Note that our production function is $Y=\alpha L$ where Y is the number of widgets produced, then our objective is to maximise profits of the of the entrepreneur so, The Profit Maximisation problem of the entrepreneur is, $$\max_{Y\geq 0}Y-wL$$ after substituting for Y, $$\max_{L\geq 0}\alpha p L-wL$$ For solving the above problem note that our objective is linear in $L$ so solving the problem will give us the unconditional labor demand function of the entrepreneur, $$L^d(\alpha,p,w)\in \begin{cases}\phi & \text{if} \ \ \alpha p > w\\\mathbb{R}^+ & \text{if} \ \ \alpha p = w\\ 0 & \text{if} \ \ \alpha p < w\\ \end{cases}$$ Substituting it back into the production function $Y=\alpha L$ will give us the optimal supply function of the entrepreneur, $$Y^s(\alpha,p,w)\in \begin{cases}\phi & \text{if} \ \ \alpha p>w \\ \mathbb{R}^+ & \text{if} \ \ \alpha p = w\\ 0 & \text{if} \ \ \alpha p < w\\ \end{cases}$$

Note that our production function is $Y=\alpha L$ where Y is the number of widgets produced, then our objective is to maximise profits of the of the entrepreneur so, The Profit Maximisation problem of the entrepreneur is, $$\max_{Y\geq 0}Y-wL$$ after substituting for Y, $$\max_{L\geq 0}\alpha p L-wL$$ For solving the above problem note that our objective is linear in $L$ so solving the problem will give us the unconditional labor demand function of the entrepreneur, $$L^d(\alpha,p,w)\in \begin{cases}\phi & \text{if} \ \ \alpha p > w\\\mathbb{R}^+ & \text{if} \ \ \alpha p = w\\ \{0\} & \text{if} \ \ \alpha p < w\\ \end{cases}$$ Substituting it back into the production function $Y=\alpha L$ will give us the optimal supply function of the entrepreneur, $$Y^s(\alpha,p,w)\in \begin{cases}\phi & \text{if} \ \ \alpha p>w \\ \mathbb{R}^+ & \text{if} \ \ \alpha p = w\\ \{0\} & \text{if} \ \ \alpha p < w\\ \end{cases}$$

Note that our production function is $Y=\alpha L$ where Y is the number of widgets produced, then our objective is to maximise profits of the of the entrepreneur so, The Profit Maximisation problem of the entrepreneur is, $$\max_{L\geq 0}Y-wL$$ after substituting for Y, $$\max_{L\geq 0}\alpha p L-wL$$ For solving the above problem note that our objective is linear in $L$ so solving the problem will give us the unconditional labor demand function of the entrepreneur, $$L^d(\alpha,p,w)\in \begin{cases}\phi & \text{if} \ \ \alpha p > w\\\mathbb{R}^+ & \text{if} \ \ \alpha p = w\\ 0 & \text{if} \ \ \alpha p < w\\ \end{cases}$$ Substituting it back into the production function $Y=\alpha L$ will give us the optimal supply function of the entrepreneur, $$Y^s(\alpha,p,w)\in \begin{cases}\phi & \text{if} \ \ \alpha p>w \\ \mathbb{R}^+ & \text{if} \ \ \alpha p = w\\ 0 & \text{if} \ \ \alpha p < w\\ \end{cases}$$

Note that our production function is $Y=\alpha L$ where Y is the number of widgets produced, then our objective is to maximise profits of the of the entrepreneur so, The Profit Maximisation problem of the entrepreneur is, $$\max_{Y\geq 0}Y-wL$$ after substituting for Y, $$\max_{L\geq 0}\alpha p L-wL$$ For solving the above problem note that our objective is linear in $L$ so solving the problem will give us the unconditional labor demand function of the entrepreneur, $$L^d(\alpha,p,w)\in \begin{cases}\phi & \text{if} \ \ \alpha p > w\\\mathbb{R}^+ & \text{if} \ \ \alpha p = w\\ 0 & \text{if} \ \ \alpha p < w\\ \end{cases}$$ Substituting it back into the production function $Y=\alpha L$ will give us the optimal supply function of the entrepreneur, $$Y^s(\alpha,p,w)\in \begin{cases}\phi & \text{if} \ \ \alpha p>w \\ \mathbb{R}^+ & \text{if} \ \ \alpha p = w\\ 0 & \text{if} \ \ \alpha p < w\\ \end{cases}$$