Consider an economy with 100 identical households and 100 identical firms. Each household is endowed with one unit of time. Half of the households are endowed with equal shares of firm, while the rest are endowed with no firm shares. A householdís utility is u (x; r) = 2 ln(x*r^2) , where x is consumption of goods, r = 1- l is leisure time and l is the householdís labour supply. Each firm hires households to produce goods according to technology y = L^1/2, where L denotes the labour input. Goods price p, wage rate w, and dividend income are all taken as given. Normalize w = 1. Do the following: (a) Derive the Marshallian demands, x (p, D) and r (p,D), where D is dividend. (b) Derive the firmís input demand function L (p) and profit function pi(p). (c) List all the market-clearing conditions.

Tim is a baker who produces doughnuts. He can access labour at a rate of $2 per hour, and capital at a rate of $0.25 per machine hour. He produces doughnuts according to Q = L+ K0.5. If the bakery is operating with an optimal factor allocation, and producing 40 doughnuts per day, determine the average total cost of a doughnut. The bakery employs one worker, Anil, who consumes doughnuts (d) and other goods (y) with utility U(d,y) = edy. (Assume y is the Marshallian good, with Py=$1). %3D Sketch Anil's utility curve, and his budget, which is the pay he receives from his job. Do Anil's preferences satisfy the rules of preference ordering? Are there any constraints on his consumption of either good? Derive the supply and demand curves based on Anil's individual demand, and Tim's costs of production.

For each of the following cases, determine whether the firm should (A) Use more labor and less capital, or (B) Use more capital and less labor. 1. The marginal rate of technical substitution is 2 (i.e., the marginal product of labor is twice the marginal product of capital) and the input price ratio is 3 (i.e., a unit of labor costs three times as much as a unit of capital) 2. The marginal rate of technical substitution is 2 (i.e., the marginal product of labor is twice the marginal product of capital) and the input price ratio is 1 (i.e., a unit of labor costs the same as a unit of capital) 3. The marginal rate of technical substitution is initially equal to the input price ratio, but the firm's machines depreciate, so that the marginal product of capital decreases. 4. The marginal rate of technical substitution is initially equal to the input price ratio, but the firm successfully negotiates a lower rate of rent for their machines, so that the price of capital decreases. 5. The…

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