploit more capital to reach the optimal production level.

Some papers have also shown that firms adjust person-hours faster than number of employees. This finding is not robustly replicated. But it can make sense since usually labour protection laws apply on the extensive margin (firing a worker) rather than on hours of work. Similarly, hiring costs apply for hiring a worker and do not vary with hours of work the person is hired for.

Also to illustrate that these costs depend on institutional context and vary greatly between countries, consider Figure 1 in Houseman and Abraham (1993). Here, the authors estimate dynamic relationships between employment in manufacturing sectors in Germany and the US, hours of work and shipments. Using the estimated coefficients, they simulate impulse response functions (IRF) of employment and hours of work following a 1-unit shock to shipments. These IRFs are plotted above. As you can see, their estimates imply that employment and hours of work fall in both countries. However, you can also see that within 2 quarters of the shock, German firms cut hours of work more severely than total employment, while the US firms layoff their workers faster than cut hours of work. This can be attributed to institutional differences in employment protection laws. In the longer run, the employment falls to the same level in Germany and the US and hours of work return to their pre-shock levels.

Note that this means adjustment patterns are estimated in the observational data. Can you think of an empirical strategy that relies on plausibly exogenous demand shocks to study this question?

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3. Labour Demand KAT.TAL.322 Advanced Course in Labour Economics Nurfatima Jandarova September 1, 2025

3. Labour Demand
Today
Labour demand
Static model
Static model
Static model
Static model
Static model
Estimations of static model
Estimations of static model
Estimations of static model
Dynamic model
Dynamic model
Dynamic model
Dynamic model
Dynamic model
Dynamic model
Dynamic model
Estimations of dynamic model
Estimations of dynamic model
Minimum wages and employment
Minimum wage and employment
Minimum wage and employment
Minimum wage and employment
Minimum wage and employment
Minimum wage and employment
Minimum wage and employment
Minimum wage and employment
Minimum wage and employment
Minimum wage and employment
Minimum wage and other margins
Summary
References