# Leontief utility with producer goods

This question may require some explanation. Also, I'm way out of my depths so please let me know if this question is misguided.

The standard Leontief utility seems to assume the goods are consumed in order to create the output (think tires and wheels to create a car). For comparison, producer goods are not consumed to create the output and the time to create the output becomes interesting. Considering the time with the producer goods seems an interesting extension of Leontief utility. Specifically, if an agent has a bundle of goods for a time less than it takes to produce the output than the bundle should provide low utility and if the agent has a bundle of goods for more than the time to produce the good than the utility should be flat (until it could produce another output).

Are you aware of any work that considers Leontief utility in reference to producer goods or that considers the time to produce the output as part of the utility?

• Seems like you're mixing up consumer theory and producer theory. The concept of utility is exclusive to consumers. Once we start talking about production the whole discussion of utility is irrelevant. – EconJohn Sep 19 '17 at 23:18
• @EconJohn Agreed. Thank you for pointing me to producer theory I will start reading that. – Wil Yegelwel Sep 20 '17 at 15:32

In microeconomics it is common to model utility decision between consumption and leisure. The latter refers to the time of a day the individual is not producing goods himself (if a yeoman producer) or to sell his labour in the market (a waged labourer). The productivity of the individual will affect how much it can produce (and earn) per unit of time. Therefore, time is directly incorporated into the model via productivity, which then affects the balance between consumption and leisure an individual will choose.

For example, consider a yeoman producer with the following utility function (yes, no Leontief, but makes the point):

$$U(C,1-L) = C + a(1-L)$$

where $L$ refers to the proportion of the day the individual is working ($L \in [0,1]$).

He can produce with the following technology:

$$Y = bL^h$$

and consumes all what he produces:

$$C = Y$$

Assuming the yeoman farmer is interested in maximising his utility, the unidimensional optimisation problem is:

$$\text{max}_{_L} U(L) = bL^h + a(1-L)$$

The first order condition is:

$$\frac{\mathrm{d}U}{\mathrm{d} L} = bhL^{h-1} - a = 0$$

Thus, optimal proportion of time allocated to work is:

$$L^* = b^{\dfrac{1}{1-h}}\left(\frac{h}{a}\right)^{\dfrac{1}{1-h}}$$

Optimal consumption is:

$$C^* = b^{\dfrac{1}{1-h}}\left(\frac{h}{a}\right)^{\dfrac{h}{1-h}}$$

The effect of (labour) productivity on hours of work depending on the returns to scale of technology ($h$). You have that:

• if $h \in (0,1)$, a ceteris paribus increase in $b$ means the individual works and consumes more.

• if $h > 1$, a ceteris paribus increase in $b$ means the individual works and consumes less.

• if $h = 1$, productivity might not affect labour at all (but does affect consumption). In this case, there is a corner solution for labour ($L^*=1$ or $L^*=0$). I leave this as an exercise.

So, you are concerned about "the time to produce output". Well, think of $b$ as an indication of how much time you can spend to produce a given amount of output with a fixed amount of labour. As I indicate above, variations in $b$ have important consequences for $L^*$, $C^*$, and ultimately for maximum utility $U(C^*,L^*)$.

Note: The specification I chose means the income and substitution effect arising from an increase in productivity ($b$) do not cancel each other. Other utility function (e.g. with $\ln C$) will mean $b$ does not affect $L$. I also leave this exercise to you.

• Thanks for your reply @luchonacho. It seems I'm quite confused about utilities and production. Everything you've written makes sense (though I will need to spend some time to work through the math myself). Is it correct to say that Leontief utility shouldn't be applied to production? When I've seen Leontief utility explained it is usually attached to things like needing 4 tires and a steering wheel to build a car. Are examples like that simply a misuse of the utility or am I missing something deeper? – Wil Yegelwel Sep 20 '17 at 15:29
• I discovered my mistake, I meant the leontief production function not utility function; though both have the same form. I will mark your answer as correct and open another once I can better formulate my question. Thanks for your help! – Wil Yegelwel Sep 20 '17 at 19:34