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Currently I am trying to figure out some old problems for an upcoming micro exam. One of them is about production under uncertainty. The exercise seems standard to me, yet I am not sure whether I have the correct solution. The question is as follows.

A firm produces goods with the cost function $C(x) = cx^2$ where $x$ is the good. The price $p_x$ is exogenously given. The utility function of the producer is given as $u(y) = \sqrt{y}$ where $y$ is the income due to profit. Determine the optimal amount of produced goods $x$ in case i.) $p_x$ is deterministic and (ii.) the price is random where it is $p^0_x$ with probability 0.5 and $p^1_x$ with probability 0.5. How will the output change if there is some future price $p^f_x > \mathbb{E}[p^s_x]$?

My solution is as follows. First, we have that the firm's profit is

$$G = p_xx - cx^2$$

and its utility

$$ u(y) = \sqrt{p_xx - cx^2}$$

Now under deterministic prices the firm owner wants to maximize expected utility, hence

$$\frac{d\mathbb{E}[U]}{dx} = (p_xx - cx^2)^{-\frac{1}{2}}(p_x - 2xc) = 0$$

and therefore $$x = \frac{p_x}{2c}$$

In case of random prices the optimality condition is now

$$ \frac{d\mathbb{E}[u]}{dx} = \frac{1}{2}((p^0_xx - cx^2)^{-\frac{1}{2}}(p^0_x - 2xc) + (p^1_xx - cx^2)^{-\frac{1}{2}}(p^1_x - 2xc) = 0$$

right? After that I have to solve for $x$ and I got the result.

Finally, if there is some future price which is higher than the expected price I am not quite sure how to proceed. Intuitively, I'd say that if the owner would produce more goods but how would I show that in an analytic manner?

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Under random prices, the producer solves

$$\max_x E[U(G(x,p)]$$

Compacting $U(G(x,p) \equiv U$, the f.o.c is

$$E[U'\cdot (p_x-2cx)]= 0 \implies x^* = \frac 1 {2c}\cdot \frac {E[U'\cdot p_x]}{E[U']} \tag{1}$$

or

$$x^* = \frac 1 {2c}\cdot \left[E(p_x) + \frac {\text{Cov}(U', p_x]}{E[U']}\right] \tag{2}$$

Now, $$U' = \frac 1{2(p_xx - cx^2)^{1/2}}$$

and $U'$ is decreasing in $p_x$. Then (see for example Egozcue, Cogent Mathematics (2015), 2: 991082), Theorem 2(2) p. 6)

$$\text{Cov}(U', p_x] <0$$

So we conclude that

$$\implies x^* < \frac {E(p_x)} {2c}$$

the RHS being the risk-neutral solution.

Plugging the distributional assumption for $p_x$ in $(1)$ gives us an implicit function for $x^*$. If one is patient enough to go through some boring algebra after that, one will obtain a very simple nice looking closed-form solution.

As for the last question, it is poorly worded and it does not make much sense, or it is a trick question, since the set up implies that the producer must commit to an output level before the price is actually realized. If this was not the case, then the producer would have no reason to solve an expected utility maximization problem, but he would wait to see the actual price and optimize accordingly.

If the last question alludes to a dynamic maximization problem, then this is a whole new situation, and additional assumptions and modelling are needed in order to proceed.

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  • $\begingroup$ Is there some way to modify the equation of $x^*$ to account for the risk attitude? $\endgroup$ – Taufi Jun 3 '17 at 16:55

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