Consider an economy where all production and all consumption is carried out by a single agent, RobinsonCrusoe.Robinson,asa consumer,is endowed withHhours of time.His utility depends in hours of leisure,h, and consumption of coconuts,

x: u(h,x)=h^1/3 x 2/3 wherex> or equal to 0 and 0< or equal to h < or equal to H. Robinson, as a firm, can transform hours of labor,z, into the production of coconutsy. His technology of production is given byy=f(z)= square root of z. Letpdenote the price of coconuts,wdenote the price of leisure/labor, andpie (symbol) denote the firm's profits. Note that the budget constraint of Robinson is px< or equal to w(H-h) + pie (symbol), and that in equilibrium in the market of coconutsx = yand in the labor marketh + z = H.

Find the competitive equilibrium.

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