1. [30 points] Consider a one-good economy with 100 identical consumers and 40 identical firms. Each consumer is endowed with one unit of time. A representative consumer's utility function is u (x, r) = log A + logx + 2logr, where A > 0 is a constant, r is consumption of goods and r is leisure. Each firm hires consumers to produce goods according to the function y = VI, where L denotes the labour input. There are 40 consumers who own equal shares of all firms. The rest of the consumers own no shares. The consumers take the dividend income as given. All consumers and firms take goods price p and wage rate was given. Normalize the goods price p = 1. (a) Set up a consumer's utility-maximization problem. Solve for the Mar- shallian demands: x (w, D) and r (w, D), where D is the consumer's dividend income. (b) Set up a representative firm's profit-maximization problem. Calculate the firm's input demand L (w) and its profit . (w). (c) Write down all the market-clearing conditions. Calculate the equilib rium wage rate w*. Then calculate the optimal choice x* for both consumers with and without firm shares